STAT

05tcentennial

VECTOR ANALYSIS

'Bicentennial publications

With the approval of the President and Fellows of Yale University, a series of volumes has been prepared by a number of the Professors and In structors, to be issued in connection with the Bicentennial Anniversary, as a partial indica tion of the character of the studies in which the University teachers are engaged.

This series of volumes is respectfully dedicated to

«raDuate$ of tfc

'

/

VECTOR ANALYSIS^

A TEXT-BOOK FOR THE USE OF STUDENTS OF MATHEMATICS AND PHYSICS

FOUNDED UPON THE LECTURES OF

J. WILLARD GIBBS, PH.D., LL.D.

Formerly Professor of Mathematical Physics in Yale University

BY

EDWIN BIDWELL WILSON, PH.D.

Professor of Vital Statistics in Harvard School of Public Health

NEW HAVEN YALE UNIVERSITY PRESS

Copyright, 1901 and 1929 BY YALE UNIVERSITY

Published, December, 1901

Second Printing, January ', 19/3

Third Printing, July, 1916

fourth Printing^ April, 1922

Fifth Printing, October, 1925

Sixth Printing, April, 1020

Seventh Printing, October, 1951

Eighth Printing, April, 1943

Ninth Printing, April, 1947

All rights reserved. This book may not be re produced, in whole or in part, in any form, ex cept by written permission from the publishers.

PRINTED IN THE UNITED STATES OF AMERICA

PKEFACB BY PROFESSOR GIBBS

SINCE the printing of a short pamphlet on the Elements of Vector Analysis in the years 1881-84, — never published, but somewhat widely circulated among those who were known to be interested in the subject, — the desire has been expressed in more than one quarter, that the substance of that trea tise, perhaps in fuller form, should be made accessible to the public.

As, however, the years passed without my finding the leisure to meet this want, which seemed a real one, I was very glad to have one of the hearers of my course on Vector Analysis in the year 1899-1900 undertake the preparation of a text-book on the subject.

I have not desired that Dr. Wilson should aim simply at the reproduction of my lectures, but rather that he should use his own judgment in all respects for the production of a text-book in which the subject should be so illustrated by an adequate number of examples as to meet the wants of stu dents of geometry and physics.

J. WILLARD GIBBS. YALE UNIVERSITY, September, 1901.

GENERAL PREFACE

WHEN I undertook to adapt the lectures of Professor Gibbs on VECTOR ANALYSIS for publication in the Yale Bicenten nial Series, Professor Gibbs himself was already so fully engaged upon his work to appear in the same series, Elementary Principles in Statistical Mechanics, that it was understood no material assistance in the composition of this book could be expected from him. For this reason he wished me to feel entirely free to use my own discretion alike in the selection of the topics to be treated and in the mode of treatment. It has been my endeavor to use the freedom thus granted only in so far as was necessary for presenting his method in text-book form.

By far the greater part of the material used in the follow ing pages has been taken from the course of lectures on Vector Analysis delivered annually at the University by Professor Gibbs. Some use, however, has been made of the chapters on Vector Analysis in Mr. Oliver Heaviside's Elec tromagnetic Theory (Electrician Series, 1893) and in Professor Foppl's lectures on Die Maxwell' sche Theorie der Electricitdt (Teubner, 1894). My previous study of Quaternions has also been of great assistance.

The material thus obtained has been arranged in the way which seems best suited to easy mastery of the subject. Those Arts, which it seemed best to incorporate in the text but which for various reasons may well be omitted at the first reading have been marked with an asterisk (*). Nu merous illustrative examples have been drawn from geometry, mechanics, and physics. Indeed, a large part of the text has to do with applications of the method. These applications have not been set apart in chapters by themselves, but have

x GENERAL PREFACE

been distributed throughout the body of the book as fast as the analysis has been developed sufficiently for their adequate treatment. It is hoped that by this means the reader may be better enabled to make practical use of the book. Great care has been taken in avoiding the introduction of unnecessary ideas, and in so illustrating each idea that is introduced as to make its necessity evident and its meaning easy to grasp. Thus the book is not intended as a complete exposition of the theory of Vector Analysis, but as a text-book from which so much of the subject as may be required for practical appli cations may be learned. Hence a summary, including a list of the more important formulae, and a number of exercises, have been placed at the end of each chapter, and many less essential points in the text have been indicated rather than fully worked out, in the hope that the reader will supply the details. The summary may be found useful in reviews and for reference.

The subject of Vector Analysis naturally divides itself into three distinct parts. First, that which concerns addition and the scalar and vector products of vectors. Second, that which concerns the differential and integral calculus in its relations to scalar and vector functions. Third, that which contains the theory of the linear vector function. The first part is a necessary introduction to both other parts. The second and third are mutually independent. Either may be taken up first. For practical purposes in mathematical physics the second must be regarded as more elementary than the third. But a student not primarily interested in physics would nat urally pass from the first part to the third, which he would probably find more attractive and easy than the second.

Following this division of the subject, the main body of the book is divided into six chapters of which two deal with each of the three parts in the order named. Chapters I. and II. treat of addition, subtraction, scalar multiplication, and the scalar and vector products of vectors. The exposition has been made quite elementary. It can readily be under stood by and is especially suited for such readers as have a knowledge of only the elements of Trigonometry and Ana-

GENERAL PREFACE xi

lytic Geometry. Those who are well versed in Quaternions or allied subjects may perhaps need to read only the sum maries. Chapters III. and IV. contain the treatment of those topics in Vector Analysis which, though of less value to the students of pure mathematics, are of the utmost impor tance to students of physics. Chapters V. and VI. deal with the linear vector function. To students of physics the linear vector function is of particular importance in the mathemati cal treatment of phenomena connected with non-isotropic media ; and to the student of pure mathematics this part of the book will probably be the most interesting of all, owing to the fact that it leads to Multiple Algebra or the Theory of Matrices. A concluding chapter, VII., which contains the development of certain higher parts of the theory, a number of applications, and a short sketch of imaginary or complex vectors, has been added.

In the treatment of the integral calculus, Chapter IV., questions of mathematical rigor arise. Although modern theorists are devoting much time and thought to rigor, and although they will doubtless criticise this portion of the book adversely, it has been deemed best to give but little attention to the discussion of this subject. And the more so for the reason that whatever system of notation be employed ques tions of rigor are indissolubly associated with the calculus and occasion no new difficulty to the student of Vector Analysis, who must first learn what the facts are and may postpone until later the detailed consideration of the restric tions that are put upon those facts.

Notwithstanding the efforts which have been made during more than half a century to introduce Quaternions into physics the fact remains that they have not found wide favor. On the other hand there has been a growing tendency espe cially in the last decade toward the adoption of some form of Vector Analysis. The works of Heaviside and Foppl re ferred to before may be cited in evidence. As yet however no system of Vector Analysis which makes any claim to completeness has been published. In fact Heaviside says : "I am in hopes that the chapter which I now finish may

xii GENERAL PREFACE

serve as a stopgap till regular vectorial treatises come to be written suitable for physicists, based upon the vectorial treat ment of vectors" (Electromagnetic Theory, Vol. I., p. 305). Elsewhere in the same chapter Heaviside has set forth the claims of vector analysis as against Quaternions, and others have expressed similar views.

The keynote, then, to any system of vector analysis must be its practical utility. This, I feel confident, was Professor Gibbs's point of view in building up his system. He uses it entirely in his courses on Electricity and Magnetism and on Electromagnetic Theory of Light. In writing this book I have tried to present the subject from this practical stand point, and keep clearly before the reader's mind the ques tions: What combinations or functions of vectors occur in physics and geometry ? And how may these be represented symbolically in the way best suited to facile analytic manip ulation ? The treatment of these questions in modern books on physics has been too much confined to the addition and subtraction of vectors. This is scarcely enough. It has been the aim here to give also an exposition of scalar and vector products, of the operator y, of divergence and curl which have gained such universal recognition since the ap pearance of Maxwell's Treatise on Electricity and Magnetism, of slope, potential, linear vector function, etc., such as shall be adequate for the needs of students of physics at the present day and adapted to them.

It has been asserted by some that Quaternions, Vector Analysis, and all such algebras are of little value for investi gating questions in mathematical physics. Whether this assertion shall prove true or not, one may still maintain that vectors are to mathematical physics what invariants are to geometry. As every geometer must be thoroughly conver sant with the ideas of invariants, so every student of physics should be able to think in terms of vectors. And there is no way in which he, especially at the beginning of his sci entific studies, can come to so true an appreciation of the importance of vectors and of the ideas connected with them as by working in Vector Analysis and dealing directly with

GENERAL PREFACE xiii

the vectors themselves. To those that hold these views the success of Professor Foppl's Vorlesungen uber Technische Mechanik (four volumes, Teubner, 1897-1900, already in a second edition), in which the theory of mechanics is devel oped by means of a vector analysis, can be but an encour aging sign.

I take pleasure in thanking my colleagues, Dr. M. B. Porter and Prof. H. A. Bumstead, for assisting me with the manu script. The good services of the latter have been particularly valuable in arranging Chapters III. and IV* in their present form and in suggesting many of the illustrations used in the work. I am also under obligations to my father, Mr. Edwin H. Wilson, for help in connection both with the proofs and the manuscript. Finally, I wish to express my deep indebt edness to Professor Gibbs. For although he has been so preoccupied as to be unable to read either manuscript or proof, he has always been ready to talk matters over with me, and it is he who has furnished me with inspiration suf ficient to carry through the work.

EDWIN BIDWELL WILSON. YALE UNIVERSITY, October, 1901.

PREFACE TO THE SECOND EDITION

THE only changes which have been made in this edition are a few corrections which my readers have been kind enough to point out to me.

E. B. W.

TABLE OF CONTENTS

PAGE PREFACE BY PROFESSOR GIBBS • vii

GENERAL PREFACE ix

CHAPTER I

ADDITION AND SCALAR MULTIPLICATION ARTS.

1-3 SCALARS AND VECTORS 1

4 EQUAL AND NULL VECTORS 4

5 THE POINT OF VIEW OF THIS CHAPTER 6

6-7 SCALAR MULTIPLICATION. THE NEGATIVE SIGN .... 7

8-10 ADDITION. THE PARALLELOGRAM LAW 8

11 SUBTRACTION 11

12 LAWS GOVERNING THE FOREGOING OPERATIONS .... 12 13-16 COMPONENTS OF VECTORS. VECTOR EQUATIONS .... 14

17 THE THREE UNIT VECTORS 1, j, k 18

18-19 APPLICATIONS TO SUNDRY PROBLEMS IN GEOMETRY. . . 21

20-22 VECTOR RELATIONS INDEPENDENT OF THE ORIGIN ... 27

23-24 CENTERS OF GRAVITY. BARYCENTRIC COORDINATES . . 39

25 THE USE OF VECTORS TO DENOTE AREAS 46

SUMMARY OF CHAPTER i 51

EXERCISES ON CHAPTER i . . 52

CHAPTER II DIRECT AND SKEW PRODUCTS OF VECTORS

27-28 THE DIRECT, SCALAR, OR DOT PRODUCT OF TWO VECTORS 55

29-30 THE DISTRIBUTIVE LAW AND APPLICATIONS 58

31-33 THE SKEW, VECTOR, OR CROSS PRODUCT OF TWO VECTORS 60

34-35 THE DISTRIBUTIVE LAW AND APPLICATIONS 63

36 THE TRIPLE PRODUCT A* B C 67

XVI

CONTENTS

ARTS. PAGE

37-38 THE SCALAR TRIPLE PRODUCT A* B X C OR [ABC] . . 68

39-40 THE VECTOR TRIPLE PRODUCT A X (B X C) 71

41-42 PRODUCTS OF MORE THAN THREE VECTORS WITH APPLI CATIONS TO TRIGONOMETRY 75

43-45 RECIPROCAL SYSTEMS OF THREE VECTORS 81

46-47 SOLUTION OF SCALAR AND VECTOR EQUATIONS LINEAR IN

AN UNKNOWN VECTOR 87

48-50 SYSTEMS OF FORCES ACTING ON A RIGID BODY .... 92

51 KINEMATICS OF A RIGID BODY 97

52 CONDITIONS FOR EQUILIBRIUM OF A RIGID BODY ... 101

53 RELATIONS BETWEEN TWO RIGHT-HANDED SYSTEMS OF

THREE PERPENDICULAR UNIT VECTORS 104

54 PROBLEMS IN GEOMETRY. PLANAR COORDINATES . . . 106

SUMMARY OF CHAPTER n 109

EXERCISES ON CHAPTER n 113

CHAPTER III THE DIFFERENTIAL CALCULUS OF VECTORS

55-56 DERIVATIVES AND DIFFERENTIALS OF VECTOR FUNCTIONS

WITH RESPECT TO A SCALAR VARIABLE 115

57 CURVATURE AND TORSION OF GAUCHE CURVES .... 120

58-59 KINEMATICS OF A PARTICLE. THE HODOGRAPH . . . 125

60 THE INSTANTANEOUS AXIS OF ROTATION 131

61 INTEGFATION WITH APPLICATIONS TO KINEMATICS . . . 133

62 SCALAR FUNCTIONS OF POSITION IN SPACE 136

63-67 THE VECTOR DIFFERENTIATING OPERATOR V • • • • 138

68 THE SCALAR OPERATOR A • V 147

69 VECTOR FUNCTIONS OF POSITION IN SPACE 149

70 THE DIVERGENCE V* AND THE CURL VX 150

71 INTERPRETATION OF THE DIVERGENCE V 152

72 INTERPRETATION OF THE CURL V X 155

73 LAWS OF OPERATION OF V> V * > V X 157

74-76 THE PARTIAL APPLICATION OF V- EXPANSION OF A VEC TOR FUNCTION ANALOGOUS TO TAYLOR'S THEOREM.

APPLICATION TO HYDROMECHANICS 159

77 THE DIFFERENTIATING OPERATORS OF THE SECOND ORDER 166

78 GEOMETRIC INTERPRETATION OF LAPLACE'S OPERATOR

V* V AS THE DISPERSION 170

SUMMARY OF CHAPTER in 172

EXERCISES ON CHAPTER in 177

CONTENTS

xvii

CHAPTER IV

THE INTEGRAL CALCULUS OF VECTORS

ARTS. PAGE 79-80 LINE INTEGRALS OF VECTOR FUNCTIONS WITH APPLICA TIONS 179

81 GAUSS'S THEOREM 184

82 STOKES'S THEOREM 187

83 CONVERSE OF STOKES'S THEOREM WITH APPLICATIONS . 193

84 TRANSFORMATIONS OF LINE, SURFACE, AND VOLUME IN

TEGRALS. GREEN'S THEOREM 197

85 REMARKS ON MULTIPLE-VALUED FUNCTIONS 200

86-87 POTENTIAL. THE INTEGRATING OPERATOR " POT " . . 205

88 COMMUTATIVE PROPERTY OF POT AND V 211

89 REMARKS UPON THE FOREGOING 215

90 THE INTEGRATING OPERATORS "NEW," "LAP," " MAX " 222

91 RELATIONS BETWEEN THE INTEGRATING AND DIFFER

ENTIATING OPERATORS 228

92 THE POTENTIAL " POT " is A SOLUTION OF POISSON'S

EQUATION 230

93-94 SOLENOIDAL AND IRROTATIONAL PARTS OF A VECTOR

FUNCTION. CERTAIN OPERATORS AND THEIR INVERSE . 234

95 MUTUAL POTENTIALS, NEWTONIANS, LAPLACIANS, AND

MAXWELLIANS 240

96 CERTAIN BOUNDARY VALUE THEOREMS 243

SUMMARY OF CHAPTER iv 249

EXERCISES ON CHAPTER iv 255

CHAFIER V

LINEAR VECTOR FUNCTIONS

97-98 LINEAR VECTOR FUNCTIONS DEFINED 260

99 DYADICS DEFINED 264

100 ANY LINEAR VECTOR FUNCTION MAY BE REPRESENTED

BY A DYADIC. PROPERTIES OF DYADICS .... 266

101 THE NONION FORM OF A DYADIC 269

102 THE DYAD OR INDETERMINATE PRODUCT OF TWO VEC

TORS IS THE MOST GENERAL. FUNCTIONAL PROPERTY

OF THE SCALAR AND VECTOR PRODUCTS 271

108-104 PRODUCTS OF DYADICS 276

105-107 DEGREES OF NULLITY OF DYADICS 282

108 THE IDEMFACTOR 288

XV111

CONTENTS

ARTS. PAGE 109-110 RECIPROCAL DYADICS. POWERS AND ROOTS OF DYADICS 290 111 CONJUGATE DYADICS. SELF-CONJUGATE AND ANTI- SELF-CONJUGATE PARTS OF A DYADIC 294

112-114 ANTI-SELF-CONJUGATE DYADICS. THE VECTOR PROD UCT. QUADRANTAL VER8ORS 297

115-116 REDUCTION OF DYADICS TO NORMAL FORM .... 302

117 DOUBLE MULTIPLICATION OF DYADICS 306

118-119 THE SECOND AND THIRD OF A DYADIC . . ... 310

120 CONDITIONS FOR DIFFERENT DEGREES OF NULLITY . 313

121 NONION FORM. DETERMINANTS 315

122 INVARIANTS OF A DYADIC. THE HAMILTON-CAYLEY

EQUATION .319

SUMMARY OF CHAPTER v 321

EXERCISES ON CHAPTER v 329

CHAPTER VI

ROTATIONS AND STRAINS

123-124 HOMOGENEOUS STRAIN REPRESENTED BY A DYADIC . 332

125-126 ROTATIONS ABOUT A FIXED POINT. VERSORS . . . 334

127 THE VECTOR SEMI-TANGENT OF VERSION 339

128 BlQUADRANTAL VERSORS AND THEIR PRODUCTS . . . 343

129 CYCLIC DYADICS 347

130 RIGHT TENSORS 351

131 TONICS AND CYCLOTONICS 353

132 REDUCTION OF DYADICS TO CANONICAL FORMS, TONICS,

CYCLOTONICS, SIMPLE AND COMPLEX SHEARERS . . 356

SUMMARY OF CHAPTER vi 368

CHAPTER VII

MISCELLANEOUS APPLICATIONS

136-142 QUADRIC SURFACES 372

143-146 THE PROPAGATION OF LIGHT IN CRYSTALS .... 392

147-148 VARIABLE DYADICS 403

149-157 CURVATURE OF SURFACES 411

158-162 HARMONIC VIBRATIONS AND BIVECTORS .... 426

VECTOR ANALYSIS

VECTOR ANALYSIS

CHAPTER I

ADDITION AND SCALAR MULTIPLICATION

1.] IN mathematics and especially in physics two very different kinds of quantity present themselves. Consider, for example, mass, time, density, temperature, force, displacement of a point, velocity, and acceleration. Of these quantities some can be represented adequately by a single number — temperature, by degrees on a thermometric scale ; time, by years, days, or seconds ; mass and density, by numerical val- . ues which are wholly determined when the unit of the scale is fixed. On the other hand the remaining quantities are not capable of such representation. Force to be sure is said to be of so many pounds or grams weight; velocity, of so many feet or centimeters per second. But in addition to this each of them must be considered as having direction as well as magnitude. A force points North, South, East, West, up, down, or in some intermediate direction. The same is true of displacement, velocity, and acceleration. No scale of num bers can represent them adequately. It can represent only their magnitude, not their direction.

2.] Definition : A vector is a quantity which is considered as possessing direction as well as magnitude.

Definition : A scalar is a quantity which is considered as pos sessing magnitude but no direction.

2 VECTOR ANALYSIS

The positive and negative numbers of ordinary algebra are the typical scalars. For this reason the ordinary algebra is called scalar algebra when necessary to distinguish it from the vector algebra or analysis which is the subject of this book.

The typical vector is the displacement of translation in space. Consider first a point P (Fig. 1). Let P be displaced in a straight line and take a new position Pf. This change of position is represented by the line PP. The magnitude of the displace ment is the length of PP1; the direction of it is the direction of the line PP1 from P to P1. Next consider a displacement not of one, but of all the points in space. Let all the points move in straight lines in the same direction and for the same distance D. This is equivalent to shifting space as a rigid body in that direction through the distance D without rotation. Such a displacement is called a translation. It possesses direction and magnitude. When space undergoes a translation T, each point of space undergoes a displacement equal to T in magnitude and direction; and conversely if the displacement PP' which any one particular point P suf fers in the translation T is known, then that of any other point Q is also known : for Q Q' must be equal and parallel to PP.

The translation T is represented geometrically or graphically by an arrow T (Fig. 1) of which the magnitude and direction are equal to those of the translation. The absolute position of this arrow in space is entirely immaterial. Technically the arrow is called a stroke. Its tail or initial point is its origin; and its head or final point, its terminus. In the figure the origin is designated by 0 and the terminus by T. This geo metric quantity, a stroke, is used as the mathematical symbol for all vectors, just as the ordinary positive and negative num bers are used as the symbols for all scalars.

ADDITION AND SCALAR MULTIPLICATION 3

* 3.] As examples of scalar quantities mass, time, den sity, and temperature have been mentioned. Others are dis tance, volume, moment of inertia, work, etc. Magnitude, however, is by no means the sole property of these quantities. Each implies something besides magnitude. Each has its own distinguishing characteristics, as an example of which its dimensions in the sense well known to physicists may be cited. A distance 3, a time 3, a work 3, etc., are very different. The magnitude 3 is, however, a property common to them all — perhaps the only one. Of all scalar quanti- tities pure number is the simplest. It implies nothing but magnitude. It is the scalar par excellence and consequently it is used as the mathematical symbol for all scalars.

As examples of vector quantities force, displacement, velo city, and acceleration have been given. Each of these has other characteristics than those which belong to a vector pure and simple. The concept of vector involves two ideas and two alone — magnitude of the vector and direction of the vector. But force is more complicated. When it is applied to a rigid body the line in which it acts must be taken into consideration; magnitude and direction alone do not suf fice. And in case it is applied to a non-rigid body the point of application of the force is as important as the magnitude or direction. Such is frequently true for vector quantities other than force. Moreover the question of dimensions is present as in the case of scalar quantities. The mathematical vector, the stroke, which is the primary object of consideration in this book, abstracts from all directed quantities their magni tude and direction and nothing but these ; just as the mathe matical scalar, pure number, abstracts the magnitude and that alone. Hence one must be on his guard lest from analogy he attribute some properties to the mathematical vector which do not belong to it ; and he must be even more careful lest he obtain erroneous results by considering the

4 VECTOR ANALYSIS

vector quantities of physics as possessing no properties other than those of the mathematical vector. For example it would never do to consider force and its effects as unaltered by shifting it parallel to itself. This warning may not be necessary, yet it may possibly save some confusion.

4.] Inasmuch as, taken in its entirety, a vector or stroke is but a single concept, it may appropriately be designated by one letter. Owing however to the fundamental difference between scalars and vectors, it is necessary to distinguish carefully the one from the other. Sometimes, as in mathe matical physics, the distinction is furnished by the physical interpretation. Thus if n be the index of refraction it must be scalar ; m, the mass, and £, the time, are also scalars ; but /, the force, and a, the acceleration, are vectors. When, however, the letters are regarded merely as symbols with no particular physical significance some typographical difference must be relied upon to distinguish vectors from scalars. Hence in this book Clarendon type is used for setting up vectors and ordinary type for scalars. This permits the use of the same letter differently printed to represent the vector and its scalar magnitude.1 Thus if C be the electric current in magnitude and direction, C may be used to represent the magnitude of that current ; if g be the vector acceleration due to gravity, g may be the scalar value of that acceleration ; if v be the velocity of a moving mass, v may be the magnitude of that velocity. The use of Clarendons to denote vectors makes it possible to pass from directed quantities to their scalar magnitudes by a mere change in the appearance of a letter without any confusing change in the letter itself.

Definition : Two vectors are said to be equal when they have the same magnitude and the same direction.

1 This convention, however, is by no means invariably followed. In some instances it would prove just as undesirable as it is convenient in others. It is chiefly valuable in the application of vectors to physics.

ADDITION AND SCALAR MULTIPLICATION 5

The equality of two vectors A and B is denoted by the usual sign =. Thus A = B

Evidently a vector or stroke is not altered by shifting it about parallel to itself in space. Hence any vector A = PPr (Fig. 1) may be drawn from any assigned point 0 as origin ; for the segment PPf may be moved parallel to itself until the point P falls upon the point 0 and P' upon some point T.

In this way all vectors in space may be replaced by directed segments radiating from one fixed point 0. Equal vectors in space will of course coincide, when placed with their ter mini at the same point 0. Thus (Fig. 1) A = PP\ and B = Q~Qf, both fall upon T = ~OT.

For the numerical determination of a vector three scalars are necessary. These may be chosen in a variety of ways. If r, </>, 0 be polar coordinates in space any vector r drawn with its origin at the origin of coordinates may be represented by the three scalars r, </>, 6 which determine the terminus of the vector. r~(r,*,0).

Or if #, y9 z be Cartesian coordinates in space a vector r may be considered as given by the differences of the coordinates a/, y'i zf of its terminus and those #, y, z of its origin.

r~ (xr — x,yr — y,zr — z).

If in particular the origin of the vector coincide with the origin of coordinates, the vector will be represented by the three coordinates of its terminus

r -(*',*,', *')•

When two vectors are equal the three scalars which repre sent them must be equal respectively each to each. Hence one vector equality implies three scalar equalities.

6 VECTOR ANALYSIS

Definition : A vector A is said to be equal to zero when its magnitude A is zero.

Such a vector A is called a null or zero vector and is written equal to naught in the usual manner. Thus

A = 0 if A = 0.

All null vectors are regarded as equal to each other without any considerations of direction.

In fact a null vector from a geometrical standpoint would be represented by a linear segment of length zero — that is to say, by a point. It consequently would have a wholly inde terminate direction or, what amounts to the same thing, none at all. If, however, it be regarded as the limit approached by a vector of finite length, it might be considered to have that direction which is the limit approached by the direction of the finite vector, when the length decreases indefinitely and ap proaches zero as a limit. The justification for disregarding this direction and looking upon all null vectors as equal is that when they are added (Art. 8) to other vectors no change occurs and when multiplied (Arts. 27, 31) by other vectors the product is zero.

5.] In extending to vectors the fundamental operations of algebra and arithmetic, namely, addition, subtraction, and multiplication, care must be exercised riot only to avoid self- contradictory definitions but also to lay down useful ones. Both these ends may be accomplished most naturally and easily by looking to physics (for in that science vectors con tinually present themselves) and by observing how such quantities are treated there. If then A be a given displace ment, force, or velocity, what is two, three, or in general x times A? What, the negative of A? And if B be another, what is the sum of A and B ? That is to say, what is the equivalent of A and B taken together ? The obvious answers to these questions suggest immediately the desired definitions.

ADDITION AND SCALAR MULTIPLICATION 1

Scalar Multiplication

6.] Definition: A vector is said to be multiplied by a positive scalar when its magnitude is multiplied by that scalar and its direction is left unaltered

Thus if v be a velocity of nine knots East by North, 2 £ times v is a velocity of twenty-one knots with the direction still East by North. Or if f be the force exerted upon the scale- pan by a gram weight, 1000 times f is the force exerted by a kilogram. The direction in both cases is vertically down ward.

If A be the vector and x the scalar the product of x and A is

denoted as usual by

x A or A x.

It is, however, more customary to place the scalar multiplier before the multiplicand A. This multiplication by a scalar is called scalar multiplication, and it follows the associative law

x (y A) = (x y) A = y (x A)

as in ordinary algebra and arithmetic. This statement is im mediately obvious when the fact is taken into consideration that scalar multiplication does not alter direction but merely multiplies the length.

Definition : A unit vector is one whose magnitude is unity.

Any vector A may be looked upon as the product of a unit vector a in its direction by the positive scalar A, its magni tude.

A = A a = a A.

The unit vector a may similarly be written as the product of A by I/A or as the quotient of A and A.

1 A

a = ^ A = -I'

A A

8 VECTOR ANALYSIS

7.] Definition : The negative sign, — prefixed to a vector reverses its direction but leaves its magnitude unchanged.

For example if A be a displacement for two feet to the right, — A is a displacement for two feet to the left. Again if the stroke A~B be A, the stroke B A, which is of the same length as A £ but which is in the direction from B to A instead of from A to 5, will be — A. Another illustration of the use of the negative sign may be taken from Newton's third law of motion. If A denote an "action," — A will denote the " reaction." The positive sign, + , may be prefixed to a vec tor to call particular attention to the fact that the direction has not been reversed. The two signs + and — when used in connection with scalar multiplication of vectors follow the same laws of operation as in ordinary algebra. These are symbolically

+ + = + ; +- = -; - + = -; = +;

— (ra A) = m (— A).

The interpretation is obvious.

Addition and Subtraction

8.] The addition of two vectors or strokes may be treated most simply by regarding them as defining translations in space (Art. 2), Let S be one vector and T the other. Let P be a point of space (Fig. 2). The trans lation S carries P into P1 such that the line PP1 is equal to S in magnitude and direction. The transformation T will then carry P1 into P11 — the line P' P" being parallel to T and equal to it in magnitude. FIG. 2. Consequently the result of S followed by

T is to carry the point P into the point P". If now Q be any other point in space, S will carry Q into Q' such that Q~Qr = S and T will then carry Qf into Q"

ADDITION AND SCALAR MULTIPLICATION

such that Q'Q" = T. Thus S followed by T carries Q into Q". Moreover, the triangle Q Qf Q" is equal to PP'P". For the two sides Q Qf and Q' Q", being equal and parallel to S and T respectively, must be likewise parallel to P P1 and P'P" respectively which are also parallel to S and T. Hence the third sides of the triangles must be equal and parallel

That is

Q Q" is equal and parallel to PP".

As Q is any point in space this is equivalent to saying that by means of S followed by T all points of space are displaced the same amount and in the same direction. This displace ment is therefore a translation. Consequently the two translations S and T are equivalent to a single translation R. Moreover

if S = PP' and T = P' P", then R = PP".

The stroke R is called the resultant or sum of the two strokes S and T to which it is equivalent. This sum is de noted in the usual manner by

R = S + T.

From analogy with the sum or resultant of two translations the following definition for the addition of any two vectors is laid down.

Definition : The sum or resultant of two vectors is found by placing the origin of the second upon the terminus of the first and drawing the vector from the origin of the first to the terminus of the second.

9.] Theorem. The order in which two vectors S and T are added does not affect the sum.

S followed by T gives precisely the same result as T followed by S. For let S carry P into P' (Fig. 3) ; and T, P' into P". S + T then carries P into P". Suppose now that T carries P into P'". The line PP'" is equal and parallel to PP". Con-

10 VECTOR ANALYSIS

sequently the points P, P'9 Pff, and Pm lie at the vertices of

a parallelogram. Hence pm pn js equal an(J par-

allel to PP. Hence S carries P"f into P". T fol lowed by S therefore car ries P into P" through P\ whereas S followed by T carries P into P" through Pm. The final result is in either case the same. This may be designated symbolically

by writing

R = S + T = T + S.

It is to be noticed that S = PP1 and T = PPm are the two sides of the parallelogram pprp»p"' which1 have the point P as common origin ; and that JL=PP" is the diagonal drawn through P. This leads to another very common way of stating the definition of the sum of two vectors.

If two vectors be drawn from the same origin and a parallelo gram be constructed upon them as sides, their sum will be that diagonal which passes through their common origin.

This is the well-known " parallelogram law " according to which the physical vector quantities force, acceleration, veloc ity, and angular velocity are compounded. It is important to note that in case the vectors lie along the same line vector addition becomes equivalent to algebraic scalar addition. The lengths of the two vectors to be added are added if the vectors have the same direction ; but subtracted if they have oppo site directions. In either case the sum has the same direction as that of the greater vector.

10.] After the definition of the sum of two vectors has been laid down, the sum of several may be found by adding together the first two, to this sum the third, to this the fourth, and so on until all the vectors have been combined into a sin-

ADDITION AND SCALAR MULTIPLICATION 11

gle one. The final result is the same as that obtained by placing the origin of each succeeding vector upon the terminus of the preceding one and then drawing at once the vector from the origin of the first to the terminus of the last. In case these two points coincide the vectors form a closed polygon and their sum is zero. Interpreted geometrically this states that if a number of displacements R, S, T • • • are such that the strokes R, S, T • • • form the sides of a closed polygon taken in order, then the effect of carrying out the displacements is nil. Each point of space is brought back to its starting point. In terpreted in mechanics it states that if any number of forces act at a point and if they form the sides of a closed polygon taken in order, then the resultant force is zero and the point is in equilibrium under the action of the forces.

The order of sequence of the vectors in a sum is of no con sequence. This may be shown by proving that any two adja cent vectors may be interchanged without affecting the result.

To show

Let A = 0 A, B = A B, C = B C, D = 0 D, E = D E. Then

_

Let now B C1 = D. Then C! B C D is a parallelogram and consequently Cf D = C. Hence

OJ£ = A + B + D + C + E,

which proves the statement. Since any two adjacent vectors may be interchanged, and since the sum may be arranged in any order by successive interchanges of adjacent vectors, the order in which the vectors occur in the sum is immaterial.

11.] Definition : A vector is said to be subtracted when it is added after reversal of direction. Symbolically,

A - B = A + (- B). By this means subtraction is reduced to addition and needs

12

VECTOR ANALYSIS

no special consideration. There is however an interesting and important way of representing the difference of two vectors geometrically. Let A = 'OA, B = 0IT(Fig. 4). Complete

the parallelogram of which A and B are the sides. Then the diagonal ~OG = C is the sum A + B of the two vectors. Next complete the parallelogram of which A and — B = OB' are the sides. Then the di

agonal 02) = !) will be the sum of A and the negative of B. But the segment OD is parallel and equal

to BA. Hence BA may be taken as the difference to the two vectors A and B. This leads to the following rule : The differ ence of two vectors which are drawn from the same origin is the vector drawn from the terminus of the vector to be sub tracted to the terminus of the vector from which it is sub tracted. Thus the two diagonals of the parallelogram, which is constructed upon A and B as sides, give the sum and dif ference of A and B.

12.] In the foregoing paragraphs addition, subtraction, and scalar multiplication of vectors have been defined and inter preted. To make the development of vector algebra mathe matically exact and systematic it would now become necessary to demonstrate that these three fundamental operations follow the same formal laws as in the ordinary scalar algebra, al though from the standpoint of the physical and geometrical interpretation _of vectors this may seem superfluous. These laws are

m (n A) = n (m A) = (m n} A,

(A + B) + C = A+ (B + C), II A + B r, B + A,

IIIa (m + n) A = m A + n A,

m (A + B) = m A + m B, III, - (A + B) = - A - B.

ADDITION AND SCALAR MULTIPLICATION 13

10 is the so-called law of association and commutation of the scalar factors in scalar multiplication.

I6 is the law of association for vectors in vector addition. It states that in adding vectors parentheses may be inserted at any points without altering the result.

11 is the commutative law of vector addition.

IIIa is the distributive law for scalars in scalar multipli cation.

III6 is the distributive law for vectors in scalar multipli cation.

Ill, is the distributive law for the negative sign.

The proofs of these laws of operation depend upon those propositions in elementary geometry which have to deal with the first properties of the parallelogram and similar triangles. They will not be given here; but it is suggested that the reader work them out for the sake of fixing the fundamental ideas of addition, subtraction, and scalar multiplication more clearly in mind. The result of the laws may be summed up in the statement :

The laws which govern addition, subtraction, and scalar multiplication of vectors are identical with those governing these operations in ordinary scalar algebra.

It is precisely this identity of formal laws which justifies the extension of the use of the familiar signs =, +, and — of arithmetic to the algebra of vectors and it is also this which ensures the correctness of results obtained by operat ing with those signs in the usual manner. One caution only need be mentioned. Scalars and vectors are entirely different sorts of quantity. For this reason they can never be equated to each other — except perhaps in the trivial case where each is zero. For the same reason they are not to be added together. So long as this is borne in mind no difficulty need be antici pated from dealing with vectors much as if they were scalars.

Thus from equations in which the vectors enter linearly with

14 VECTOR ANALYSIS

scalar coefficients unknown vectors may be eliminated or found by solution in the same way and with the same limita tions as in ordinary algebra; for the eliminations and solu tions depend solely on the scalar coefficients of the equations and not at all on what the variables represent. If for

instance

aA + &B + cC + dD = 0,

then A, B, C, or D may be expressed in terms of the other three

as D = --:OA + &B + cC).

a

And two vector equations such as

3 A+ 4B=E

and 2 A + 3 B = F

yield by the usual processes the solutions

A=3E-4F and B = 3 F - 2 E.

Components of Vectors

13.] Definition : Vectors are said to be collinear when they are parallel to the same line; coplanar, when parallel to the same plane. Two or more vectors to which no line can be drawn parallel are said to be non-collinear. Three or more vectors to which no plane can be drawn parallel are said to be non-coplanar. Obviously any two vectors are coplanar.

Any vector b collinear with a may be expressed as the product of a and a positive or negative scalar which is the ratio of the magnitude of b to that of a. The sign is positive when b and a have the same direction ; negative, when they have opposite directions. If then OA = a, the vector r drawn

ADDITION AND SCALAR MULTIPLICATION 15

from the origin 0 to any point of the line 0 A produced in

either direction is

r = x a. (1)

If x be a variable scalar parameter this equation may there fore be regarded as the (vector) equation of all points in the line OA. Let now B be any point not upon the line OA or that line produced • in either direction (Fig. 5).

Let OB = b. The vector b is surely not of the form x a. Draw through B Flo 5 "

a line parallel to OA and Let R be any point upon it. The vector BE is collinear with a and is consequently expressible as #a. Hence the vector drawn from 0 to R is

0~E=0~B + ITR

or r = b + #a. (2)

This equation may be regarded as the (vector) equation of all the points in the line which is parallel to a and of which B is one point.

14.] Any vector r coplanar with two non-collinear vectors a and b may be resolved into two components parallel to a and b respectively. This resolution may be accomplished by constructing the par allelogram (Fig. 6) of which the sides are parallel to a and b and of which the di agonal is r. Of these components one is x a ; the other, y b. x and y are respec tively the scalar ratios (taken with the proper sign) of the lengths of these components to the lengths

of a and b, Hence

r = x a + y b (2)'

is a typical form for any vector coplanar with a and b. If several vectors rx, r2, r3 • • • may be expressed in this form as

16

VECTOR ANALYSIS

their sum r is then

rl = xl a + yl b, r2 = #2 a + 2/2 b, r3 = xz a + 2/3 b.

+ (ft + ft + ft + •••) »•

This is the well-known theorem that the components of a sum of vectors are the sums of the components of those vectors. If the vector r is zero each of its components must be zero. Consequently the one vector equation r = 0 is equivalent to the two scalar equations

y\ + ft + ft + • • • =

(3)

15.] Any vector r in space may be resolved into three components parallel to any three given non-coplanar vectors.

Let the vectors be a, b, and c. The resolution may then be accom plished by constructing the parallelepiped (Fig. 7) of which the edges are parallel to a, b, and c and of which the di agonal is r. This par- allelopiped may be drawn easily by passing three planes parallel re spectively to a and b, b and c, c and a through the origin 0 of the vector r ; and a similar set of three planes through its terminus It. These six planes will then be parallel in pairs

FIG. 7.

ADDITION AND SCALAR MULTIPLICATION 17

and hence form a parallelepiped. That the intersections of the planes are lines which are parallel to a, or b, or c is obvious. The three components of r are x a, y b, and zc; where x, y, and z are respectively the scalar ratios (taken with the proper sign) of the lengths of these components to the length of a, b, and c. Hence

r = # a + 7/b + zc (4)

is a typical form for any vector whatsoever in space. Several vectors rlf r2, r3 . . . may be expressed in this form as

rx = xl a + yl b + zl c, r2 = #2 a + y2 b + *2 c»

1*3 = XZ a + 2/3 b

Their sum r is then

1 =rl + r2 + F3 + * ' ' = 0*1 + *2 + XZ + ' • 0 a

+ (2/i + 2/2+ 3/3 + •••)!>

+ Ol +^2 + ^3+ "O0-

If the vector r is zero each of its three components is zero. Consequently the one vector equation r = 0 is equivalent to the three scalar equations

xl + #2 + #3 + • - • = 0 v

2/i + 2/2 + 2/3 + • • • = ° y r = 0. (5)

*i + *2 + % + • • • = 0 /

Should the vectors all be coplanar with a and b, all the com ponents parallel to c vanish. In this case therefore the above equations reduce to those given before.

16.] If two equal vectors are expressed in terms of the same three non-coplanar vectors, the corresponding scalar co efficients are equal.

18 VECTOR ANALYSIS

Let r = r',

r' = x9 a + y1 b + z' c,

'

Then x = x', y = y

For r - r' = 0 = (x - xf) a + (y - y') b + (* - z1) c.

Hence x - *' = 0, y - y' = 0, z - *' = 0.

But this would not be true if a, b, and c were coplanar. In that case one of the three vectors could be expressed in terms

of the other two as

c = m a + n b.

Then r = #a + y b + s c = (a + m z) a + (y + TI z) b, r' = x! a + y1 b + z;c = (x1 + m z') a + (y' + n z') b, r — r' = [(x + m z ) — (x' + m z')] a,

Hence the individual components of r — r' in the directions a and b (supposed different) are zero.

Hence x + mz = xr + mzr

y -f n z = yf + n z1.

But this by no means necessitates x, y, z to be equal respec tively to x\ y\ z1. In a similar manner if a and b were col- linear it is impossible to infer that their coefficients vanish individually. The theorem may perhaps be stated as follows : . In case two equal vectors are expressed in terms of one vector, or two non-collinear vectors, or three non-coplanar vectors, the corresponding scalar coefficients are equal. But this is not ne cessarily true if the two vectors be collinear ; or the three vectors, coplanar. This principle will be used in the applications (Arts. 18 et seq.).

The Three Unit Vectors i, j, k.

17.] In the foregoing paragraphs the method of express ing vectors in terms of three given non-coplanar ones has been explained. The simplest set of three such vectors is the rect-

ADDITION AND SCALAR MULTIPLICATION 19

angular system familiar in Solid Cartesian Geometry. This rectangular system may however be either of two very distinct types. In one case (Fig. 8, first part) the Z-axis l lies upon that side of the X Y- plane on which rotation through a right angle from the X-axis to the F-axis appears counterclockwise or positive according to the convention adopted in Trigonome try. This relation may be stated in another form. If the X axis be directed to the right and the F-axis vertically, the ^-axis will be directed toward the observer. Or if the X- axis point toward the observer and the F-axis to the right, the ^-axis will point upward. Still another method of state-

Z

,,k

Right-handed

FIG. 8.

Left-handed

ment is common in mathematical physics and engineering. If a right-handed screw be turned from the Xaxis to the F- axis it will advance along the (positive) Z-axis. Such a sys tem of axes is called right-handed, positive, or counterclock wise.2 It is easy to see that the F-axis lies upon that side of the ^X-plane on which rotation from the ^-axis to the X- axis is counterclockwise ; and the X-axis, upon that side of

1 By the X-, Y-, or Z-axis the positive half of that axis is meant. The X Y- plane means the plane which contains the X- and Y-axis, i. e., the plane z = 0.

2 A convenient right-handed system and one which is always available consists of the thumb, first finger, and second finger of the right hand. If the thumb and first finger be stretched out from the palm perpendicular to each other, and if the second finger be bent over toward the palm at right angles to first finger, a right- handed system is formed by the fingers taken in the order thumb, first finger, second finger.

20 VECTOR ANALYSTS

the F^-plane on which rotation from the F-axis to the Z- axis is counterclockwise. Thus it appears that the relation between the three axes is perfectly symmetrical so long as the same cyclic order XYZXY is observed. If a right-handed screw is turned from one axis toward the next it advances along the third.

In the other case (Fig. 8, second part) the ^-axis lies upon that side of the X F-plane on which rotation through a right angle from the JT-axis to the F-axis appears clockwise or neg ative. The F-axis then lies upon that side of the ^X-plane on which rotation from the ^-axis to the X-axis appears clockwise and a similar statement may be made concerning the X-axis in its relation to the F^-plane. In this case, too, the relation between the three axes is S3rmmetrical so long as the same cyclic order X YZX Y is preserved but it is just the opposite of that in the former case. If a fe/Mianded screw is turned from one axis toward the next it advances along the third. Hence this system is called left-handed, negative, or clockwise.1

The two systems are not superposable. They are sym metric. One is the image of the other as seen in a mirror. If the JT- and F-axes of the two different systems be superimposed, the ^-axes will point in opposite directions. Thus one system may be obtained from the other by reversing the direction of one of the axes. A little thought will show that if two of the axes be reversed in direction the system will not be altered, but if all three be so reversed it will be.

Which of the two systems be used, matters little. But in asmuch as the formulae of geometry and mechanics differ slightly in the matter of sign, it is advisable to settle once for all which shall be adopted. In this book the right-handed or counterclockwise system will be invariably employed.

1 A left-handed system may be formed by the left hand just as a right-handed one was formed by the right.

ADDITION AND SCALAR MULTIPLICATION 21

Definition : The three letters i, j, k will be reserved to de note three vectors of unit length drawn respectively in the directions of the JT-, T-, and Z- axes of a right-handed rectan gular system.

In terms of these vectors, any vector may be expressed as

r = xi + y] + zk. (6)

The coefficients xy y, z are the ordinary Cartesian coordinates of the terminus of r if its origin be situated at the origin of coordinates. The components of r parallel to the X-, F-, and ^f-axes are respectively

x i, y j, z k.

The rotations about i from j to k, about j from k to i, and about k from i to j are all positive.

By means of these vectors i, j, k such a correspondence is established between vector analysis and the analysis in Car tesian coordinates that it becomes possible to pass at will from either one to the other. There is nothing contradic tory between them. On the contrary it is often desirable or even necessary to translate the formulae obtained by vector methods into Cartesian coordinates for the sake of comparing them with results already known and it is still more frequently convenient to pass from Cartesian analysis to vectors both on account of the brevity thereby obtained and because the vector expressions show forth the intrinsic meaning of the formulae.

Applications

*18.J Problems in plane geometry may frequently be solved easily by vector methods. Any two non-collinear vectors in the plane may be taken as the fundamental ones in terms of which all others in that plane may be expressed. The origin may also be selected at pleasure. Often it is possible to

22 VECTOR ANALYSIS

make such an advantageous choice of the origin and funda mental vectors that the analytic work of solution is materially simplified. The adaptability of the vector method is about the same as that of oblique Cartesian coordinates with differ ent scales upon the two axes.

Example 1 : The line which joins one vertex of a parallelo gram to the middle point of an opposite side trisects the diag onal (Fig. 9).

Let A BCD be the parallelogram, BE the line joining the vertex B to the middle point E of the side AD, R the point in which this line cuts the diagonal A C. To show A R is one third of FlG 9 AC. Choose A as origin, A B and AD as the

two fundamental vectors S and T. Then A C is the sum of S and T. Let AR = R. To show

R = 1 (S + T).

-

where x is the ratio of ER to EB — an unknown scalar.

And R = y (S + T),

where y is the scalar ratio of A R to A C to be shown equal

to.

Hence \ T + x (S -i T) = y (S + T)

or * S + 1 (1 - X) T = y S + y T.

Hence, equating corresponding coefficients (Art. 16),

2 (1 - x) = y.

ADDITION AND SCALAR MULTIPLICATION 23 From which y = « .

Inasmuch as x is also «- the line j&2? must be trisected as

o

well as the diagonal A C.

Example 2 : If through any point within a triangle lines be drawn parallel to the sides the sum of the ratios of these lines to their corresponding sides is 2.

Let ABC be the triangle, R the point within it. Choose A as origin, A B and A C as the two fundamental vectors S and T. Let

AR = R = w S + 7i T. (a)

m S is the fraction of A B which is cut off by the line through R parallel to A C. The remainder of A B must be the frac tion (1 — m) S. Consequently by similar triangles the ratio of the line parallel to A C to the line A C itself is (1 — ra). Similarly the ratio of the line parallel to A B to the line A B itself is (1 — n ). Next express R in terms of S and T — S the third side of the triangle. Evidently from (a)

R = (m + ri) S + n (T - S).

Hence (m + ri) S is the fraction of A B which is cut off by the line through R parallel to B C. Consequently by similar tri angles the ratio of this line to BC itself is (m + n). Adding the three ratios

(1 - m) + (1 - n) + (m + ri) = 2,

and the theorem is proved.

Example 3 : If from any point within a parallelogram lines be drawn parallel to the sides, the diagonals of the parallelo grams thus formed intersect upon the diagonal of the given parallelogram.

Let A B CD be a parallelogram, R a point within it, KM and LN two lines through R parallel respectively to AB and

24 VECTOR ANALYSIS

AD, the points K, Z, M, N lying upon the sides DA, AS, B C, CD respectively. To show that the diagonals KN and LM of the two parallelograms KRND and LBME meet on A C. Choose A as origin, A B and A D as the two funda mental vectors S and T. Let

R = AB = m S 4- ft T, and let P be the point of intersection of KN with LM.

Then KN=KR + BN = m S + (1 - rc) T,

=(1 -m) S + 7i T,

Hence P = n T + x [m S + (1 — n) T],

and P = m S + y [(1 - m) S + n T].

Equating coefficients,

x m = m + y (1 — m)

By solution, ;

m + n — 1

m ~ m + n — 1

Substituting either of these solutions in the expression for P, the result is

P^-^-^S + T),

which shows that P is collinear with A C.

* 19.] Problems in three dimensional geometry may be solved in essentially the same manner as those in two dimen sions. In this case there are three fundamental vectors in terms of which all others can be expressed. The method of solution is analogous to that in the simpler case. Two

ADDITION AND SCALAR MULTIPLICATION 25

expressions for the same vector are usually found. The co efficients of the corresponding terms are equated. In this way the equations between three unknown scalars are obtained from which those scalars may be determined by solution and then substituted in either of the expressions for the required vector. The vector method has the same degree of adapta bility as the Cartesian method in which oblique axes with different scales are employed. The following examples like those in the foregoing section are worked out not so much for their intrinsic value as for gaining a familiarity with vectors.

Example 1 : Let A B CD be a tetrahedron and P any point within it. Join the vertices to P and produce the lines until they intersect the opposite faces in A\ B', C1, Df. To

show

PA' PB' PC1 PD'

A~Af TTB' ~C~Of

' "

Choose A as origin, and the edges A J?, A C, AD as the three fundamental vectors B, C, D. Let the vector A P be

P = A P=IE + raC + 7i D,

Also A' = A A' = A B + BA'.

The vector BA1 is coplanar with WC = C — B and BD D — B. Hence it may be expressed in terms of them.

A' = B + ^1(C-B)+y1(D~B). Equating coefficients Jcl m = xv

Hence &., =

PA' _ V

ZZ7 ~~&

1 I + m + n

PA* JL-1 and — "• ^7 '

26 VECTOR ANALYSIS

In like manner A B' = #2 C + y2 D

and A B' = ^t £ + B B' = B + &2 (P - B).

Hence o;2C + y2D = B + A:2(ZB + mC + ^D-B and 0 = 1 + *, (J - 1),

Hence

2 -i __ ' -

and

In the same way it may be shown that PC' .PL1

CC* 3D'

Adding the four ratios the result is

•i d JL vn -4- <w ^ _L 7 J_ w -I- 77 1

Example % : To find a line which passes through a given point and cuts two given lines in space.

Let the two lines be fixed respectively by two points A and B, C and D on each. Let 0 be the given point. Choose it as origin and let

C = ~OC, D=d~D.

Any point P of A B may be expressed as

P= OP= 0~A + xA£ = A + x (B- A). Any point Q of CD may likewise be written

If the points P and Q lie in the same line through 0, P and are collinear That is

ADDITION AND SCALAR MULTIPLICATION

27

Before it is possible to equate coefficients one of the four vectors must be expressed in terms of the other three.

Then P = A + x (B - A)

— & Tf _1_ ( 1 A _J_ m Tl _1_ >w P I^^T

Hence 1 — x = z y /,

x = zy m, 0 = z [1 + y (n - 1)J.

Hence

m

x =

y =

2 =

i i- »

_________ •

I + m

Substituting in P and ft

I A+ m B

+ m

ft =

Either of these may be taken as defining a line drawn from 0 and cutting A B and CD.

Vector Relations independent of the Origin

20.] Example 1 : To divide a line A B in a given ratio m : n (Fig. 10).

Choose any arbitrary point 0 as origin. Let OA = A and OB = B. To find the vector P = ~OP of which the terminus P divides AB in the ratio m : n.

m

B

FIG. 10.

That is,

P =

B = A

-f- 7i

n A + m B

n

(B - A). (7)

28 VECTOR ANALYSIS

The components of P parallel to A and B are in inverse ratio to the segments A P and PB into which the line A B is divided by the point P. If it should so happen that P divided the line AB externally, the ratio A P / PE would be nega tive, and the signs of m and n would be opposite, but the formula would hold without change if this difference of sign in m and n be taken into account.

Example 2 : To find the point of intersection of the medians of a triangle.

Choose the origin 0 at random. Let A BC be the given triangle. Let 0~A = A, ()B = B, and "00 = C. Let Af,£',C' be respectively the middle points of the sides opposite the vertices A, B, (7. Let M be the point of intersection of the medians and M = 0 M the vector drawn to it. Then

and

~< = B

Assuming that 0 has been chosen outside of the plane of the triangle so that A, B, C are non-coplanar, corresponding coeffi cients may be equated.

Hence x = y — - •

9 3

Hence M =4 (A + B + C).

ADDITION AND SCALAR MULTIPLICATION 29

The vector drawn to the median point of a triangle is equal to one third of the sum of the vectors drawn to the vertices.

In the problems of which the solution has just been given the origin could be chosen arbitrarily and the result is in dependent of that choice. Hence it is even possible to disre gard the origin entirely and replace the vectors A, B, C, etc., by their termini A, B, C, etc. Thus the points themselves become the subjects of analysis and the formulae read

n A + m B m + n

and M=~(A + B + C).

This is typical of a whole class of problems soluble by vector methods. In fact any purely geometric relation between the different parts of a figure must necessarily be independent of the origin assumed for the analytic demonstration. In some cases, such as those in Arts. 18, 19, the position of the origin may be specialized with regard to some crucial point of the figure so as to facilitate the computation ; but in many other cases the generality obtained by leaving the origin un- specialized and undetermined leads to a symmetry which renders the results just as easy to compute and more easy to remember.

Theorem : The necessary and sufficient condition that a vector equation represent a relation independent of the origin is that the sum of the scalar coefficients of the vectors on one side of the sign of equality is equal to the sum of the coefficients of the vectors upon the other side. Or if all the terms of a vector equation be transposed to one side leaving zero on the other, the sum of the scalar coefficients must be zero.

Let the equation written in the latter form be

30 VECTOR ANALYSIS

Change the origin from 0 to 0' by adding a constant vector B = OO1 to each of the vectors A, B, C, D ---- The equation then becomes

a (A 4- B) + 6 (B + B) + c (C + B) + d (D + R) + • - • = 0

If this is to be independent of the origin the coefficient of B must vanish. Hence

That this condition is fulfilled in the two examples cited is obvious.

if

m + n

If M = \ (A f B + C),

m + n m + n

l

3

* 21.] The necessary and sufficient condition that two vectors satisfy an equation, in which the sum of the scalar coefficients is zero, is that the vectors be equal in magnitude and in direction.

First let a A + 6 B = 0

and a + 6 = 0.

It is of course assumed that not both the coefficients a and b vanish. If they did the equation would mean nothing. Sub stitute the value of a obtained from the second equation into the first.

-&A + 6B = 0.

Hence A = B.

ADDITION AND SCALAR MULTIPLICATION 31

Secondly if A and B are equal in magnitude and direction

the equation

A-B = 0

subsists between them. The sum of the coefficients is zero.

The necessaiy and sufficient condition that three vectors satisfy an equation, in which the sum of the scalar coefficients is zero, is that when drawn from a common origin they termi nate in the same straight line.1

First let aA + 6B + cC = 0

and a + b + c = 0.

Not all the coefficients a, J, c, vanish or the equations would be meaningless. Let c be a non-vanishing coefficient. Substitute the value of a obtained from the second equation into the first.

or

Hence the vector which joins the extremities of C and A is collinear with that which joins the extremities of A and B. Hence those three points -4, -B, C lie on a line. Secondly suppose three vectors A= OA, B = OB,G= 00 drawn from the same origin 0 terminate in a straight line. Then the vectors

AB = B - A and A~C = C - A

are collinear. Hence the equation

subsists. The sum of the coefficients on the two sides is the same.

The necessary and sufficient condition that an equation, in which the sum of the scalar coefficients is zero, subsist

1 Vectors which have a common origin and terminate in one line are called by Hamilton " termino-collinear:'

82 VECTOR ANALYSIS

between four vectors, is that if drawn from a common origin they terminate in one plane.1

First let a A + 6B + cC + dV = 0

and a + b + c + d = Q.

Let d be a non-vanishing coefficient. Substitute the value of a obtained from the last equation into the first.

or d (D - A) = 6 (A - B) + c (A - C).

The line A D is coplanar with A B and A C. Hence all four termini A, B, (7, D of A, B, C, D lie in one plane. Secondly suppose that the termini of A, B, C, D do lie in one plane. Then AZ) = D - A, ~AC = C - A, and ~AB = B - A are co planar vectors. One of them may be expressed in terms of the other two. This leads to the equation

/ (B - A) + m (C - A) + n (D - A) = 0,

where /, m, and n are certain scalars. The sum of the coeffi cients in this equation is zero.

Between any five vectors there exists one equation the sum of whose coefficients is zero.

Let A, B, C,D,E be the five given vectors. Form the differences

E-A, E--B, E-C, E-D.

One of these may be expressed in terms of the other three - or what amounts to the same thing there must exist an equation between them.

ft (E - A) + / (E - B) + m (E - C) + n (E - D) = 0. The sum of the coefficients of this equation is zero.

1 Vectors which have a common origin and terminate in one plane are called by Hamilton " termino-complanar."

ADDITION AND SCALAR MULTIPLICATION 33

*22.] The results of the foregoing section afford simple solutions of many problems connected solely with the geo metric properties of figures. Special theorems, the vector equations of lines and planes, and geometric nets in two and three dimensions are taken up in order.

Example 1: If a line be drawn parallel to the base of a triangle, the line which joins the opposite vertex to the inter section of the diagonals of the trapezoid thus formed bisects the base (Fig. 11).

Let ABC be the triangle, ED the line parallel to the base CB, G the point of intersection of the diagonals EB and DC of the tra pezoid CBDE, and Fthe intersec tion of A G with CB. To show FI(J n that F bisects CB. Choose the

origin at random. Let the vectors drawn from it to the various points of the figure be denoted by the corresponding Clarendons as usual. Then since ED is by hypothesis paral lel to CB, the equation

E - D = n (C - B)

holds true. The sum of the coefficients is evidently zero as it should be. Rearrange the terms so that the equation

takes on the form

E — nC = "D — 7i B.

The vector E — n C is coplanar with E and C. It must cut the line EC. The equal vector D — 7&B is coplanar with D and B. It must cut the line DB. Consequently the vector represented by either side of this equation must pass through the point A. Hence

E — 7iC = D — ?iB = #A.

34 VECTOR ANALYSIS

However the points E, 0, and A lie upon the same straight

line. Hence the equation which connects the vectors E,C,

and A must be such that the sum of its coefficients is zero.

This determines x as 1 — n.

Hence B - »C = D - »B = (1 - w) A.

By another rearrangement and similar reasoning

E + 7i B =D + 7iC= (1 + n)Qt. Subtract the first equation from the second :

n (B + C) = (1 + n) G - (1 - n) A.

This vector cuts EC and AQ. It must therefore be a multiple of F and such a multiple that the sum of the coeffi cients of the equations which connect B, C, and F or 0, A, and F shall be zero.

Hence n (B + C) = (1 + »)G - (1 - ») A = 2 nf.

„•

Hence F =

and the theorem has been proved. The proof has covered considerable space because each detail of the reasoning has been given. In reality, however, the actual analysis has con sisted of just four equations obtained simply from the first.

Example % : To determine the equations of the line and plane.

Let the line be fixed by two points A and B upon it. Let P be any point of the line. Choose an arbitrary origin. The vectors A, B, and P terminate in the same line. Hence

aA + 6B and a + I + p = 0.

™ ,

Therefore P =

a + b

ADDITION AND SCALAR MULTIPLICATION 35

For different points P the scalars a and b have different values. They may be replaced by x and y, which are used more generally to represent variables. Then

x + y

Let a plane be determined by three points -4, B, and C. Let P be any point of the plane. Choose an arbitrary origin. The vectors A, B, C, and P terminate in one plane. Hence

6B + cC

and a + b + c+p = Q.

aA + 6B + cC

Therefore P =

-f c

As a, 6, c, vary for different points of the plane, it is more customary to write in their stead x, yt z.

+ y + z

Example 3 ; The line which joins one vertex of a com plete quadrilateral to the intersection of two diagonals divides the opposite sides har monically (Fig. 12).

Let A, B, C, D be four vertices of a quadrilateral. Let A B meet CD in a fifth vertex E, and AD meet BC in the sixth vertex F. Let the two diagonals AC and p 12

BD intersect in G. To show

that FG intersects A B in a point i" and CD in a point E1' such that the lines AB and (7I> are divided internally at E1 and 2?" in the same ratio as they are divided externally by E. That is to show that the cross ratios

86 VECTOR ANALYSIS

Choose the origin at random. The four vectors A, B, C, D drawn from it to the points A, B, C, D terminate in one plane. Hence

and a + b + e + d = 0.

Separate the equations by transposing two terms :

Divide : 0 =

a + c = — (b + d). • cC 6B +

a + c b + d

aA + d'D __ 6B + cC

a + d b + c

(a + C)G — (a + d)F cC — di

In like manner F = Form:

(a + c) - (a + d) " (a + c) — (a + d) (a + c)Q — (a + rf)F cC —

or

c — a c — a

Separate the equations again and divide :

aA + EB _ cC + a -f b c + d

(6)

Hence 2? divides A B in the ratio a : b and CD in the ratio c / d. But equation (a) shows that JEff divides C D in the ratio — c:d. Hence E and E" divide CD internally and externally in the same ratio. Which of the two divisions is internal and which external depends upon the relative signs of c and d. If they have the same sign the internal point of division is E; if opposite signs, it is E1'. In a similar way E1 and E may be shown to divide A B harmonically.

Example 4 - To discuss geometric nets.

By a geometric net in a plane is meant a figure composed of points and straight lines obtained in the following manner. Start with a certain number of points all of which lie in one

ADDITION AND SCALAR MULTIPLICATION 37

plane. Draw all the lines joining these points in pairs. These lines will intersect each other in a number of points. Next draw all the lines which connect these points in pairs. This second set of lines will determine a still greater number of points which may in turn be joined in pairs and so on. The construction may be kept up indefinitely. At each step the number of points and lines in the figure increases. Probably the most interesting case of a plane geometric net is that in which four points are given to commence with. Joining these there are six lines which intersect in three points different from the given four. Three new lines may now be drawn in the figure. These cut out six new points. From these more lines may be obtained and so on.

To treat this net analytically write down the equations

= 0 (c)

and a + b + c + d = Q

which subsist between the four vectors drawn from an unde termined origin to the four given points. From these it is possible to obtain

a A + 6B cC + dD

Tjl

a + b c + d

A + cC Z>B + dD

a + c b + d

A + dJ) &B + cC

a + d b + c

by splitting the equations into two parts and dividing. Next four vectors such as A, D, E, F may be chosen and the equa tion the sum of whose coefficients is zero may be determined. This would be

— aA + dV + (a + b) E-f (a + c) P = 0.

By treating this equation as (c) was treated new points may be obtained*

38 VECTOR ANALYSIS

— a A + dD (a + 6)E + (a + c)F

H = 1 =

— a + d 2a + b + c

aA + (a + ft)E __ <?D+ (a + c)F

a 4- c + d (a + 6) E

c a + b + d

Equations between other sets of four vectors selected from A, B, C, D, E, F, 0 may be found ; and from these more points obtained. The process of finding more points goes forward indefinitely. A fuller account of geometric nets may be found in Hamilton's " Elements of Quaternions," Book I.

As regards geometric nets in space just a word may be said. Five points are given. From these new points may be obtained by finding the intersections of planes passed through sets of three of the given points with lines connecting the remaining pairs. The construction may then be carried for ward with the points thus obtained. The analytic treatment is similar to that in the case of plane nets. There are five vectors drawn from an undetermined origin to the given five points. Between these vectors there exists an equation the sum of whose coefficients is zero. This equation may be separated into parts as before and the new points may thus be obtained.

+ 6B cC + dD +

then F =

a + b c + d + e

A + cC 6B + dV + e

a + b b + d + c

are two of the points and others may be found in the same way. Nets in space are also discussed by Hamilton, loc. cit.

ADDITION AND SCALAR MULTIPLICATION 39

Centers of Gravity

* 23.] The center of gravity of a system of particles may be found very easily by vector methods. The two laws of physics which will be assumed are the following:

1°. The center of gravity of two masses (considered as situated at points) lies on the line connecting the two masses and divides it into two segments which are inversely pro portional to the masses at the extremities.

2°. In finding the center of gravity of two systems of masses each system may be replaced by a single mass equal in magnitude to the sum of the masses in the system and situated at the center of gravity of the system.

Given two masses a and b situated at two points A and B. Their center of gravity G is given by

where the vectors are referred to any origin whatsoever. This follows immediately from law 1 and the formula (7) for division of a line in a given ratio.

The center of gravity of three masses a, J, c situated at the three points -4, B, C may be found by means of law 2. The masses a and b may be considered as equivalent to a single mass a + b situated at the point

a A + &B

a + b

Then G = (a + 6) "A + 6B + c C

a -f- b

TT aA-h&B-f-cC

Hence G = •

a -f b + c

40 VECTOR ANALYSIS

Evidently the center of gravity of any number of masses a, &, c, d, ... situated at the points A, B, C, D, ... may be found in a similar manner. The result is

aA + ftB + cO + rfD + ... ^ a + b + c + d + ...

Theorem 1 : The lines which join the center of gravity of a triangle to the vertices divide it into three triangles which are proportional to the masses at the op posite vertices (Fig. 13). Let A, B, C be the vertices of a triangle weighted with masses a, &, c. Let G be the cen ter of gravity. Join A, B, C to G and produce the lines until they intersect the opposite sides in Af, B\ C1 respectively. To show that the areas

G B C : G C A : G A B : A B C = a : b : c : a + b + c .

The last proportion between ABC and a + b + c comes from compounding the first three. It is, however, useful in the demonstration.

ABC AA' A G . GA' b + c

+ 1.

Hence

GBC~ GA! ' CTA' G~Af

ABC a + b + c

In a similar manner and

GBC a

BCA a + I + c

GCA~ b

CAB _ a + b + c GAB ~ ~~c '

Hence the proportion is proved.

Theorem 2 : The lines which join the center of gravity of a tetrahedron to the vertices divide the tetrahedron into four

ADDITION AND SCALAR MULTIPLICATION 41

tetrahedra which are proportional to the masses at the oppo site vertices.

Let -4, B, C, D be the vertices of the tetrahedron weighted respectively with weights a, &, c, d. Let 0 be the center of gravity. Join A, B, C, D to G and produce the lines until they meet the opposite faces in A', B\ G\ D'. To show that the volumes

BCDG:CDAG:DABG:ABCG:ABCD

BCDA

BCDG

In like manner

and

and

A A'		A	G . j_	G	A' b + c + d ,	1
GA'		G a	A> + b +	GA' a c + d
CDA	G	a a + 6 + c + d
CD DA	AB BG	a -\	-b	b + c + d
DA AB	B C	C G	«4	-b	c + c + d

ABCD d

which proves the proportion.

* 24.] By a suitable choice of the three masses, a, J, c lo cated at the vertices A, B, (7, the center of gravity G may be made to coincide with any given point P of the triangle. If this be not obvious from physical considerations it cer tainly becomes so in the light of the foregoing theorems. For in order that the center of gravity fall at P, it is only necessary to choose the masses a, 6, c proportional to the areas of the triangles PEG, PCA^ and PAB respectively. Thus not merely one set of masses a, &, c may be found, but an infinite number of sets which differ from each other only by a common factor of proportionality. These quantities

42 VECTOR ANALYSIS

a, 6, c may therefore be looked upon as coordinates of the points P inside of the triangle ABC. To each set there corresponds a definite point P, and to each point P there corresponds an infinite number of sets of quantities, which however do not differ from one another except for a factor of proportionality.

To obtain the points P of the plane ABC which lie outside of the triangle ABC one may resort to the conception of negative weights or masses. The center of gravity of the masses 2 and — 1 situated at the points A and B respectively would be a point G dividing the line A B externally in the ratio 1 : 2. That is

Any point of the line A B produced may be represented by a suitable set of masses a, b which differ in sign. Similarly any point P of the plane ABC may be represented by a suitable set of masses a, 6, c of which one will differ in sign from the other two if the point P lies outside of the triangle ABC. Inasmuch as only the ratios of a, 6, and c are im portant two of the quantities may always be taken positive.

The idea of employing the masses situated at the vertices as coordinates of the center of gravity is due to Mobius and was published by him in his book entitled " Der barycentrische Calcul" in 1827. This may be fairly regarded as the starting point of modern analytic geometry.

The conception of negative masses which have no existence in nature may be avoided by replacing the masses at the vertices by the areas of the triangles GBC, GO A, and GAB to which they are proportional. The coordinates of a point P would then be three numbers proportional to the areas of the three triangles of which P is the common vertex ; and the sides of a given triangle ABC, the bases. The sign of these areas is determined by the following definition.

ADDITION AND SCALAR MULTIPLICATION 43

Definition: The area ABC of a triangle is said to be positive when the vertices A, B, C follow each other in the positive or counterclockwise direction upon the circle de scribed through them. The area is said to be negative when the points follow in the negative or clockwise direction.

Cyclic permutation of the letters therefore does not alter the sign of the area.

Interchange of two letters which amounts to a reversal of the cyclic order changes the sign.

A CB = BA 0 = CBA = -A B C. If P be any point within the triangle the equation PAB+PBC+PCA=ABC

must hold. The same will also hold if P be outside of the triangle provided the signs of the areas be taken into con sideration. The areas or three quantities proportional to them may be regarded as coordinates of the point P.

The extension of the idea of " barycentric " coordinates to space is immediate. The four points A, B, C, D situated at the vertices of a tetrahedron are weighted with mass a, J, c, d respectively. The center of gravity G is represented by these quantities or four others proportional to them. To obtain points outside of the tetrahedron negative masses may be employed. Or in the light of theorem 2, page 40, the masses may be replaced by the four tetrahedra which are proportional to them. Then the idea of negative vol umes takes the place of that of negative weights. As this idea is of considerable importance later, a brief treatment of it here may not be out of place.

Definition : The volume A B CD of a tetrahedron is said to be positive when the triangle ABC appears positive to

44 VECTOR ANALYSIS

the eye situated at the point D. The volume is negative if the area of the triangle appear negative.

To make the discussion of the signs of the various tetrahedra perfectly clear it is almost necessary to have a solid modeL A plane drawing is scarcely sufficient. It is difficult to see from it which triangles appear positive and which negative. The following relations will be seen to hold if a model be examined.

The interchange of two letters in the tetrahedron A BCD changes the sign.

ACBD = CBAD=BACD=DBCA

The sign of the tetrahedron for any given one of the pos sible twenty-four arrangements of the letters may be obtained by reducing that arrangement to the order A B C D by means of a number of successive interchanges of two letters. If the number of interchanges is even the sign is the same as that of A B CD ; if odd, opposite. Thus

If P is any point inside of the tetrahedron A B CD the equation

ABCP-BCDP+ CDAP-DABP=ABCD

holds good. It still is true if P be without the tetrahedron provided the signs of the volumes be taken into considera tion. The equation may be put into a form more symmetri cal and more easily remembered by transposing all the terms to one number. Then

The proportion in theorem 2, page 40, does not hold true if the signs of the tetrahedra be regarded. It should read

BCDG:CDGA:DGAB:GABC:ABCD

ADDITION AND SCALAR MULTIPLICATION 45

If the point G- lies inside the tetrahedron a, J, c, d repre sent quantities proportional to the masses which must be located at the vertices A,B,C,D respectively if G is to be the center of gravity. If G lies outside of the tetrahedron they may still be regarded as masses some of which are negative — or perhaps better merely as four numbers whose ratios determine the position of the point Gr. In this manner a set of "bary- centric " coordinates is established for space.

The vector P drawn from an indeterminate origin to any point of the plane A B C is (page 35)

aA + yB + zC

x + y + z

Comparing this with the expression

aA + &B + cC a + b + c

it will be seen that the quantities x, y, z are in reality nothing more nor less than the barycentric coordinates of the point P with respect to the triangle ABO. In like manner from

equation

__#A + yB + 2C + wD

x + y + z + w

which expresses any vector P drawn from an indeterminate origin in terms of four given vectors A, B, C, D drawn from the same origin, it may be seen by comparison with

+ &B + c C + rfD

0 =

a + b + c + d

that the four quantities x, y, 2, w are precisely the bary centric coordinates of P, the terminus of P, with respect to the tetrahedron A B CD. Thus the vector methods in which the origin is undetermined and the methods of the " Bary centric Calculus " are practically co-extensive.

It was mentioned before and it may be well to repeat here

46

VECTOR ANALYSIS

that the origin may be left wholly out of consideration and the vectors replaced by their termini. The vector equations then become point equations

x A + y B 4- z 0

and

x + y + z

xA + yB + zC + wD

• w.

At 0

x + y + z

This step brings in the points themselves as the objects of analysis and leads still nearer to the " Barycentrische Calcul " of Mobius and the "Ausdehnungslehre " of Grassmann.

The Use of Vectors to denote Areas

25.] Definition: An area lying in one plane MN and bounded by a continuous curve PQR which nowhere cuts itself is said to appear positive from the point 0 when the

letters PQR follow each other in the counterclockwise or positive order; negative, when they follow in the negative or clockwise order (Fig. 14).

It is evident that an area can have no determined sign per se, but only in reference to that direction in which its

boundary is supposed to be traced and to some point 0 out side of its plane. For the area P R Q is negative relative to PQR; and an area viewed from 0 is negative relative to the same area viewed from a point Of upon the side of the plane opposite to 0. A circle lying in the X F-plane and described in the positive trigonometric order appears positive from every point on that side of the plane on which the positive £axis lies, but negative from all points on the side upon which

ADDITION AND SCALAR MULTIPLICATION 47

the negative ^-axis lies. For this reason the point of view and the direction of description of the boundary must be kept clearly in mind.

Another method of stating the definition is as follows : If a person walking upon a plane traces out a closed curve, the area enclosed is said to be positive if it lies upon his left- hand side, negative if upon his right. It is clear that if two persons be considered to trace out together the same curve by walking upon opposite sides of the plane the area enclosed will lie upon the right hand of one and the left hand of the other. To one it will consequently appear positive ; to the other, negative. That side of the plane upon which the area seems positive is called the positive side ; the side upon which it appears negative, the negative side. This idea is familiar to students of electricity and magnetism. If an electric current flow around a closed plane curve the lines of magnetic force through the circuit pass from the negative to the positive side of the plane. A positive magnetic pole placed upon the positive side of the plane will be repelled by the circuit.

A plane area may be looked upon as possessing more than positive or negative magnitude. It may be considered to possess direction, namely, the direction of the normal to the positive side of the plane in which it lies. Hence a plane area is a vector quantity. The following theorems concerning areas when looked upon as vectors are important.

Theorem 1 : If a plane area be denoted by a vector whose magnitude is the numerical value of that area and whose direction is the normal upon the positive side of the plane, then the orthogonal projection of that area upon a plane will be represented by the component of that vector in the direction normal to the plane of projection (Fig. 15).

Let the area A lie in the plane MN. Let it be projected orthogonally upon the plane M ' N'. Let M N&nd M* Nr inter-

48

VECTOR ANALYSIS

sect in the line I and let the diedral angle between these two planes be x. Consider first a rectangle PQJRS in MN whose sides, PQ, RS and QR, SP are respectively parallel and perpendicular to the line /. This will project into a rectangle P'Q'R'S1 in M'N'. The sides P'Qf and JR'S' will be equal to PQ and US; but the sides Q1 R' and S'P' will be equal to QR and SP multiplied by the cosine of #, the angle between the planes. Consequently the rectangle

At'

FIG. 15.

Hence rectangles, of which the sides are respectively parallel and perpendicular to I, the line of intersection of the two planes, project into rectangles whose sides are likewise respectively parallel and perpendicular to I and whose area is equal to the area of the original rectangles multiplied by the cosine of the angle between the planes.

From this it follows that any area A is projected into an area which is equal to the given area multiplied by the cosine of the angle between the planes. For any area A may be di vided up into a large number of small rectangles by drawing a series of lines in MN parallel and perpendicular to the line I.

ADDITION AND SCALAR MULTIPLICATION 49

Each of these rectangles when projected is multiplied by the cosine of the angle between the planes and hence the total area is also multiplied by the cosine of that angle. On the other hand the component A' of the vector A, which repre sents the given area, in the direction normal to the plane MfNf of projection is equal to the total vector A multiplied by the cosine of the angle between its direction which is the normal to the plane M ^and the normal to M'Nr. This angle is x ; for the angle between the normals to two planes is the same as the angle between the planes. The relation between the magnitudes of A and A' is therefore

A1 = A cos x,

which proves the theorem.

26.] Definition : Two plane areas regarded as vectors are said to be added when the vectors which represent them are added.

A vector area is consequently the sum of its three com ponents obtainable by orthogonal projection upon three mutually perpendicular planes. Moreover in adding two areas each may be resolved into its three components, the corresponding components added as scalar quantities, and these sums compounded as vectors into the resultant area. A generalization of this statement to the case where the three planes are not mutually orthogonal and where the projection is oblique exists.

A surface made up of several plane areas may be repre sented by the vector which is the sum of all the vectors representing those areas. In case the surface be looked upon as forming the boundary or a portion of the boundary of a solid, those sides of the bounding planes which lie outside of the body are conventionally taken to be positive. The vec tors which represent the faces of solids are always directed out from the solid, not into it

4

50 VECTOR ANALYSIS

Theorem 2 : The vector which represents a closed polyhedral surface is zero.

This may be proved by means of certain considerations of hydrostatics. Suppose the polyhedron drawn in a body of fluid assumed to be free from all external forces, gravity in cluded.1 The fluid is in equilibrium under its own internal pressures. The portion of the fluid bounded by the closed surface moves neither one way nor the other. Upon each face of the surface the fluid exerts a definite force proportional to the area of the face and normal to it. The resultant of all these forces must be zero, as the fluid is in equilibrium. Hence the sum of all the vector areas in the closed surface is zero.

The proof may be given in a purely geometric manner. Consider the orthogonal projection of the closed surface upon any plane. This consists of a double area. The part of the surface farthest from the plane projects into positive area ; the part nearest the plane, into negative area. Thus the surface projects into a certain portion of the plane which is covered twice, once with positive area and once with negative. These cancel each other. Hence the total projection of a closed surface upon a plane (if taken with regard to sign) is zero. But by theorem 1 the projection of an area upon a plane is equal to the component of the vector representing that area in the direction perpendicular to that plane. Hence the vector which represents a closed surf ace has no component along the line perpendicular to the plane of projection. This, however, was any plane whatsoever. Hence the vector is zero.

The theorem has been proved for the case in which the closed surface consists of planes. In case that surface be

1 Such a state of affairs is realized to all practical purposes in the case of a polyhedron suspended in the atmosphere and consequently subjected to atmos pheric pressure. The force of gravity acts but is counterbalanced by the tension in the suspending string.

ADDITION AND SCALAR MULTIPLICATION 51

curved it may be regarded as the limit of a polyhedral surface whose number of faces increases without limit. Hence the vector which represents any closed surface polyhedral or curved is zero. If the surface be not closed but be curved it may be represented by a vector just as if it were polyhedral. That vector is the limit l approached by the vector which represents that polyhedral surface of which the curved surface is the limit when the number of faces becomes indefinitely great.

SUMMARY OF CHAPTER I

A vector is a quantity considered as possessing magnitude and direction. Equal vectors possess the same magnitude and the same direction. A vector is not altered by shifting it parallel to itself. A null or zero vector is one whose mag nitude is zero. To multiply a vector by a positive scalar multiply its length by that scalar and leave its direction unchanged. To multiply a vector by a negative scalar mul tiply its length by that scalar and reverse its direction.

Vectors add according to the parallelogram law. To subtract a vector reverse its direction and add. Addition, subtrac tion, and multiplication of vectors by a scalar follow the same laws as addition, subtraction, and multiplication in ordinary algebra. A vector may be resolved into three components parallel to any three non-coplanar vectors. This resolution can be accomplished in only one way.

r = x* + yb + zc. (4)

The components of equal vectors, parallel to three given non-coplanar vectors, are equal, and conversely if the com ponents are equal the vectors are equal. The three unit vectors i, j, k form a right-handed rectangular system. In

1 This limit exists and is unique. It is independent of the method in which the polyhedral surface approaches the curved surface.

52 VECTOR ANALYSIS

terms of them any vector may be expressed by means of the Cartesian coordinates #, y, z.

r = xi + yj+zk. (6)

Applications. The point which divides a line in a given ratio m : n is given by the formula

(7)

m + n

The necessary and sufficient condition that a vector equation represent a relation independent of the origin is that the sum of the scalar coefficients in the equation be zero. Between any four vectors there exists an equation with scalar coeffi cients. If the sum of the coefficients is zero the vectors are termino-coplanar. If an equation the sum of whose scalar coefficients is zero exists between three vectors they are termino-collinear. The center of gravity of a number of masses a, &, c • • • situated at the termini of the vectors A, B, C • • • supposed to be drawn from a common origin is given by the formula

A vector may be used to denote an area. If the area is plane the magnitude of the vector is equal to the magnitude of the area, and the direction of the vector is the direction of the normal upon the positive side of the plane. The vector representing a closed surface is zero.

EXERCISES ON CHAPTER I

1. Demonstrate the laws stated in Art. 12.

2. A triangle may be constructed whose sides are parallel and equal to the medians of any given triangle.

ADDITION AND SCALAR MULTIPLICATION 53

3. The six points in which the three diagonals of a com* plete quadrangle l meet the pairs of opposite sides lie three by three upon four straight lines.

4. If two triangles are so situated in space that the three points of intersection of corresponding sides lie on a line, then the lines joining the corresponding vertices pass through a common point and conversely.

5. Given a quadrilateral in space. Find the middle point of the line which joins the middle points of the diagonals. Find the middle point of the line which joins the middle points of two opposite sides. Show that these two points are the same and coincide with the center of gravity of a system of equal masses placed at the vertices of the quadrilateral.

6. If two opposite sides of a quadrilateral in space be divided proportionally and if two quadrilaterals be formed by joining the two points of division, then the centers of gravity of these two quadrilaterals lie on a line with the center of gravity of the original quadrilateral. By the center of gravity is meant the center of gravity of four equal masses placed at the vertices. Can this theorem be generalized to the case where the masses are not equal ?

7. The bisectors of the angles of a triangle meet in a point.

8. If the edges of a hexahedron meet four by four in three points, the four diagonals of the hexahedron meet in a point. In the special case in which the hexahedron is a parallelepiped the three points are at an infinite distance

9. Prove that the three straight lines through the middle points of the sides of any face of a tetrahedron, each parallel to the straight line connecting a fixed point P with the mid dle point of the opposite edge of the tetrahedron, meet in a

1 A complete quadrangle consists of the six straight lines which may he passed through four points no three of which are collinear. The diagonals are the lines which join the points of intersection of pairs of sides

54 VECTOR ANALYSIS

point E and that this point is such that PE passes through and is bisected by the center of gravity of the tetrahedron.

10. Show that without exception there exists one vector equation with scalar coefficients between any four given vectors A, B, C, D.

11. Discuss the conditions imposed upon three, four, or five vectors if they satisfy two equations the sum of the co efficients in each of which is zero.

CHAPTER II

DIRECT AND SKEW PRODUCTS OF VECTORS

Products of Two Vectors

27.] THE operations of addition, subtraction, and scalar multiplication have been defined for vectors in the way suggested by physics and have been employed in a few applications. It now becomes necessary to introduce two new combinations of vectors. These will be called products because they obey the fundamental law of products ; i. e., the distributive law which states that the product of A into the sum of B and C is equal to the sum of the products of A into B and A into C.

Definition : The direct product of two vectors A and B is the scalar quantity obtained by multiplying the product of the magnitudes of the vectors by the cosine of the angle be tween them.

The direct product is denoted by writing the two vectors with a dot between them as

A-B.

This is read A dot B and therefore may often be called the dot product instead of the direct product. It is also called the scalar product owing to the fact that its value is sca lar. If A be the magnitude of A and B that of B, then by definition

A-B = ^£cos (A,B). (1)

Obviously the direct product follows the commutative law

A-B = B'A. (2)

56 VECTOR ANALYSIS

If either vector be multiplied by a scalar the product is multiplied by that scalar. That is

(x A) • B = A • (x B) = x (A • B).

In case the two vectors A and B are collinear the angle be tween them becomes zero or one hundred and eighty degrees and its cosine is therefore equal to unity with the positive or negative sign. Hence the scalar product of two parallel vectors is numerically equal to the product of their lengths. The sign of the product is positive when the directions of the vectors are the same, negative when they are opposite. The product of a vector by itself is therefore equal to the square

of its length

A.A=^42. (3)

Consequently if the product of a vector by itself vanish the vector is a null vector.

In case the two vectors A and B are perpendicular the angle between them becomes plus or minus ninety degrees and the cosine vanishes. Hence the product A • B vanishes. Conversely if the scalar product A • B vanishes, then

A B cos (A, B) = 0.

Hence either A or B or cos (A, B) is zero, and either the vectors are perpendicular or one of them is null. Thus the condition for the perpendicularity of two vectors, neither of which vanishes, is A • B = 0.

28.] The scalar products of the three fundamental unit vectors i, j, k are evidently

i»i = j»j = k«k = l, (4)

i . j = j . k = k . i = 0.

If more generally a and b are any two unit vectors the product

a • b = cos (a, b).

DIRECT AND SKEW PRODUCTS OF VECTORS 57

Thus the scalar product determines the cosine of the angle between two vectors and is in a certain sense equivalent to it. For this reason it might be better to give a purely geometric definition of the product rather than one which depends upon trigonometry. This is easily accomplished as follows : If a and b are two unit vectors, a • b is the length of the projection of either upon the other. If more generally A and B are any two vectors A • B is the product of the length of either by the length of projection of the other upon it. From these definitions the facts that the product of a vector by itself is the square of its length and the product of two perpendicular vectors is zero follow immediately. The trigo nometric definition can also readily be deduced.

The scalar product of two vectors will appear whenever the cosine of the included angle is of importance. The following examples may be cited. The projection of a vector B upon a vector A is

A«B AB

A = — — A a cos (A, B) = B cos (A, B) a, (5)

A • A

where a is a unit vector in the direction of A. If A is itself a unit vector the formula reduces to

(A-B) A = £cos (A,B) A.

If A be a constant force and B a displacement the work done

by the force A during the displacement is A • B. If A repre

sent a plane area (Art. 25), and if B be a

vector inclined to that plane, the scalar prod

uct A • B will be the volume of the cylinder

of which the area A is the base and of

which B is the directed slant height. For

the volume (Fig. 16) is equal to the base FlG

A multiplied by the altitude h. This is

the projection of B upon A or B cos (A, B). Hence

v = A h = A B cos (A, B) = A • B.

58 VECTOR ANALYSIS

29.] The scalar or direct product follows the distributive law of multiplication. That is

(A + B) .C = A-C + B.C. (6)

This may be proved by means of projections. Let C be equal to its magnitude C multiplied by a unit vector c in its direc tion. To show

(A + B) • (<7c) = A • (0o) + B. (0o) or (A + B) • c = A • c + B • c.

A • c is the projection of A upon c ; B • c, that of B upon c ; (A + B) • c, that of A + B upon c. But the projection of the sum A + B is equal to the sum of the projections. Hence the relation (6) is proved. By an immediate generalization

(A + B + ...)'(P + a+-") = A-P + A.Q+...

+ B.P + B.Q + ... <ey

The scalar product may be used just as the product in ordi nary algebra. It has no peculiar difficulties.

If two vectors A and B are expressed in terms of the three unit vectors i, j, k as

A = ^[1i + ^2j + ^8k, and B = ^ i + JB2 j + £B k,

then A- B = (Al i + Az j + AB k) . (^i + £2 j + ^k) = Al Bl i . i + Al £2 i • j + Al B% i • k

By means of (4) this reduces to

A-* = A1£l + A2E, + ABJB,. (7)

If in particular A and B are unit vectors, their components Al,A^,A3 and B19S29SB are the direction cosines of the lines A and B referred to X, Y, Z.

DIRECT AND SKEW PRODUCTS OF VECTORS 59

Al = cos (A, JT), A<i = cos (A, F), Az = cos (A, ^f), ^! = cos (B, JT), .#2 = cos (B, T), £3 = cos (B, £).

Moreover A • B is the cosine of the included angle. Hence the equation becomes

cos (A, B) = cos (A, X) cos (B, X) + cos (A, T) cos (B, T)

+ cos (A,£) cos (B,Z).

In case A and B are perpendicular this reduces to the well- known relation

0 = cos (A, JT) cos (B, X) + cos (A, Y) cos (B, F)

+ cos (A,^) cos (B,£)

between the direction cosines of the line A and the line B.

30.] If A and B are two sides 0 A

and OB of a triangle OAB, the third o

side A£isG = -B-JL (Fig. 17). PlG' 17*

C*C = (B-A). (B-A)=B-B

or (7 2 = A2 + J52-2 A^cos(AB).

That is, the square of one side of a triangle is equal to the sum of the squares of the other two sides diminished by twice their product times the cosine of the angle between them. Or, the square of one side of a triangle is equal to the sum of the squares of the other two sides diminished by twice the product of either of those sides by the projection of the other upon it — the generalized Pythagorean theorem.

If A and B are two sides of a parallelogram, C = A + B and D = A — B are the diagonals. Then

C.C = (A + B).(A + B)=A.A + 2A.B + B.B, D.D=(A-B).(A-B)=A-A-2A.B + B.B,

C-C + D.D = 2(A-A + B«B), or a2 + 7)2 = 2 (A*

60 VECTOR ANALYSIS

That is, the sum of the squares of the diagonals of a parallelo gram is equal to twice the sum of the squares of two sides. In like manner also

or C*-D2 = 4A£ cos (A, B).

That is, the difference of the squares of the diagonals of a parallelogram is equal to four times the product of one of the sides by the projection of the other upon it.

If A is any vector expressed in terms of i, j, k as

A = Al i + A2 j + AB k, then A • A = A* = A* + A* + A*. (8)

But if A be expressed in terms of any three non-coplanar unit vectors a, b, c as

+ 2 J c b«c + 2 c a e «a

A2 = a? + 62 + c2 + 2 a b cos (a, b) + 2 b c cos (b, c)

+ 2 ca cos (c, a).

This formula is analogous to the one in Cartesian geometry which gives the distance between two points referred to oblique axes. If the points be xv yv zv and #2, yv z% the distance squared is

D2 = (*2 - x^ + (ya - yi)2 + (z2 - zj* + 2 (a, - xj (ya - 2/0 cos (X, Y) + 2 (yt- ft) («,-*!> cos (F.S) + 2 (z2 -24) (x2 - xj cos (^,-T).

31.] Definition: The skew product of the vector A into the vector B is the vector quantity C whose direction is the normal upon that side of the plane of A and B on which

DIRECT AND SKEW PRODUCTS OF VECTORS 61

rotation from A to B through an angle of less than one hundred and eighty degrees appears positive or counter clockwise ; and whose magnitude is obtained by multiplying the product of the magnitudes of A and B by the sine of the angle from A to B.

The direction of A x B may also be defined as that in which an ordinary right-handed screw advances as it turns so as c= AXB to carry A toward B (Fig. 18).

The skew product is denoted by a cross as the direct product was ^

by a dot. It is written FIG. 18.

C = A x B

and read A cross B. For this reason it is often called the cross product. More frequently, however, it is called the vector prod uct, owing to the fact that it is a vector quantity and in con trast with the direct or scalar product whose value is scalar. The vector product is by definition

C = A x B = ^J5sin (A,B)c, (9)

when A and B are the magnitudes of A and B respectively and where c is a unit vector in the direction of C. In case A and B are unit vectors the skew product A X B reduces to the unit vector c multiplied by the sine of the angle from A to B. Obviously also if either vector A or B is multiplied by a scalar x their product is multiplied by that scalar.

0 A) X B = A X (zB) = xC.

If A and B are parallel the angle between them is either zero or one hundred and eighty degrees. In either case the sine vanishes and consequently the vector product A X B is a null vector. And conyersely if A X B is zero

A B sin (A, B) = 0.

62 VECTOR ANALYSIS

Hence A or B or sin (A, B) is zero. Thus the condition for parallelism of two vectors neither of which vanishes is A X B = 0. As a corollary the vector product of any vector into itself vanishes.

32.] The vector product of two vectors will appear wher ever the sine of the included angle is of importance, just as the scalar product did in the case of the cosine. The two prod ucts are in a certain sense complementary. They have been denoted by the two common signs of multiplication, the dot and the cross. In vector analysis they occupy the place held by the trigonometric functions of scalar analysis. They are at the same time amenable to algebraic treatment, as will be seen later. At present a few uses of the vector product may be cited.

If A and B (Fig. 18) are the two adjacent sides of a parallel ogram the vector product

C = A x B = A B sin (A, B) c

represents the area of that parallelogram in magnitude and direction (Art. 25). This geometric representation of A X B is of such common occurrence and importance that it might well be taken as the definition of the product. From it the trigonometric definition follows at once. The vector product appears in mechanics in connection with couples. If A and — A are two forces forming a couple, the moment of the couple is A X B provided only that B is a vector drawn from any point of A to any point of — A. The product makes its appearance again in considering the velocities of the individ ual particles of a body which is rotating with an angular ve locity given in magnitude and direction by A. If R be the radius vector drawn from any point of the axis of rotation A the product A X & will give the velocity of the extremity of B (Art. 51). This velocity is perpendicular alike to the axis of rotation and to the radius vector B.

DIRECT AND SKEW PRODUCTS OF VECTORS 63

33.] The vector products A X B and B x A are not the

same. They are in fact the negatives of each other. For if rotation from A to B appear positive on one side of the plane of A and B, rotation from B to A will appear positive on the other. Hence A X B is the normal to the plane of A and B upon that side opposite to the one upon which B x A is the normal. The magnitudes of A X B and B X A are the same.

Hence

AxB = -BxA. (10)

The factors in a vector product can be interchanged if and only if the sign of the product be reversed.

This is the first instance in which the laws of operation in vector analysis differ essentially from those of scalar analy sis. It may be that at first this change of sign which must accompany the interchange of factors in a vector product will give rise to some difficulty and confusion. Changes similar to this are, however, very familiar. No one would think of inter changing the order of x and y in the expression sin (x — y) without prefixing the negative sign to the result. Thus

sin (y — x) = — sin (x — y),

although the sign is not required for the case of the cosine, cos (y — x) = cos ( x — y).

Again if the cyclic order of the letters ABC in the area of a triangle be changed, the area will be changed in sign (Art.

25).

AB C = -ACB.

In the same manner this reversal of sign, which occurs when the order of the factors in a vector product is reversed, will appear after a little practice and acquaintance just as natural and convenient as it is necessary.

34.] The distributive law of multiplication holds in the case of vector products just as in ordinary algebra — except

64

VECTOR ANALYSTS

that the order of the factors must be carefully maintained when expanding.

A very simple proof may be given by making use of the ideas developed in Art. 26. Suppose that C is not coplanar with A and B. Let A and B be two sides of a triangle taken in order. Then — (A + B) will be the third side (Fig. 19). Form the prism of which this triangle is the base and of which C is the slant height or edge. The areas of the lateral faces of this prism are

A x C, B x C, — (A -f B) x C. The areas of the bases are

5 (A x B) and - - (A x B).

But the sum of all the faces of the prism is zero; for the prism is a closed surface. Hence

4

FIG. 19.

AxC + BxC-(A + B)xC = 0, or A X C + B X C = (A + B) X C. (11)

The relation is therefore proved in case C is non-coplanar with A and B. Should C be coplanar with A and B, choose D, any vector out of that plane. Then C + D also will lie out of that plane. Hence by (11)

A X (C + D) + B X (C + D) = (A + B) x (C + D).

Since the three vectors in each set A, C, D, and B, C, D, and A + B, C, D will be non-coplanar if D is properly chosen, the products may be expanded.

DIRECT AND SKEW PRODUCTS OF VECTORS 65

AxC + AxD-fBxC + BxD

= (A + B) x C + (A + B) x D. But by (11) AxD + BxD = (A + B)xD. Hence AxC + BxC = (A + B)xC.

This completes the demonstration. The distributive law holds for a vector product. The generalization is immediate.

(A + B+---)x(P + a + ---) = AxP + Axa + --- (11)'

+ B x P + B x <J + •••

35.] The vector products of the three unit vectors i, j, k are easily seen by means of Art. 17 to be

ixi = jxj = kxk = 0,

ixj=-j xi = k, (12)

jxk = — k x j = i, kxi = — ixk=j.

The skew product of two equal l vectors of the system i, j, k is zero. The product of two unequal vectors is the third taken with the positive sign if the vectors follow in the cyclic order i j k but with the negative sign if they do not.

If two vectors A and B are expressed in terms of i, j, k, their vector product may be found by expanding according to the distributive law and substituting.

A x B ' = (Al i + -42 j + ^3k) x (^i + £2j + £3*)

= Al£lixi + AlB2ixj + AlBzixTt

+ A2£l j x i + AI £2 j x j + AZ BB j x k,

+ Az S1 k x i + AB BZ k x j + Az Bz k x k.

Hence A x B = (A^B^ - AB£2) i + (AZB1 -A,BB)j

4- (A, £z - A2 BJ k.

1 This follows also from the fact that the sign is changed when the order of factors is reversed. Hence j X j = — j Xj=0.

5

66 VECTOR ANALYSIS

This may be written in the form of a determinant as

Ax B =

The formulae for the sine and cosine of the sum or dif ference of two angles follow immediately from the dot and cross products. Let a and b be two unit vectors lying in the i j-plane. If x be the angle that a makes with i, and y the angle b makes with i, then

a =

Hence If

Hence

Hence

Hence

a • b : a • b :

cos (y — x) : V

a.V:

cos (y + x) :

a x b :

a x b =

sin (y — x) -

axb' =

ax b' =

sin (y + x) -

cos x i + sin x j,

cos y i -f sin y j,

cos (a, b) = cos (y — x),

cos x cos y + sin x sin y.

cos y cos x + sin y sin x.

cos y i — sin y j,

cos (a, b') = cos (y + x).

cos y cos x — sin y sin x.

k sin (a, b) = k sin (y — x),

k (sin y cos x — sin x cos y).

sin y cos x — sin x cos y.

k sin (a, b') = k sin (y + x)9

k (sin y cos x + sin x cos y).

sin y cos x + sin # cos y.

If /, m, 7i and Z', w', TI' are the direction cosines of two unit vectors a and a' referred to JT, F, £, then

a = li + m j + 7i k,

m'j

a • a' = cos (a, a') = IV + m mr + n nf,

as has already been shown in Art. 29. The familiar formula for the square of the sine of the angle between a and a' may be found.

DIRECT AND SKEW PRODUCTS OF VECTORS 67

a x a' = sin (a, a') e = (mnf — m' ri) i + (n V — n? I) j + (Jm'-f'm) k,

where e is a unit vector perpendicular to a and a'.

(a x a') • (a x a') = sin2 (a, af ) e • e = sin2 (a, a'). sin2 (a, a') = (mn'—mrn)*+ (nl'—nf /)2 +(lm' — I'm)*.

This leads to an easy way of establishing the useful identity

= (72 + w2 + 7i2) (V* + m'2 + n'*) — (ll' + mm' + n n')2.

Products of More than Two Vectors

36.] Up to this point nothing has been said concerning products in which the number of vectors is greater than two. If three vectors are combined into a product the result is called a triple product. Next to the simple products A-B and AxB the triple products are the most important. All higher products may be reduced to them.

The simplest triple product is formed by multiplying the scalar product of two vectors A and B into a third C as

(A-B) C.

This in reality does not differ essentially from scalar multi plication (Art. 6). The scalar in this case merely happens to be the scalar product of the two vectors A and B. Moreover inasmuch as two vectors cannot stand side by side in the form of a product as BC without either a dot or a cross to unite them, the parenthesis in (A»B) C is superfluous. The expression « ^ n

cannot be interpreted in any other way * than as the product of the vector C by the scalar A«B.

i Later (Chap. V.) the product BC, where no sign either dot or cross occurs, will be defined. But it will be seen there that (A.B) C and A-(B C) are identical and consequently no ambiguity can arise from the omission of the parenthesis.

68 VECTOR ANALYSIS

37.] The second triple product is the scalar product of two vectors, of which one is itself a vector product, as A-(BxC) or (AxB>C.

This sort of product has a scalar value and consequently is

often called the scalar triple prod uct. Its properties are perhaps most easily deduced from its commonest geometrical interpretation. Let A, B, and C be any three vectors drawn from the same origin (Fig. 20). Then BxC is the area of the par

allelogram of which B and C are two adjacent sides. The

scalar . * (14)

will therefore be the volume of the parallelepiped of which BxC is the base and A the slant height or edge. See Art. 28. This volume v is positive if A and BxC lie upon the same side of the B C-plane ; but negative if they lie on opposite sides. In other words if A, B, C form a right-handed or positive system of three vectors the scalar A* (BxC) is posi tive; but if they form a left-handed or negative system, it is negative.

In case A, B, and C are coplanar this volume will be neither positive nor negative but zero. And conversely if the volume is zero^the three edges A, B, C of the parallelo- piped must lie in one plane. Hence the necessary and suffi cient condition for the coplanarity of three vectors A, B, C none of which vanishes is A-(BxC) = 0. As a corollary the scalar triple product of three vectors of which two are equal or collinear must vanish ; for any two vectors are coplanar.

The two products A«(BxC) and (AxB)-C are equal to the same volume v of the parallelepiped whose concurrent edges are A, B, C. The sign of the volume is the same in both cases. Hence (AxB).c = A.(BxC) = ,. (14)

DIRECT AND SKEW PRODUCTS OF VECTORS 69

This equality may be stated as a rule of operation. The dot and the cross in a scalar triple product may be interchanged without altering the value of the product.

It may also be seen that the vectors A, B, C may be per muted cyclicly without altering the product

A-(BxC) = B-(CxA) = C-(AxB). (15)

For each of the expressions gives the volume of the same parallelepiped and that volume will have in each case the same sign, because if A is upon the positive side of the B C- plane, B will be on the positive side of the C A-plane and C upon the positive side of the A B-plane. The triple product may therefore have any one of six equivalent forms

A<BxC) = B-(CxA) =: C.(AxB) (35)'

= (AxB)-C = (BxC)-A = (CxA)-B

If however the cyclic order of the letters is changed the product will change sign.

A-(BxC) = - A<CxBV (16)

This may be seen from the figure or from the fact that BxC = — CxB.

Hence : A scalar triple product is not altered by interchanging the dot or the cross or by permuting cyclicly the order of the vectors, but it is reversed in sign if the cyclic order be changed.

38.] A word is necessary upon the subject of parentheses in this triple product. Can they be omitted without am biguity ? They can. The expression

A-BxC

can have only the one interpretation

A<BxC).

For the expression (A-B)xC is meaningless. It is impos sible to form the skew product of a scalar A«B and a vector

70 VECTOR ANALYSIS

C. Hence as there is only one way in which A»BxC may be interpreted, no confusion can arise from omitting the parentheses. Furthermore owing to the fact that there are six scalar triple products of A, B, and C which have the same value and are consequently generally not worth distinguish ing the one from another, it is often convenient to use the symbol

[ABC]

to denote any one of the six equal products.

[A B C] = A.BxC = B*CxA = C'AxB = AxB.C = BxC-A = CxA-B then [A B C] = - [A 0 B]. (16)'

The scalar triple products of the three unit vectors i, j, k all vanish except the two which contain the three different vectors.

[ijk] = _[ikj] = l. (17)

Hence if three vectors A, B, C be expressed in terms of i, j, k as

B = ^ i + A, j + £8 k,

C = C'1i+C'2j + C'3k, then [ABC] =A1£ZC3 + £, C2 A.+

This may be obtained by actually performing the multiplica tions which are indicated in the triple product. The result may be written in the form of a determinant.1

[A B C] =

-4i

\

B

1 This is the formula given in solid analytic geometry for the volume of a tetrahedron one of whose vertices is at the origin. For a more general formula see exercises.

DIRECT AND SKEW PRODUCTS OF VECTORS 71

If more generally A, B, C are expressed in terms of any three non-coplanar vectors a, b, c which are not necessarily unit

vectors,

A = ax a + a2 b + «8 c

B = &! a + &2 b + J8 c C = cl a

c2 b

where a^ #2, #3," ftp &2, stants, then

[A B 0] = (al &2 CB +

are certain con

[a b c].

or

[A B C] =

[a be]

'

(19)'

39.] The third type of triple product is the vector product of two vectors of which one is itself a vector product. Such

are

Ax(BxC) and (AxB)xC.

The vector Ax(BxC) is perpendicular to A and to (BxC). But (BxC) is perpendicular to the plane of B and C. Hence Ax (BxC), being perpendicular to (BxC) must lie in the plane of B and C and thus take the form

Ax(BxC) = x B + y C,

where x and y are two scalars. In like manner also the vector (AxB)xC, being perpendicular to (AxB) must lie in the plane of A and B. Hence it will be of the form

(AxB)xC = ra.A + n B

where m and n are two scalars. From this it is evident that in general

(AxB)xC is not equal to Ax(BxC).

The parentheses therefore cannot be removed or inter changed. It is essential to know which cross product is

72 VECTOR ANALYSIS

formed first and which second. This product is termed the vector triple product in contrast to the scalar triple product.

The vector triple product may be used to express that com ponent of a vector B which is perpendicular to a given vector A. This geometric use of the product is valuable not only in

itself but for the light it sheds

AXB AXB*

B

upon the properties of the product. Let A (Fig. 21) be a given vector and B another vector whose com ponents parallel and perpendicular to A are to be found. Let the components of B parallel and per- A X (AXB) pendicular to A be B'and B" re-

2i spectively. Draw A and B from a

common origin. The product AxB

is perpendicular to the plane of A and B. The product Ax (AxB) lies in the plane of A and B. It is furthermore perpendicular to A. Hence it is collinear with B". An examination of the figure will show that the direction of Ax (AxB) is opposite to that of B". Hence

Ax(AxB) = — cB", where c is some scalar constant.

Now Ax (AxB) = - A* B sin (A, B) V

but - c B"-^= - c B sin (A, B) b",

if b" be a unit vector in the direction of B". Hence c — A2 — A* A.

Hence B" = - Ax(AxB) . (20)

The component of B perpendicular to A has been expressed in terms of the vector triple product of A, A, and B. The component B' parallel to A was found in Art 28 to be

DIRECT AND SKEW PRODUCTS OF VECTORS 73 B'=£?A (21)

B = B'+B» = £?A-±^>. (22)

A«A A«A

40.] The vector triple product Ax (BxC) may be expressed as the sum of two terms as

Ax(BxC)=A-C B-A-B C

In the first place consider the product when two of the vectors are the same. By equation (22)

A-A B = A-B A - Ax(AxB) (22)

or Ax(AxB) = A*B A - A- A B (23)

This proves the formula in case two vectors are the same. To prove it in general express A in terms of the three non-coplanar vectors B, C, and BxC.

A = bE + cC + a (BxC), (I)

where #, &, c are scalar constants. Then

Ax(BxC) = SBx(BxC) + cCx(BxC) (II)

+ a (BxC)x(BxC).

The vector product of any vector by itself is zero. Hence

(BxC)x(BxC) = 0

Ax(BxC) = 6Bx(BxC) + c Cx(BxC). (II)' By (23) Bx(BxC) = B-C B - B-B C

Cx(BxC) = - Cx(CxB) = - C-B C + C-C B. Hence Ax(BxC) = [(&B-C + cC-C)B- (6B-B + cC«B)C]. (II)" But from (I) A-B = JB«B + cC-B + a (BxC>B and A-C = b B-C + c C*C + a (BxC)-O.

By Art. 37 (BxC)-B = 0 and (BxC)-C = 0. Hence A-B = JB-B + cC-B,

A-C = 5B-C + cC-C.

74 VECTOR ANALYSIS

Substituting these values in (II)",

Ax(BxC) = A.C B - A.B C. (24)

The relation is therefore proved for any three vectors A, B, C.

Another method of giving the demonstration is as follows.

It was shown that the vector triple product Ax(BxC) was

of the form

Ax(BxC) = #B + yC.

Since Ax(AxC) is perpendicular to A, the direct product of it by A is zero. Hence

A-[Ax(BxC)] = a; A*B + yA«C = 0 and x : y = A*C : — A»B.

Hence Ax(BxC) = n (A-0 B - A-B C),

where n is a scalar constant. It remains to show n = 1. Multiply by B.

Ax(BxC>B = n (A-C B.B-A-B C-B).

The scalar triple product allows an interchange of dot and cross. Hence

Ax(BxC>B = A<BxC)xB = - A-[Bx(BxC)], if the order of the factors (BxC) and B be inverted.

-A-[Bx(BxC)] = -A-[B.C B-B.BC] = — B-C A«B + B-B A«C.

Hence n = 1 and Ax(BxC) = A.C B — A-B C. (24)

From the three letters A, B, C by different arrangements, four allied products in each of which B and C are included in parentheses may be formed. These are

Ax(BxC), Ax(CxB), (CxB)xA, (BxC)xA.

As a vector product changes its sign whenever the order of two factors is interchanged, the above products evidently satisfy the equations

Ax (BxC) = - Ax(CxB) = (CxB)xA = - (BxC)xA.

DIRECT AND SKEW PRODUCTS OF VECTORS 75

The expansion for a vector triple product in which the parenthesis comes first may therefore be obtained directly from that already found when the parenthesis comes last.

(AxB)xC = - Cx(AxB) = - C-B A + C«A B.

The formulae then become

Ax(BxC) = A-C B - A.B C (24)

and (AxB)xC = A*C B - C-B A. ' (24)'

These reduction formulae are of such constant occurrence and great importance that they should be committed, to memory. Their content may be stated in the following rule. To expand a vector triple product first multiply the exterior factor into the remoter term in the parenthesis to form a scalar coefficient for the nearer one, then multiply the exterior factor into the nearer term in the parenthesis to form a scalar coefficient for the remoter one, and subtract this result from the first.

41.] As far as the practical applications of vector analysis are concerned, one can generally get along without any formulae more complicated than that for the vector triple product. But it is frequently more convenient to have at hand other reduction formulae of which all may be derived simply by making use of the expansion for the triple product Ax(BxC) and of the rules of operation with the triple pro duct A«BxC.

To reduce a scalar product of two vectors each of which is itself a vector product of two vectors, as

(AxB>(CxD).

Let this be regarded as a scalar triple product of the three vectors A, B, and CxD — thus

AxB-(CxD).

Interchange the dot and the cross.

76 VECTOR ANALYSIS

AxB.(CxD) = A-Bx(CxD) Bx(CxD) = B-D C - B-C D.

Hence (AxB>(CxD) = A-C B-D - A-D B.C. (25)

This may be written in determinantal form.

(25)'

If A and D be called the extremes ; B and C the means ; A and C the antecedents: B and D the consequents in this product according to the familiar usage in proportions, then the expansion may be stated in words. The scalar product of two vector products is equal to the (scalar) product of the antecedents times the (scalar) product of the consequents diminished by the (scalar) product of the means times the (scalar) product of the extremes.

To reduce a vector product of two vectors each of which is itself a vector product of two vectors, as

(AxB)x(CxD). Let CxD = E. The product becomes

(AxB)xE = A-E B - B-E A. Substituting the value of E back into the equation :

(AxB)x(CxD) = (A-CxD)B - (B-CxD) A. (26) Let F = AxB. The product then becomes

Fx(CxD) = F«D C — F-C D (AxB)x(CxD) = (AxB-D)C - (AxB-C) D. (26)'

By equating these two equivalent results and transposing all the terms to one side of the equation,

[B C D] A - [C D A] B + [D A B] C - [A B C] D - 0. (27)

This is an equation with scalar coefficients between the four vectors A, B, C, D. There is in general only one such equa-

DIRECT AND SKEW PRODUCTS OF VECTORS 11

tion, because any one of the vectors can be expressed in only one way in terms of the other three : thus the scalar coeffi cients of that equation which exists between four vectors are found to be nothing but the four scalar triple products of those vectors taken three at a time. The equation may also be written in the form

[A B C] D = [B C D] A + [C A D] B + [A B D] C. (27)'

More examples of reduction formulae, of which some are important, are given among the exercises at the end of the chapter. In view of these it becomes fairly obvious that the combination of any number of vectors connected in any legitimate way by dots and crosses or the product of any number of such combinations can be ultimately reduced to a sum of terms each of which contains only one cross at most. The proof of this theorem depends solely upon analyzing the possible combinations of vectors and showing that they all fall under the reduction formulae in such a way that the crosses may be removed two at a time until not more than one remains.

* 42.] The formulae developed in the foregoing article have interesting geometric interpretations. They also afford a simple means of deducing the formulae of Spherical Trigo nometry. These do not occur in the vector analysis proper. Their place is taken by the two quadruple products,

(AxB>(CxD) = A-C B-D - B-C A-D (25) and (AxB)x(CxD) = [ACD] B - [BCD] A

= [ABD] C - [ABC] D, (26)

which are now to be interpreted.

Let a unit sphere (Fig. 22) be given. Let the vectors A, B, C, D be unit vectors drawn from a common origin, the centre of the sphere, and terminating in the surface of the sphere at the points A,B, (7, D. The great circular arcs

78

VECTOR ANALYSIS

FIG. 22.

AB, A C) etc., give the angles between the vectors A and B, A and C, etc. The points A, B, C, D determine a quadrilateral upon the sphere. A C and BD are one pair of opposite sides ; A D and B C> the other. A B and CD are the diagonals.

(AxB).(CxD) = A-C B-D - A-D B-C AxB = sin (A, B), CxD = sin (C, D).

The angle between AxB and CxD is the angle between the normals to the AB- and CD-planes. This is the same as

the angle between the planes themselves. Let it be denoted

by x. Then

(AxB). (CxD) = sin (A,B) sin (C,D) cos a:.

The angles (A, B), (C, D) may be replaced by the great circular arcs AB, CD which measure them. Then

(AxB).(CxD) = sin A B sin CD cos#, A-C B-D- A.D B*C = cos AC cosBD - cos AD cos BC.

Hence

sin A B sin CD cos x = cos A C cos B D — cos AD cos B C.

In words : The product of the cosines of two opposite sides of a spherical quadrilateral less the product of the cosines of the other two opposite sides is equal to the product of the sines of the diagonals multiplied by the cosine of the angle between them. This theorem is credited to Gauss.

Let A, B, C (Fig. 23) be a spherical tri angle, the sides of which are arcs of great circles. Let the sides be denoted by a, 6, c respectively. Let A, B, C be the unit vectors drawn from the center of the sphere to the points -A, B, C. Furthermore let pa, pb, pe be the great circular arcs dropped

FIG. 23.

DIRECT AND SKEW PRODUCTS OF VECTORS 79

perpendicularly from the vertices -4, J9, C to the sides a, 6, £. Interpret the formula

(AxB)-(CxA) = A-C B-A - B.C A-A.

(AxB) = sin (A, B) = sin c, (CxA) = sin (C, A) = sin 6. Then (AxB) • (CxA) = sin c sin b cos #,

where x is the angle between AxB and CxA. This angle is equal to the angle between the plane of A, B and the plane of C, A. It is, however, not the interior angle A which is one of the angles of the triangle : but it is the exterior angle 180° — A, as an examination of the figure will show.

Hence

(AxB). (CxA) = sin c sin b cos (180° — A)

= — sin c sin 6 cos A A»C B«A — B«C A- A = cos & cos c — cos a 1.

By equating the results and transposing,

cos a = cos 6 cos c — sin 6 sin c cos A cos 6 = cos c cos a — sin c sin a cos B cos c = cos a cos 6 — sin a sin 6 cos C.

The last two may be obtained by cyclic permutation of the letters or from the identities

(BxC).(AxB) = B-A C'B - C-A, (CxAHBxC) = C-B A.C - B-C.

Next interpret the identity (AxB)x(CxD) in the special cases in which one of the vectors is repeated.

(AxB)x(AxC) = [A B C] A.

Let the three vectors a, b, c be unit vectors in the direction of BxC, CxA, AxB respectively. Then

AxB = c sin c, AxC = — b sin 6

(AxB)x(AxC) = — cxb sin c sin & = A sin c sin 6 sin A [A B C] = (AxB)-C = c«C sin c = cos (90° —pc) sin c [ABC] A = sin c sin pc A.

80 VECTOR ANALYSIS

By equating the results and cancelling the common factor,

sin^c = sin b sin A sin^?a = sin c sin B sin pb = sin a sin C.

The last two may be obtained by cyclic permutation of the letters. The formulae give the sines of the altitudes of the triangle in terms of the sines of the angle and sides. Again

write

(AxB)x(AxC) = [ABC]A

(BxC)x(BxA) = [BCA]B (CxA)x(CxB) = [CAB]C.

Hence sin c sin b sin A = [A B C]

sin a sin c sin B = [B C A] sin b sin a sin C = [C A B].

The expressions [ABC], [BCA], [CAB] are equal. Equate the results in pairs and the formulae

sin b sin A = sin a sin B sin c sin B = sin b sin C sin a sin C = sin c sin A

are obtained. These may be written in a single line.

sin A sin B sin C sin a sin b sin c

The formulae of Plane Trigonometry are even more easy to obtain. If A B C be a triangle, the sum of the sides taken as vectors is zero — for the triangle is a closed polygon. From this equation

a + b-f c = 0

almost all the elementary formulae follow immediately. It is to be noticed that the angles from a to b, from b to c, from

DIRECT AND SKEW PRODUCTS OF VECTORS 81

o to a are not the interior angles A, B, (7, but the exterior angles 180° -A, 180° - B, 180° - C.

— a = b + c a»a = (b + c)*(b + c) = b-b + c-c + 2 b«c.

If a, J, c be the length of the sides a, b, c, this becomes

c2 = a2 + 62 - 2 a 6 cos C.

The last two are obtained in a manner similar to the first one or by cyclic permutation of the letters. The area of the triangle is

^axb = ^bxc = 2cxa = 2 a b sin C = % b c sin A = ^ c a sin B.

If each of the last three equalities be divided by the product £ a b c, the fundamental relation

sin A sin B sin 0

is obtained. Another formula for the area may be found from

the product

(bxc)«(bxc) = (cxa)-(axb)

2 Area (6 c sin A} = (c a sin B) (a b sin (7) a2 sin -Z?sin C

2 Area =

sin A

Reciprocal Systems of Three Vectors. Solution of Equations

43.] The problem of expressing any vector r in terms of three non-coplanar vectors a, b, c may be solved as follows. Let

82 VECTOR ANALYSIS

where a, J, c are three scalar constants to be determined Multiply by • b x c.

r.bxc = a a»bxc + 6 b«bxc + cc-bxo or [rbc] = a [a be].

In like manner by multiplying the equation by • c x a and . a X b the coefficients b and c may be found.

[r c a] = I [b c a] [r a b] = c [c a b]

Hence r = a+b+, (28)

[be a] [c a b]

The denominators are all equal. Hence this gives the equation

[a b c] r — [b c r] a + [c r a] b - [r a b] c = 0

which must exist between the four vectors r, a, b, c. The equation may also be written

r»b x c r»o x a, r«a x b r = -r . - a + r v .. b + e

[abe] [abc] [a be]

bxc cxa, axb

or r = r» r _ a + r • r b + r • o.

[abc] [abc] [abc]

The three vectors which appear here multiplied by !•, namely

bxc cxa axb

_ * _ - > _

[a be] [a b c] [a b c]

are very important. They are perpendicular respectively to the planes of b and c, c and a, a and b. They occur over and over again in a large number of important relations. For this reason they merit a distinctive name and notation. Definition : The system of three vectors

b x c ^ cxa axb [abc]' [abc] ' [abc]

DIRECT AND SKEW PRODUCTS OF VECTORS 83

which are found by dividing the three vector products bxc, c x a, a x b of three non-coplanar vectors a, b, c by the scalar product [abc] is called the reciprocal system to a, b, c.

The word non-coplanar is important. If a, b, c were co planar the scalar triple product [a b c] would vanish and consequently the fractions

bxc cxa axb

j j ________

[a be] [a b c] [a b c]

would all become meaningless. Three coplanar vectors have no reciprocal system. This must be carefully remembered. Hereafter when the term reciprocal system is used, it will be understood that the three vectors a, b, c are not coplanar. The system of three vectors reciprocal to system a, b, c will be denoted by primes as a', b', c',

,_bxc? h, _ c x a , , __ a x b (29) ""[abc]' [abc]' ~[a¥c]'

The expression for r reduces then to the very simple form r = r-a' a + r-b'b + r.c' c. (30)

The vector r may be expressed in terms of the reciprocal system a', b', c' instead of in terms of a, b, c. In the first place it is necessary to note that if a, b, c are non-coplanar, a', b', c' which are the normals to the planes of b and c, c and a, a and b must also be non-coplanar. Hence r may be expressed in terms of them by means of proper scalar coefficients #, y, z.

r — x a' + ?/b' + z c'

Multiply successively by -a, -b, -c. This gives [a b c] r-a = x [b c a], x = r-a [abc]r-b = y [cab], y = r-b [a b c] r-c = z [a b c], z = r-c

Hence r = r-a a ' + r-b b ' + r-c c '. (31)

84 VECTOR ANALYSIS

44.] If a', V, c' be the system reciprocal to a, b, c the scalar product of any vector of the reciprocal system into the corresponding vector of the given system is unity ; but the product of two non-corresponding vectors is zero. That is

a'.a = bM>=:c'.c = l (32)

a'.b = a'.c = b'-a = b'*c = c'-a = c'-b = 0.

This may be seen most easily by expressing a', V, c' in terms of themselves according to the formula (31)

r = r«aa' + r*bb' + r»cc'. Hence a' = a'-aa' + a'«b V + a'«cc'

b' = b'.aa' + b'-bb' + bW c' = c'«aa' + c'-bb' + c'.cc'.

Since a', b', c' are non-coplanar the corresponding coeffi cients on the two sides of each of these three equations must be equal. Hence from the first

1 = a '*a 0 = a'-b 0 = a' c. From the second 0 = b'«a l=b'«b 0 = b'«e. From the third 0 = c'«a 0 = c'«b l = c'«o.

This proves the relations. They may also be proved directly from the definitions of a', b', c'.

bxc bxc«a [be a]

a'» a = • a = = = 1

[abc] [abc] [abc]

bxc bxc-b 0

a' . b = • b = = =0

[abc] [abc] [abc]

and so forth.

Conversely if two sets of three vectors each, say A, B, C, and a, b, c, satisfy the relations

A«a = B«b = C«c = 1 A-b = A«c = B«a = B-c = C«a = C«b = 0

DIRECT AND SKEW PRODUCTS OF VECTORS 85

then the set A, B, C is the system reciprocal to a, b, c. By reasoning similar to that before

A = A-a a' + A«b b' + A-c c' B = B-a a' + B-b b' + B-c c' C = C«aa' + C-bb' + C-c c'.

Substituting in these equations the given relations the re sult is

A = a', B = b', C = c'. Hence

Theorem : The necessary and sufficient conditions that the set of vectors a', b', c' be the reciprocals of a, b, c is that they satisfy the equations

a'.a = b'.b = c'.c = l (32)

a'-b = a'-c = b'«a = b'.c = c'-a = c'.b = 0.

As these equations are perfectly symmetrical with respect to a', b', c' and a, b, c it is evident that the system a, b, c may be looked upon as the reciprocal of the system a', b', c' just as the system a', b', c' may be regarded as the reciprocal of a, b, c. That is to say,

Theorem: If a', b', c' be the reciprocal system of a, b, c, then a, b, c will be the reciprocal system of a', V, c'.

V x c' c' x a' a' x b' (29V

- — • b=- - - . v /

[a'b'e'] [a'b'c'] [a'b'c']

These relations may be demonstrated directly from the definitions of a', b', c'. The demonstration is straightfor ward, but rather long and tedious as it depends on compli cated reduction formulae. The proof given above is as short as could be desired. The relations between a', b',c'and a, b, c are symmetrical and hence if a', b', c' is the reciprocal system of a, b, c, then a, b, c must be the reciprocal system of

86

VECTOR ANALYSIS

45.] Theorem : If a', V, c' and a, b, c be reciprocal systems the scalar triple products [a'b'c'] and [a b c] are numerical reciprocals. That is

t.'b'.']=[i

[a'bV] [abc]=l

xc cxa axb"| [a "be] [abc] [abc] J

[bxc cxa axb].

(33)

But Hence

Hence

~[abc]3 [bxc cxa axb] = (bxc)x(cxa>(axb).

(bxc) x (cxa) = [abc]c. [bxc cxa axb] = [abc] c-axb = [abc]2. 1 1

[a'bV] =

[abc]:

[abc]2 =

[abc]

(33)'

By means of this relation between [a' b' c'] and [a b c] it is possible to prove an important reduction formula,

(P.axE)(A«BxC) =

P-A P.B p.c Q.A Q.B a-c

B*A £*B £*C

(34)

which replaces the two scalar triple products by a sum of nine terms each of which is the product of three direct pro ducts. Thus the two crosses which occur in the two scalar products are removed. To give the proof let P, ft, B be expressed as

P = P-A A' + P.B B' + P.C C'

Then But

B = B-A A' + B.B B' + B«C C'.

P.A P.B P.C

[POB] =

a-A Q.B a-c

R-A R.B R.C

1 [ABC]'

[A'B'C'].

[A'B'C'J =

DIRECT AND SKEW PRODUCTS OF VECTORS 87

Hence [PQE] [ABC] =

P.A P-B P-C

Q.A a-B Q.C

R.A R.B B*C

The system of three unit vectors i, j, k is its own reciprocal system.

jxki kxi ,, i x j

J'k==k- (35)

For this reason the primes i', j', k' are not needed to denote a system of vectors reciprocal to i, j, k. The primes will therefore be used in the future to denote another set of rect angular axes i, j, k , just as X* , F', Z* are used to denote a set of axes different from X, F, Z.

The only systems of three vectors which are their own reciprocals are the right-handed and left-handed systems of three unit vectors. That is the system i, j, k and the system i, j, — k.

Let A, B, C be a set of vectors which is its own reciprocal.

Then by (32)

A«A = B-B = CNC = 1.

Hence the vectors are all unit vectors.

A-B = A-C = 0. Hence A is perpendicular to B and C.

B-A = B-C = 0. Hence B is perpendicular to A and C.

C-A =C.B = O.

Hence C is perpendicular to A and B.

Hence A, B, C must be a system like i, j, k or like i, j, — k.

* 46.] A scalar equation of the first degree in a vector r is an equation in each term of which r occurs not more than once. The value of each term must be scalar. As an exam ple of such an equation the following may be given.

a a-bxr + 6(oxd)»(exr) + c f«r + d = 0,

88 VECTOR ANALYSIS

where a, b, c, d, e, f are known vectors ; and a, &, c, d, known scalars. Obviously any scalar equation of the first degree in an unknown vector r may be reduced to the form

r-A = a

where A is a known vector ; and a, a known scalar. To ac complish this result in the case of the given equation proceed

as follows.

a axbor + "b (cxd)xe-r + c f»r + d = 0

{a axb + b (cxd)xe + c f}«r = — d.

In more complicated forms it may be necessary to make use of various reduction formulae before the equation can be made

to take the desired form,

]>A = a.

As a vector has three degrees of freedom it is clear that one scalar equation is insufficient to determine a vector. Three scalar equations are necessary.

The geometric interpretation of the equa tion

r.A => a (36)

is interesting. Let r be a variable vector (Fig. 24) drawn from a fixed origin. Let A be a fixed vector drawn from the same origin. The equation then becomes

r A cos (r,A) = a,

a or T cos (r,A) =— ,

if r be the magnitude of r ; and A that of A. The expression

r cos (r, A)

is the projection of r upon A. The equation therefore states that the projection of r upon a certain fixed vector A must

DIRECT AND SKEW PRODUCTS OF VECTORS 89

always be constant and equal to a/ A. Consequently the ter minus of r must trace out a plane perpendicular to the vector A at a distance equal to a/ A from the origin. The projec tion upon A of any radius vector drawn from the origin to a point of this plane is constant and equal to a/ A. This gives the following theorem.

Theorem : A scalar equation in an unknown vector may be regarded as the equation of a plane, which is the locus of the terminus of the unknown vector if its origin be fixed.

It is easy to see why three scalar equations in an unknown vector determine the vector completely. Each equation de termines a plane in which the terminus of r must lie. The three planes intersect in one common point. Hence one vec tor r is determined. The analytic solution of three scalar equations is extremely easy. If the equations are

r«A = a

r-B = b (37)

r-C = c9

it is only necessary to call to mind the formula

r = r.A A' + r-BB' + r-C C'.

Hence r = a A' + 6 B' + c C'. (38)

The solution is therefore accomplished. It is expressed in terms A', B', C' which is the reciprocal system to A, B, C. One caution must however be observed. The vectors A, B, C will have no reciprocal system if they are coplanar. Hence the solution will fail. In this case, however, the three planes de termined by the three equations will be parallel to a line. They will therefore either not intersect (as in the case of the lateral faces of a triangular prism) or they will intersect in a common line. Hence there will be either no solution for r or there will be an infinite number.

90 VECTOR ANALYSIS

From four scalar equations

r-A = a

r.B = 6 (39)

r«C = c

r«D =d

the vector r may be entirely eliminated. To accomplish this solve three of the equations and substitute the value in the

fourth.

r = aA' + 6B' + cC'

a A'«D + &B'.D + cC'-D = d

or a [BCD] + b [CAD] + c [ABD] = d [ABC]. (40)

* 47.] A vector equation of the first degree in an unknown vector is an equation each term of which is a vector quantity containing the unknown vector not more than once. Such an equation is

(AxB)x(Cxr) + D ET + n r + F =0,

where A, B, C, D, E, F are known vectors, n a known scalar, and r the unknown vector. One such equation may in gen eral be solved for r. That is to say, one vector equation is in general sufficient to determine the unknown vector which is contained in it to the first degree.

The method of solving a vector equation is to multiply it with a dot successively by three arbitrary known non-coplanar vectors. Thus three scalar equations are obtained. These may be solved by the methods of the foregoing article. In the first place let the equation be

A a«r + B b»r + C c-r = D,

where A, B, C, D, a, b, c are known vectors. No scalar coeffi cients are written in the terms, for they may be incorporated in the vectors. Multiply the equation successively by A', B', C'. It is understood of course that A, B, C are non-coplanar.

DIRECT AND SKEW PRODUCTS OF VECTORS 91

a-r = D-Ar

b-r = D-B'

c-r = D-C'.

But r = a' a-r + b' b-r + c' c-r.

Hence r = D-A' a' + D-B' b' + D-C' c'.

The solution is therefore accomplished in case A, B, C are non- coplanar and a, b, c also non-coplanar. The special cases in which either of these sets of three vectors is coplanar will not be discussed here.

The most general vector equation of the first degree in an unknown vector r contains terms of the types

A a-r, n r, Exr, D.

That is it will contain terms which consist of a known vector multiplied by the scalar product of another known vec tor and the unknown vector ; terms which are scalar multi ples of the unknown vector; terms which are the vector product of a known and the unknown vector ; and constant terms. The terms of the type A a-r may always be reduced to three in number. For the vectors a, b, c, • • • which are multiplied into r may all be expressed in terms of three non- coplanar vectors. Hence all the products a-r, b-r, or, • • • may be expressed in terms of three. The sum of all terms of the type A a-r therefore reduces to an expression of three terms, as

A a-r + B b-r + C c-r.

The terms of the types n r and Exr may also be expressed in this form.

n r = 7i a' a-r + n b'b-r + n c'c-r Exr = Exa' a-r + Exb' b-r+Exc' c-r.

Adding all these terms together the whole equation reduces to the form

L a-r + M b-r + N c-r = K.

92 VECTOR ANALYSIS

This has already been solved as

r = K.L' a' + K-M' b' + XJT c'.

The solution is in terms of three non-coplanar vectors a', V, cf. These form the system reciprocal to a, b, c in terms of which the products containing the unknown vector r were expressed.

* SUNDRY APPLICATIONS OF PKODUCTS Applications to Mechanics

48.] In the mechanics of a rigid body a force is not a vector in the sense understood in this book. See Art. 3. A force has magnitude and direction ; but it has also a line of application. Two forces which are alike in magnitude and direction, but which lie upon different lines in the body do not produce the same effect. Nevertheless vectors are sufficiently like forces to be useful in treating them.

If a number of forces fx, f2, f3, ---act on a body at the same point 0, the sum of the forces added as vectors is called

the resultant R.

E = f1 + f2 + f8 + ...

In the same way if f x, f 2, f 8 • • • do not act at the same point the term resultant is still applied to the sum of these forces added just as if they were vectors.

B = f1 + fa + f8 + ... (41)

The idea of the resultant therefore does not introduce the line of action of a force. As far as the resultant is concerned a force does not differ from a vector.

Definition: The moment of a force f about the point 0 is equal to the product of the force by the perpendicular dis tance from 0 to the line of action of the force. The moment however is best looked upon as a vector quantity. Its mag nitude is as defined above. Its direction is usually taken to

DIRECT AND SKEW PRODUCTS OF VECTORS 93

be the normal on that side of the plane passed through the point 0 and the line f upon which the force appears to pro duce a tendency to rotation about the point 0 in the positive trigonometric direction. Another method of defining the moment of a force t = PQ about the point 0 is as follows : The moment of the force f = PQ about the point 0 is equal to twice the area of the triangle 0 PQ. This includes at once both the magnitude and direction of the moment (Art. 25). The point P is supposed to be the origin ; and the point Q, the terminus of the arrow which represents the force f. The letter M will be used to denote the moment. A subscript will be attached to designate the point about which the moment is taken.

The moment of a number of forces f x, f2, • • • is the (vector) sum of the moments of the individual forces.

If

This is known as the total or resultant moment of the forces

* v *&••••

49.] If f be a force acting on a body and if d be the vector drawn from the point 0 to any point in the line of action of the force, the moment of the force about the point 0 is the vector product of d into f .

Mo W = dxf (42)

For dxf = d f sin (d, f) e,

if e be a unit vector in the direction of dxf.

•

dxf = dsm (d, f)/e.

Now d sin (d, f) is the perpendicular distance from 0 to f. The magnitude of dxf is accordingly equal to this perpen dicular distance multiplied by/, the magnitude of the force.

94 VECTOR ANALYSIS

This is the magnitude of the moment MO {f} . The direction of dxf is the same as the direction of the moment. Hence the relation is proved.

Mo {f} = dxf.

The sum of the moments about 0 of a number of forces f p f 2, • • • acting at the same point P is equal to the moment of the resultant B of the forces acting at that point. For let d be the vector from 0 to P. Then

Mo {f x> = dxf l Mo {fa| = dxfa

+ ... (43)

= dx(£1 + fa + -..)=dxB

The total moment about 0 f of any number of forces f x, f 2, • • • acting on a rigid body is equal to the total moment of those forces about 0 increased by the moment about 0' of the resultant BO considered as acting at 0.

M<x {f i> f 2 >•••} = Mo {f r f 2, • • •} + Mo< {Bo \. (44)

Let dj, d2, • • • be vectors drawn from 0 to any point in f r f2, • • • respectively. Let d/, d2', • • • be the vectors drawn from Of to the same points in f x, f 2, • • • respectively. Let o be the vector from 0 to_0'. Then

d^d/H-c, d2 = d2' + c, •••

Mo {f i, f2, • • •} = dxxf j + d2xf2 + • • • Mo'{f!,f2, •••J=d1'xf1 + da'xfa + ...

^ = (dx - c)xf ! + (d2 - c)xf a + • - . = dxxf x+d2xf 2 + ---- cx(f j + f a + . . .)

But — c is the vector drawn from 0' to 0. Hence — c x f, is the moment about 0' of a force equal in magnitude and parallel in direction to f 1 but situated at 0. Hence

DIRECT AND SKEW PRODUCTS OF VECTORS 95

fa + ...) = - cxBo = Mo' {Bo}. Hence MO/ {f x, f2, • • •} = M0 {fr fa ;,•••} + MCX {Bo |. (44)

The theorem is therefore proved.

The resultant £ is of course the same at all points. The subscript 0 is attached merely to show at what point it is supposed to act when the moment about Of is taken. For the point of application of E affects the value of that moment.

The scalar product of the total moment and the resultant is the same no matter about what point the moment be taken. In other words the product of the total moment, the result ant, and the cosine of the angle between them is invariant for all points of space.

E • MO' {f i, f 2 »•••}= B • MO {f ! , f 2 9 • • •}

where Of and 0 are any two points in space. This -important relation follows immediately from the equation

Mo' {*i, fa, • • •} = Mo {fj , f2, • • •} + Mo' {Eo}. For E.Mo'if!,f2, •••}=*• M0 {f^, • • •} + E- M0' {B0}.

But the moment of E is perpendicular to E no matter what the point 0 of application be. Hence

E-MO* IEO} = o

and the relation is proved. The variation in the total moment due to a variation of the point about which the moment is taken is always perpendicular to the resultant.

50.] A point Or may be found such that the total moment about it is parallel to the resultant. The condition for parallelism is

{fx, fa, •-•}=<)

=0

96 VECTOR ANALYSIS

where 0 is any point chosen at random. Replace Mo'{Eo} by its value and for brevity omit to write the f v f 2, • • • in the braces { }. Then

RxMcy = ExMo - Ex(cxE) = 0. The problem is to solve this equation for c.

ExMo — E«E c + R.c E = 0.

Now R is a known quantity. Mo is also supposed to be known. Let c be chosen in the plane through 0 perpen dicular to E. Then E»c = 0 and the equation reduces to

ExM0 •= E«E c ExMo E-E

If c be chosen equal to this vector the total moment about the point Or, which is at a vector distance from 0 equal to c, will be parallel to E. Moreover, since the scalar product of the total moment and the resultant is constant and since the resultant itself is constant it is clear that in the case where they are parallel the numerical value of the total moment will be a minimum.

The total moment is unchanged by displacing the point about which it is taken in the direction of the resultant.

For Mo' jf !, f2, • • •} = Mo {f ! , f 2, • • •} - cxE.

If c = 0 Of is parallel to E, cxE vanishes and the moment about Of is equal to that about 0. Hence it is possible to find not merely one point Or about which the total moment is parallel to the resultant ; but the total moment about any point in the line drawn through 0' parallel to E is parallel to E. Furthermore the solution found in equation for c is the only one which exists in the plane perpendicular to E — unless the resultant E vanishes. The results that have been obtained may be summed up as follows :

DIRECT AND SKEW PRODUCTS OF VECTORS 97

If any system of forces f 19 f 2, ••• whose resultant is not zero act upon a rigid body, then there exists in space one and only one line such that the total moment about any point of it is parallel to the resultant. This line is itself parallel to the resultant. The total moment about all points of it is the same and is numerically less than that about any other point in space.

This theorem is equivalent to the one which states that any system of forces acting upon a rigid body is equivalent to a single force (the resultant) acting in a definite line and a couple of which the plane is perpendicular to the resultant and of which the moment is a minimum. A system of forces may be reduced to a single force (the resultant) acting at any desired point 0 of space and a couple the moment of which (regarded as a vector quantity) is equal to the total moment about 0 of the forces acting on the body. But in general the plane of this couple will not be perpendicular to the result ant, nor will its moment be a minimum.

Those who would pursue the study of systems of forces acting on a rigid body further and more thoroughly may consult the Traite de Mecanique Rationnelle l by P. APPELL. The first chapter of the first volume is entirely devoted to the discussion of systems of forces. Appell defines a vector as a quantity possessing magnitude, direction, and point of application. His vectors are consequently not the same as those used in this book. The treatment of his vectors is carried through in the Cartesian coordinates. Each step however may be easily converted into the notation of vector analysis. A number of exercises is given at the close of the chapter.

51.] Suppose a body be rotating about an axis with a con stant angular velocity a. The points in the body describe circles concentric with the axis in planes perpendicular to

1 Paris, Gauthier-Villars et Fils, 1893.

7

98

VECTOR ANALYSIS

FIG. 25.

the axis. The velocity of any point in its circle is equal to the product of the angular velocity and the radius of the circle. It is therefore equal to the product of the angular

velocity and the perpendicular dis tance from the point to the axis. The direction of the velocity is perpendicular to the axis and to the radius of the circle described by the point.

Let a (Fig. 25) be a vector drawn along the axis of rotation in that direction in which a right-handed screw would advance if turned in the direction in which the body is rotating. Let the magnitude of a

be a, the angular velocity. The vector a may be taken to represent the rotation of the body. Let r be a radius vector drawn from any point of the axis of rotation to a point in the body. The vector product

axr = a rsin(a,r)

is equal in magnitude and direction to the velocity v of the terminus of r. For its direction is perpendicular to a and r and its magnitude is the product of a and the perpendicular distance r sin (a, r) from the point to the line a. That is

v = axr. (45)

If the body be rotating simultaneously about several axes ai* a2> aa •' ' which pass through the same point as in the case of the gyroscope, the velocities due to the various

rotations are

vi -=a1xr1

v8 = a8xr8

DIRECT AND SKEW PRODUCTS OF VECTORS 99

where rx, r2, r3, • • • are the radii vec tores drawn from points on the axis a19 a2, a3, • • • to the same point of the body. Let the vectors rx, r2, r8, • • • be drawn from the common point of intersection of the axes. Then

TJ = ra = r8 = • • • = r and

v = vt + v2 + v3 + • • • == axxr + a2xr + a8xr + • • •

This shows that the body moves as if rotating with the angular velocity which is the vector sum of the angular velocities a19 a2, a8, • • • This theorem is sometimes known as the parallelogram law of angular velocities.

It will be shown later (Art.) 60 that the motion of any rigid body one point of which is fixed is at each instant of time a rotation about some axis drawn through that point. This axis is called the instantaneous axis of rotation. The axis is not the same for all time, but constantly changes its position. The motion of a rigid body one point of which is fixed is therefore represented by

v = axr (45)

where a is the instantaneous angular velocity; and r, the radius vector drawn from the fixed point to any point of the body.

The most general motion of a rigid body no point of which is fixed may be treated as follows. Choose an arbitrary point 0. At any instant this point will have a velocity v0. Relative to the point 0 the body will have a motion of rotation about some axis drawn through 0. Hence the velocity v of any point of the body may be represented by the sum of V0 the velocity of 0 and axr the velocity of that point

relative to 0.

v = v0 + axr. (46)

100 VECTOR ANALYSIS

In case v0 is parallel to a, the body moves around a and along a simultaneously. This is precisely the motion of a screw advancing along a. In case v0 is perpendicular to a, it is possible to find a point, given by the vector r, such that its velocity is zero. That is

This may be done as follows. Multiply by xa.

(axr)xa = — v0xa or a«a r — a-r a = — v0xa.

Let r be chosen perpendicular to a. Then a»r is zero and

a»a r = — v0 x a f = - v0 x a a«a

The point r, thus determined, has the property that its veloc ity is zero. If a line be drawn through this point parallel to a, the motion of the body is one of instantaneous rotation about this new axis.

In case v0 is neither parallel nor perpendicular to a it may be resolved into two components

v — v ' 4- v "

•n — vn r »n

which are respectively parallel and perpendicular to a.

v = v0' + v0" + axr A point may now be found such that

v0" = — axr.

Let the different points of the body referred to this point be denoted by r'. Then the equation becomes

v = v0' + axr'. (46)'

The motion here expressed consists of rotation about an axis a and translation along that axis. It is therefore seen that the most general motion of a rigid body is at any instant

DIRECT AND SKEW PRODUCTS OF VECTORS 101

the motion of a screw advancing at a certain rate along a definite axis a in space. The axis of the screw and its rate of advancing per unit of rotation (i. e. its pitch) change from instant to instant.

52.] The conditions for equilibrium as obtained by the principle of virtual velocities may be treated by vector methods. Suppose any system of forces f x, f 2, • • • act on a rigid body. If the body be displaced through a vector dis tance D whether this distance be finite or infinitesimal the work done by the forces is

The total work done is therefore

W^^il + D.f2 + ...

If the body be in equilibrium under the action of the forces the work done must be zero.

W= D-fj + D-f2 + • • • = D-Cfj + f2 + • • 0 = D.E = 0.

The work done by the forces is equal to the work done by their resultant. This must be zero for every displacement

D. The equation

D-E = 0

holds for all vectors D. Hence

E = 0.

The total resultant must be zero if the body be in equilibrium.

The work done by a force f when the rigid body is dis

placed by a rotation of angular velocity a for an infinitesimal

time t is approximately

a-dxf t,

where d is a vector drawn from any point of the axis of rota tion a to any point of f. To prove this break up f into two components f ', f " parallel and perpendicular respectively to a.

a-dxf = a-dxf ' + a-dxf ".

102 VECTOR ANALYSIS

As f ' is parallel to a the scalar product [a d f '] vanishes. a-dxf = a-dxf ".

On the other hand the work done by t" is equal to the work done by f during the displacement. For f ' being parallel to a is perpendicular to its line of action. If h be the common vector perpendicular from the line a to the force f ", the work done by f " during a rotation of angular velocity a for time t is approximately

The vector d drawn from any point of a to any point of f may be broken up into three components of which one is h, another is parallel to a, and the third is parallel to f ". In the scalar triple product [adf] only that component of d which is perpendicular alike to a and f " has any effect. Hence

W= a-hxf " t = a-dxf ' tf = a-dxf t.

If a rigid body upon which the forces f v f 2, • • • act be dis placed by an angular velocity a for an infinitesimal time t and if dx, d2, • • • be the vectors drawn from any point 0 of a to any points of f v f 2, • - • respectively, then the work done by the forces f v f2, - • • will be approximately

W= (a-djXfj + a«d2xf2 + • • •) t

= a.(d1xf1 + d2xf 2 + .-.)* = a.M0{f1,f2,...} t.

If the body be in equilibrium this work must be zero. Hence a*M0 \tl9 f2, • • •} t = 0.

The scalar product of the angular velocity a and the total moment of the forces tv f2, • • • about any point 0 must be zero. As a may be any vector whatsoever the moment itself must vanish.

Mo {fr f r - • •} = 0.

DIRECT AND SKEW PRODUCTS OF VECTORS 103

The necessary conditions that a rigid body be in equilib rium under the action of a system of forces is that the result ant of those forces and the total moment about any point in space shall vanish.

Conversely if the resultant of a system of forces and the moment of those forces about any one particular point in space vanish simultaneously, the body will be in equilibrium.

If E = 0, then for any displacement of translation D

D«E = o.

JF=D-f1 + D.f2 + ... = 0

and the total work done is zero, when the body suffers any displacement of translation.

Let Mo {fp f2> • • •} be zero for a given point 0. Then for any other point O1

Mo< {fx, f2, • • -\ = Mo 1 flf f2, - • •} + M0' {Bo}- But by hypothesis E is also zero. Hence

Hence

where a is any vector whatsoever. But this expression is equal to the work done by the forces when the body is rotated for a time t with an angular velocity a about the line a passing through the point O1. This work is zero.

Any displacement of a rigid body may be regarded as a translation through a distance D combined with a rotation for a time t with angular velocity a about a suitable line a in space. It has been proved that the total work done by the forces during this displacement is zero. Hence the forces must be in equilibrium. The theorem is proved.

104 VECTOR ANALYSIS

Applications to Geometry

53.] Relations between two right-handed systems of three mutually perpendicular unit vectors. — Let i, j, k and i', j', k' be two such systems. They form their own reciprocal systems. Hence

r = ™/I+r*^,tr^*v, (47)

and r = r«i' i' + r«j' j' + r«k' k'.

From this / i' = i'-i i + i'.j j + i'-k k = a1 i + a2 j + a3 k

I k' = k'-i i + k'-j j + k'-k k = cl i + c2 j + c3 k.

The scalarsflj, a2, a3; blt Z>2, b 3; cv 02, c3 are respectively the direction cosines of i'; j'; k' with respect to i, j, k. That is

<&]_ — cos (i', i) a2 = cos (i', j) a3 = cos (i', k)

0j "==- COS (j ) l) Ot^'==- COS (j i j) t>3 —- COS (J , Kj ( 4o)

cx = cos (k', i) c2 = cos (k', j) c3 = cos (k', k). In the same manner

^ i = i-i'i' + i-j'j7 + i-k' k'= ^ i' + \ y + GI k'

j j - j-i'i' + j-j'j' + j-k'k' = aa i' + 62 j' + c2 k' (47)"

( k = k.i'i' + k.j'j' + k-k'k' = aa if + J8 J' + C3 k/

»!/!_/ "1 9i 9i 9

t fcf •!? — I — — /t ^ I n & j /• 4

and ) j.j = 1 = «22 + J22 + c22 (49)'

( k-k = 1 = as2 + 632 + c32

and ] j'.k' = 0 = \ cj + &2 c2 + bB cs (50)

I i_^ •! f\

\ K •! = U = Cj ttj -f- C<£ dy -f- C3 a%

DIRECT AND SKEW PRODUCTS OF VECTORS 105

and \ j-k = 0 = a* a9 + K I* + c0 c, (50)'

j.k = 0 = #2 a3 + 62 63 +

and

But Hence

k' = i'xj'= (a2 53 - a3 62) i

yl "2

= (a2 &3 - a3 62),

(51)

(52)

Or

Co =

and similar relations may be found for the other six quantities av a2, a3 ; bv &2, &3. All these scalar relations between the coefficients of a transformation which expresses one set of orthogonal axes X1, F', Z* in terms of another set JT, F, Z are important and well known to students of Cartesian methods. The ease with which they are obtained here may be note worthy.

A number of vector relations, which are perhaps not so well known, but nevertheless important, may be found by multi plying the equations

i' = al i + a2 j + a3 k

in vector multiplication.

&! k' — Cj j ' = a3 j — a2 k.

(53)

The quantity on either side of this equality is a vector. From its form upon the right it is seen to possess no component in

106 VECTOR ANALYSIS

the i direction but to lie wholly in the jk-plane ; and from its form upon the left it is seen to lie in the j'k '-plane. Hence it must be the line of intersection of those two planes. Its magnitude is V af + a£ or V b^ + c^. This gives the scalar relations

af + a* = V + *!2 = 1 - a*.

The magnitude 1 — a^ is the square*of the sine of the angle between the vectors i and i'. Hence the vector

^k'-cj'^sj-aak (53)

is the line of intersection of the j'k'- and jk-planes, and its magnitude is the sine of the angle between the planes. Eight other similar vectors may be found, each of which gives one of the nine lines of intersection of the two sets of mu tually orthogonal planes. The magnitude of the vector is in each case the sine of the angle between the planes.

54.] Various examples in Plane and Solid Geometry may be solved by means of products.

Example 1 : The perpendiculars from the vertices of a trian gle to the opposite sides meet in a point. Let A B 0 be the triangle. Let the perpendiculars from A to BC and from B to CA meet in the point 0. To show 0 0 is perpendicular to A B. Choose 0 as origin and let OA = A, OB = B, and =C. Then

= C-B,

By hypothesis

A.(C - B) = 0

and B<A - C) = 0.

Add; C<B - A) = 0,

which proves the theorem.

Example 2 : To find the vector equation of a line drawn through the point B parallel to a given vector A.

DIRECT AND SKEW PRODUCTS OF VECTORS 107

Let 0 be the origin and B the vector OS. Let £ be the ra dius vector from 0 to any point of the required line. Then E — B is parallel to A. Hence the vector product vanishes.

Ax(B-B) = 0.

This is the desired equation. It is a vector equation in the unknown vector B. The equation of a plane was seen (page 88) to be a scalar equation such as

B«C = c in the unknown vector B.

The point of intersection of a line and a plane may be found at once. The equations are

( Ax(B - B) = 0 i B-C = e AxB = AxB

A-C B - C-B A = (AxB)xC A-C B - c A = (AxB)xC

Hence (AxB)xC + c A .

A-C

The solution evidently fails when A»C = 0. In this case how ever the line is parallel to the plane and there is no solution ; or, if it lies in the plane, there are an infinite number of solu tions.

Example 3: The introduction of vectors to represent planes.

Heretofore vectors have been used to denote plane areas of definite extent. The direction of the vector was normal to the plane and the magnitude was equal to the area to be re presented. But it is possible to use vectors to denote not a plane area but the entire plane itself, just as a vector represents a point. The result is analogous to the plane coordinates of analytic geometry. Let 0 be an assumed origin. Let M N be a plane in space. The plane MN is to be denoted b^ a vector

108 VECTOR ANALYSIS

whose direction is the direction of the perpendicular dropped upon the plane from the origin 0 and whose magnitude is the reciprocal of the length of that perpendicular. Thus the nearer a plane is to the origin the longer will be the vector which represents it.

If r be any radius vector drawn from the origin to a point in the plane and if p be the vector which denotes the plane,

then

r-p = 1

is the equation of the plane. For

r»p = r cos (r, p) p.

Now p, the length of p is the reciprocal of the perpendicular distance from 0 to the plane. On the other hand r cos (r, p) is that perpendicular distance. Hence r«p must be unity. If r and p be expressed in terms of i, j, k

r = #i + yj + zk p = ui + vj + wit Hence r»p = xu + yv + zw = 'L.

The quantities u, v, w are the reciprocals of the intercepts of the plane p upon the axes.

The relation between r and p is symmetrical. It is a rela tion of duality. If in the equation

r-p = 1

r be regarded as variable, the equation represents a plane p which is the locus of all points given by r. If however p be regarded as variable and r as constant, the equation repre sents a point r through which all the planes p pass. The development of the idea of duality will not be carried out. It is familiar to all students of geometry. The use of vec tors to denote planes will scarcely be alluded to again until Chapter VII.

DIRECT AND SKEW PRODUCTS OF VECTORS 109

SUMMARY OF CHAPTER II

The scalar product of two vectors is equal to the product of their lengths multiplied by the cosine of the angle between

them.

A-B = A B cos (A, B) (1)

A-B = B.A (2)

A.A = ^«. (3)

The necessary and sufficient condition for the perpendicularity of two vectors neither of which vanishes is that their scalar product vanishes. The scalar products of the vectors i, j, k are

^=J!Uk!=o (4)

A.B = A1B1 + A,, B2 + AS Bz (7)

H = A* = A* + A* + A*. (8)

If the projection of a vector B upon a vector A is Bf,

-R' — A'B A (*\

XA ' '

The vector product of two vectors is equal in magnitude to the product of their lengths multiplied by the sine of the an gle between them. The direction of the vector product is the normal to the plane of the two vectors on that side on which a rotation of less than 180° from the first vector to the second

appears positive.

AxB = A B sin (A, B) c. (9)

The vector product is equal in magnitude and direction to the vector which represents the parallelogram of which A and B are the two adjafcent sides. The necessary and sufficient con dition for the parallelism of two vectors neither of which

110

VECTOR ANALYSIS

vanishes is that their vector product vanishes. The com mutative laws do not hold.

AxB =

AxB = -BxA ixi =jxj = kxk = ixj =— jxi = k jxk = — kxj =i kxi = — ixk = j £2) i + (Aa Bl - A1

(10) (12)

AxB =

Bn Bo

(13) (13)'

The scalar triple product of three vectors [A B C] is equal to the volume of the parallelepiped of which A, B, C are three edges which meet in a point.

[AB C] = A-BxC = B.CxA = C-AxB = AxB-C = BxC«A = Cx A-B [ABC] =- [A OB].

(15)'

(16)'

The dot and the cross in a scalar triple product may be inter changed and the order of the letters may be permuted cyclicly without altering the value of the product ; but a change of cyclic order changes the sign.

[ABC] =

(18)'

[ABC] =

[a be]

(19)'

DIRECT AND SKEW PRODUCTS OF VECTORS 111 If the component of B perpendicular to A be B",

B,, = _AX(AXB)

A*A

Ax(BxC) = A-C B - A-B C (24)

(AxB)xC = A-C B - C-B A (24)'

(AxB>(CxD) = A.C B-D - A-D B-C (25)

(AxB)x(CxD) = [A CD] B- [BCD] A

= [ABD] C-[ABC] D. (26)

The equation which subsists between four vectors A, B, C, D is

[BCD] A-[CDA]B + [DAB] C- [ABC] D = 0. (27)

Application of formulae of vector analysis to obtain the for mulae of Plane and Spherical Trigonometry.

The system of vectors a', V, c' is said to be reciprocal to the system of three non-coplanar vectors a, b, c

bxc cxa axb

when a'= _ . ., b' = =— -=> *' = ——• (29)

[a be] [abc] [abc]

A vector r may be expressed in terms of a set of vectors and its reciprocal in two similar ways

r = r.a' a + r.V b + r-c' c (30)

r = r-aa' + r.bb' + r.cc'. (31)

The necessary and sufficient conditions that the two systems of non-coplanar vectors a, b, c and a', b', c' be reciprocals is that

a'.a = V»b = c'«c = 1 a'.b = a'-c = b'.c = b'.a = c'-a = e'-b = 0.

If a', b', c' form a system reciprocal to a, b, c ; then a, b, c will form a system reciprocal to a', b', c'.

112 VECTOR ANALYSIS

P.A P.B P.C

a-A ft-B a-c

R.A B-B R-C

[PaK][ABC] =

(34)

The system i, j, k is its own reciprocal and if conversely a system be its own reciprocal it must be a right or left handed system of three mutually perpendicular unit vectors. Appli cation of the theory of reciprocal systems to the solution of scalar and vector equations of the first degree in an unknown vector. The vector equation of a plane is

r-A = a. (36)

Applications of the methods developed in Chapter II., to the treatment of a system of forces acting on a rigid body and in particular to the reduction of any system of forces to a single force and a couple of which the plane is perpendicular to that force. Application of the methods to the treatment of instantaneous motion of a rigid body obtaining

v = v0 + a x r (46)

where v is the velocity of any point, v0 a translational veloc ity in the direction a, and a the vector angular velocity of ro tation. Further application of the methods to obtain the conditions for equilibrium by making use of the principle of virtual velocities. Applications of the method to obtain the relations which exist between the nine direction cosines of the angles between two systems of mutually orthogonal axes. Application to special problems in geometry including the form under which plane coordinates make their appear ance in vector analysis and the method by which planes (as distinguished from finite plane areas) may be represented by vectors.

DIRECT AND SKEW PRODUCTS OF VECTORS 113

EXERCISES ON CHAPTER II Prove the following reduction formulae

1. Ax{Bx(CxD)} = [ACD]B-A-BCxD

= B«D AxC — B-C AxD.

2. [AxB CxD ExF] = [ABD] [CEF]- [ABC] [DBF]

= [ABE] [FCD] - [ABF] [BCD] = [CD A] [BEF] - [CDB] [AEF].

3. [AxB BxC CxA] = [ABC]2.

P.A P.B P

4 [PQE] (AxB) =

Q.B Q

R»A R«B R

5. Ax(BxC) + Bx(CxA) + Cx(AxB) = 0.

6. [AxP Bxtt CxR] + [Axtt BxR CxP]

+ [AxR BxP Cxtt] = 0.

7. Obtain formula (34) in the text by expanding

[(AxB)xP].[Cx(ttxR)] in two different