Aula de Gibbs

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Aula de Gibbs

  1. 1. STAT
  2. 2. 05tcentennialVECTOR ANALYSIS
  3. 3. Bicentennial publicationsWith the approval of the President and Fellowsof Yale University, a seriesof volumes has beenprepared by a number of the Professors and Instructors, to be issued in connection with theBicentennial Anniversary, as a partial indication of the character of the studies in which theUniversity teachers are engaged.This series of volumes isrespectfully dedicated toraDuate$ of tfc
  4. 4. /VECTOR ANALYSIS^A TEXT-BOOK FOR THE USE OF STUDENTSOF MATHEMATICS AND PHYSICSFOUNDED UPON THE LECTURES OFJ. WILLARD GIBBS, PH.D., LL.D.Formerly Professor of Mathematical Physics in Yale UniversityBYEDWIN BIDWELL WILSON, PH.D.Professor of Vital Statistics inHarvard School of Public HealthNEW HAVENYALE UNIVERSITY PRESS
  5. 5. Copyright, 1901 and 1929BY YALE UNIVERSITYPublished, December, 1901Second Printing, January, 19/3Third Printing, July, 1916fourth Printing^ April, 1922Fifth Printing, October, 1925Sixth Printing, April, 1020Seventh Printing, October, 1951Eighth Printing, April, 1943Ninth Printing, April, 1947All rights reserved. This book may not be reproduced, in whole or in part, in any form, except by written permission from the publishers.PRINTED IN THE UNITED STATES OF AMERICA
  6. 6. PKEFACB BY PROFESSOR GIBBSSINCE the printing of a short pamphlet on the Elements ofVector Analysis in the years 1881-84, never published, butsomewhat widely circulated among those who were known tobe interested in the subject, the desire has been expressedin more than one quarter, that the substance of that treatise, perhaps in fuller form, should be made accessible tothe public.As, however, the years passed without my finding theleisure to meet this want, which seemed a real one, I wasvery glad to have one of the hearers of my course on VectorAnalysis in the year 1899-1900 undertake the preparation ofa text-book on the subject.I have not desired that Dr. Wilson should aim simplyat the reproduction of my lectures, but rather that he shoulduse his own judgment in all respects for the production of atext-book in which the subject should be so illustrated by anadequate number of examples as to meet the wants of students of geometry and physics.J. WILLARD GIBBS.YALE UNIVERSITY, September, 1901.
  7. 7. GENERAL PREFACEWHEN I undertook to adapt the lectures of Professor Gibbson VECTOR ANALYSIS for publication in the Yale Bicentennial Series, Professor Gibbs himself was already so fullyengaged upon his work to appear in the same series, ElementaryPrinciples in Statistical Mechanics, that it was understood nomaterial assistance in the composition of this book could beexpected from him. For this reason he wished me to feelentirely free to use my own discretion alike in the selectionof the topics to be treated and in the mode of treatment.It has been my endeavor to use the freedom thus grantedonly in so far as was necessary for presenting his method intext-book form.By far the greater part of the material used in the following pages has been taken from the course of lectures onVector Analysis delivered annually at the University byProfessor Gibbs. Some use, however, has been made of thechapters on Vector Analysis in Mr. Oliver Heaviside s Electromagnetic Theory (Electrician Series, 1893) and in ProfessorFoppl s lectures on Die Maxwell sche Theorie der Electricitdt(Teubner, 1894). My previous study of Quaternions hasalso been of great assistance.The material thus obtained has been arranged in the waywhich seems best suited to easy mastery of the subject.Those Arts, which it seemed best to incorporate in thetext but which for various reasons may well be omitted atthe firstreading have been marked with an asterisk (*). Numerous illustrative examples have been drawn from geometry,mechanics, and physics. Indeed, a large part of the text hasto do with applications of the method. These applicationshave not been set apart in chapters by themselves, but have
  8. 8. x GENERAL PREFACEbeen distributed throughout the body of the book as fast asthe analysis has been developed sufficiently for their adequatetreatment. It is hoped that by this means the reader may bebetter enabled to make practical use of the book. Great carehas been taken in avoiding the introduction of unnecessaryideas, and in so illustrating each idea that is introduced asto make its necessity evident and its meaning easy to grasp.Thus the book is not intended as a complete exposition ofthe theory of Vector Analysis, but as a text-book from whichso much of the subject as may be required for practical applications may be learned. Hence a summary, including a listof the more important formulae, and a number of exercises,have been placed at the end of each chapter, and many lessessential points in the text have been indicated rather thanfully worked out, in the hope that the reader will supply thedetails. The summary may be found useful in reviews andfor reference.The subject of Vector Analysis naturally divides itself intothree distinct parts. First, that which concerns addition andthe scalar and vector products of vectors. Second, that whichconcerns the differential and integral calculus in its relationsto scalar and vector functions. Third, that which containsthe theory of the linear vector function. The first part isa necessary introduction to both other parts. The secondand third are mutually independent. Either may be takenup first. For practical purposes in mathematical physics thesecond must be regarded as more elementary than the third.But a student not primarily interested in physics would naturally pass from the first part to the third, which he wouldprobably find more attractive and easy than the second.Following this division of the subject, the main body ofthe book is divided into six chapters of which two deal witheach of the three parts in the order named. Chapters I. andII. treat of addition, subtraction, scalar multiplication, andthe scalar and vector products of vectors. The expositionhas been made quite elementary. It can readily be understood by and isespecially suited for such readers as have aknowledge of only the elements of Trigonometry and Ana-
  9. 9. GENERAL PREFACE xilytic Geometry. Those who are well versed in Quaternionsor allied subjects may perhaps need to read only the summaries. Chapters III. and IV. contain the treatment ofthose topics in Vector Analysis which, though of less valueto the students of pure mathematics, are of the utmost importance to students of physics. Chapters V. and VI. deal withthe linear vector function. To students of physics the linearvector function is of particular importance in the mathematical treatment of phenomena connected with non-isotropicmedia ; and to the student of pure mathematics this part ofthe book will probably be the most interesting of all, owingto the fact that it leads to Multiple Algebra or the Theoryof Matrices. A concluding chapter, VII., which contains thedevelopment of certain higher parts of the theory, a numberof applications, and a short sketch of imaginary or complexvectors, has been added.In the treatment of the integral calculus, Chapter IV.,questions of mathematical rigor arise. Although moderntheorists are devoting much time and thought to rigor, andalthough they will doubtless criticise this portion of the bookadversely, it has been deemed best to give but little attentionto the discussion of this subject. And the more so for thereason that whatever system of notation be employed questions of rigor are indissolubly associated with the calculusand occasion no new difficulty to the student of VectorAnalysis, who must first learn what the facts are and maypostpone until later the detailed consideration of the restrictions that are put upon those facts.Notwithstanding the efforts which have been made duringmore than half a century to introduce Quaternions intophysics the fact remains that they have not found wide favor.On the other hand there has been a growing tendency especially in the last decade toward the adoption of some form ofVector Analysis. The works of Heaviside and Foppl referred to before may be cited in evidence. As yet howeverno system of Vector Analysis which makes any claim tocompleteness has been published. In fact Heaviside says :"I am in hopes that the chapter which I now finish may
  10. 10. xiiGENERAL PREFACEserve as a stopgap till regular vectorial treatises come to bewritten suitable for physicists, based upon the vectorial treatment of vectors" (Electromagnetic Theory, Vol. I., p. 305).Elsewhere in the same chapter Heaviside has set forth theclaims of vector analysis as against Quaternions, and othershave expressed similar views.The keynote, then, to any system of vector analysis mustbe its practical utility. This, I feel confident, was ProfessorGibbs s point of view in building up his system. He uses itentirely in his courses on Electricity and Magnetism and onElectromagnetic Theory of Light. In writing this book Ihave tried to present the subject from this practical standpoint, and keep clearly before the reader s mind the questions: What combinations or functions of vectors occur inphysics and geometry ? And how may these be representedsymbolically in the way best suited to facile analytic manipulation ? The treatment of these questions in modern bookson physics has been too much confined to the addition andsubtraction of vectors. This is scarcely enough. It hasbeen the aim here to give also an exposition of scalar andvector products, of the operator y, of divergence and curlwhich have gained such universal recognition since the appearance of Maxwell s Treatise on Electricity and Magnetism,of slope, potential, linear vector function, etc., such as shallbe adequate for the needs of students of physics at thepresent day and adapted to them.It has been asserted by some that Quaternions, VectorAnalysis, and all such algebras are of little value for investigating questions in mathematical physics. Whether thisassertion shall prove true or not, one may still maintain thatvectors are to mathematical physics what invariants are togeometry. As every geometer must be thoroughly conversant with the ideas of invariants, so every student of physicsshould be able to think in terms of vectors. And there isno way in which he, especially at the beginning of his scientific studies, can come to so true an appreciation of theimportance of vectors and of the ideas connected with themas by working in Vector Analysis and dealing directly with
  11. 11. GENERAL PREFACE xiiithe vectors themselves. To those that hold these views thesuccess of Professor Foppl s Vorlesungen uber TechnischeMechanik (four volumes, Teubner, 1897-1900, already in asecond edition), in which the theory of mechanics is developed by means of a vector analysis, can be but an encouraging sign.I take pleasure in thanking my colleagues, Dr. M. B. Porterand Prof. H. A. Bumstead, for assisting me with the manuscript. The good services of the latter have been particularlyvaluable in arranging Chapters III. and IV* in their presentform and in suggesting many of the illustrations used in thework. I am also under obligations to my father, Mr. EdwinH. Wilson, for help in connection both with the proofs andthe manuscript. Finally, I wish to express my deep indebtedness to Professor Gibbs. For although he has been sopreoccupied as to be unable to read either manuscript orproof, he has always been ready to talk matters over withme, and it is he who has furnished me with inspiration sufficient to carry through the work.EDWIN BIDWELL WILSON.YALE UNIVERSITY, October, 1901.PREFACE TO THE SECOND EDITIONTHE only changes which have been made in this edition area few corrections which my readers have been kind enough topoint out to me.E. B. W.
  12. 12. TABLE OF CONTENTSPAGEPREFACE BY PROFESSOR GIBBS viiGENERAL PREFACE ixCHAPTER IADDITION AND SCALAR MULTIPLICATIONARTS.1-3 SCALARS AND VECTORS 14 EQUAL AND NULL VECTORS 45 THE POINT OF VIEW OF THIS CHAPTER 66-7 SCALAR MULTIPLICATION. THE NEGATIVE SIGN .... 78-10 ADDITION. THE PARALLELOGRAM LAW 811 SUBTRACTION 1112 LAWS GOVERNING THE FOREGOING OPERATIONS .... 1213-16 COMPONENTS OF VECTORS. VECTOR EQUATIONS .... 1417 THE THREE UNIT VECTORS 1, j, k 1818-19 APPLICATIONS TO SUNDRY PROBLEMS IN GEOMETRY. . . 2120-22 VECTOR RELATIONS INDEPENDENT OF THE ORIGIN ... 2723-24 CENTERS OF GRAVITY. BARYCENTRIC COORDINATES . . 3925 THE USE OF VECTORS TO DENOTE AREAS 46SUMMARY OF CHAPTER i 51EXERCISES ON CHAPTER i . . 52CHAPTER IIDIRECT AND SKEW PRODUCTS OF VECTORS27-28 THE DIRECT, SCALAR, OR DOT PRODUCT OF TWO VECTORS 5529-30 THE DISTRIBUTIVE LAW AND APPLICATIONS 5831-33 THE SKEW, VECTOR, OR CROSS PRODUCT OF TWO VECTORS 6034-35 THE DISTRIBUTIVE LAW AND APPLICATIONS 6336 THE TRIPLE PRODUCT A* B C 67
  13. 13. XVI CONTENTSARTS. PAGE37-38 THE SCALAR TRIPLE PRODUCT A* B X C OR [ABC] . . 6839-40 THE VECTOR TRIPLE PRODUCT A X (B X C) 7141-42 PRODUCTS OF MORE THAN THREE VECTORS WITH APPLICATIONS TO TRIGONOMETRY 7543-45 RECIPROCAL SYSTEMS OF THREE VECTORS 8146-47 SOLUTION OF SCALAR AND VECTOR EQUATIONS LINEAR INAN UNKNOWN VECTOR 8748-50 SYSTEMS OF FORCES ACTING ON A RIGID BODY .... 9251 KINEMATICS OF A RIGID BODY 9752 CONDITIONS FOR EQUILIBRIUM OF A RIGID BODY ... 10153 RELATIONS BETWEEN TWO RIGHT-HANDED SYSTEMS OFTHREE PERPENDICULAR UNIT VECTORS 10454 PROBLEMS IN GEOMETRY. PLANAR COORDINATES . . . 106SUMMARY OF CHAPTER n 109EXERCISES ON CHAPTER n 113CHAPTER IIITHE DIFFERENTIAL CALCULUS OF VECTORS55-56 DERIVATIVES AND DIFFERENTIALS OF VECTOR FUNCTIONSWITH RESPECT TO A SCALAR VARIABLE 11557 CURVATURE AND TORSION OF GAUCHE CURVES .... 12058-59 KINEMATICS OF A PARTICLE. THE HODOGRAPH . . . 12560 THE INSTANTANEOUS AXIS OF ROTATION 13161 INTEGFATION WITH APPLICATIONS TO KINEMATICS . . . 13362 SCALAR FUNCTIONS OF POSITION IN SPACE 13663-67 THE VECTOR DIFFERENTIATING OPERATOR V 13868 THE SCALAR OPERATOR A V 14769 VECTOR FUNCTIONS OF POSITION IN SPACE 14970 THE DIVERGENCE V* AND THE CURL VX 15071 INTERPRETATION OF THE DIVERGENCE V 15272 INTERPRETATION OF THE CURL V X 15573 LAWS OF OPERATION OF V> V *> V X 15774-76 THE PARTIAL APPLICATION OF V- EXPANSION OF A VECTOR FUNCTION ANALOGOUS TO TAYLOR S THEOREM.APPLICATION TO HYDROMECHANICS 15977 THE DIFFERENTIATING OPERATORS OF THE SECOND ORDER 16678 GEOMETRIC INTERPRETATION OF LAPLACE S OPERATORV* V AS THE DISPERSION 170SUMMARY OF CHAPTER in 172EXERCISES ON CHAPTER in 177
  14. 14. CONTENTS xviiCHAPTER IVTHE INTEGRAL CALCULUS OF VECTORSARTS. PAGE79-80 LINE INTEGRALS OF VECTOR FUNCTIONS WITH APPLICATIONS 17981 GAUSS S THEOREM 18482 STOKES S THEOREM 18783 CONVERSE OF STOKES S THEOREM WITH APPLICATIONS . 19384 TRANSFORMATIONS OF LINE, SURFACE, AND VOLUME INTEGRALS. GREEN S THEOREM 19785 REMARKS ON MULTIPLE-VALUED FUNCTIONS 20086-87 POTENTIAL. THE INTEGRATING OPERATOR " POT ". . 20588 COMMUTATIVE PROPERTY OF POT AND V 21189 REMARKS UPON THE FOREGOING 21590 THE INTEGRATING OPERATORS "NEW," "LAP,"" MAX "22291 RELATIONS BETWEEN THE INTEGRATING AND DIFFERENTIATING OPERATORS 22892 THE POTENTIAL " POT "is A SOLUTION OF POISSON SEQUATION 23093-94 SOLENOIDAL AND IRROTATIONAL PARTS OF A VECTORFUNCTION. CERTAIN OPERATORS AND THEIR INVERSE . 23495 MUTUAL POTENTIALS, NEWTONIANS, LAPLACIANS, ANDMAXWELLIANS 24096 CERTAIN BOUNDARY VALUE THEOREMS 243SUMMARY OF CHAPTER iv 249EXERCISES ON CHAPTER iv 255CHAFIER VLINEAR VECTOR FUNCTIONS97-98 LINEAR VECTOR FUNCTIONS DEFINED 26099 DYADICS DEFINED 264100 ANY LINEAR VECTOR FUNCTION MAY BE REPRESENTEDBY A DYADIC. PROPERTIES OF DYADICS .... 266101 THE NONION FORM OF A DYADIC 269102 THE DYAD OR INDETERMINATE PRODUCT OF TWO VECTORS IS THE MOST GENERAL. FUNCTIONAL PROPERTYOF THE SCALAR AND VECTOR PRODUCTS 271108-104 PRODUCTS OF DYADICS 276105-107 DEGREES OF NULLITY OF DYADICS 282108 THE IDEMFACTOR 288
  15. 15. XV111 CONTENTSARTS. PAGE109-110 RECIPROCAL DYADICS. POWERS AND ROOTS OF DYADICS 290111 CONJUGATE DYADICS. SELF-CONJUGATE AND ANTI-SELF-CONJUGATE PARTS OF A DYADIC 294112-114 ANTI-SELF-CONJUGATE DYADICS. THE VECTOR PRODUCT. QUADRANTAL VER8ORS 297115-116 REDUCTION OF DYADICS TO NORMAL FORM .... 302117 DOUBLE MULTIPLICATION OF DYADICS 306118-119 THE SECOND AND THIRD OF A DYADIC . . ... 310120 CONDITIONS FOR DIFFERENT DEGREES OF NULLITY . 313121 NONION FORM. DETERMINANTS 315122 INVARIANTS OF A DYADIC. THE HAMILTON-CAYLEYEQUATION .319SUMMARY OF CHAPTER v 321EXERCISES ON CHAPTER v 329CHAPTER VIROTATIONS AND STRAINS123-124 HOMOGENEOUS STRAIN REPRESENTED BY A DYADIC . 332125-126 ROTATIONS ABOUT A FIXED POINT. VERSORS . . . 334127 THE VECTOR SEMI-TANGENT OF VERSION 339128 BlQUADRANTAL VERSORS AND THEIR PRODUCTS . . . 343129 CYCLIC DYADICS 347130 RIGHT TENSORS 351131 TONICS AND CYCLOTONICS 353132 REDUCTION OF DYADICS TO CANONICAL FORMS, TONICS,CYCLOTONICS, SIMPLE AND COMPLEX SHEARERS . . 356SUMMARY OF CHAPTER vi 368CHAPTER VIIMISCELLANEOUS APPLICATIONS136-142 QUADRIC SURFACES 372143-146 THE PROPAGATION OF LIGHT IN CRYSTALS .... 392147-148 VARIABLE DYADICS 403149-157 CURVATURE OF SURFACES 411158-162 HARMONIC VIBRATIONS AND BIVECTORS .... 426
  16. 16. VECTOR ANALYSIS
  17. 17. VECTOR ANALYSISCHAPTER IADDITION AND SCALAR MULTIPLICATION1.]IN mathematics and especially in physics two verydifferent kinds of quantity present themselves. Consider, forexample, mass, time, density, temperature, force, displacementof a point, velocity, and acceleration. Of these quantitiessome can be represented adequately by a single numbertemperature, by degrees on a thermometric scale ; time, byyears, days, or seconds ; mass and density, by numerical val- .ues which are wholly determined when the unit of the scaleis fixed. On the other hand the remaining quantities are notcapable of such representation. Force to be sure is said to beof so many pounds or grams weight; velocity, of so manyfeet or centimeters per second. But in addition to this eachof them must be considered as having direction as well asmagnitude. A force points North, South, East, West, up,down, or in some intermediate direction. The same is trueof displacement, velocity, and acceleration. No scale of numbers can represent them adequately. It can represent onlytheir magnitude, not their direction.2.] Definition : A vector is a quantity which is consideredas possessing direction as well as magnitude.Definition : A scalar is a quantity which is considered as possessing magnitude but no direction.
  18. 18. 2 VECTOR ANALYSISThe positive and negative numbers of ordinary algebra are thetypical scalars. For this reason the ordinary algebra is calledscalar algebra when necessary to distinguish it from the vectoralgebra 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 astraight line and take a new position Pf.This change of position is represented by theline PP. The magnitude of the displacement is the length of PP1; the direction ofit is the direction of the line PP1from P toP1. Next consider a displacement not of one,but of all the points in space. Let all thepoints move in straight lines in the same direction and for thesame distance D. This is equivalent to shifting space as arigid body in that direction through the distance D withoutrotation. Such a displacement is called a translation. Itpossesses direction and magnitude. When space undergoesa translation T, each point of space undergoes a displacementequal to T in magnitude and direction; and conversely ifthe displacement PP which any one particular point P suffers in the translation T is known, then that of any otherpoint Q is also known : for Q Q must be equal and parallelto PP.The translation T isrepresented geometrically or graphicallyby an arrow T (Fig. 1) of which the magnitude and directionare equal to those of the translation. The absolute positionof this arrow in space isentirely immaterial. Technically thearrow is called a stroke. Its tail or initial point is its origin;and its head or final point, its terminus. In the figure theorigin isdesignated by and the terminus by T. This geometric quantity, a stroke, is used as the mathematical symbolfor all vectors, just as the ordinary positive and negative numbers are used as the symbols for all scalars.
  19. 19. ADDITION AND SCALAR MULTIPLICATION 3*3.] As examples of scalar quantities mass, time, density, and temperature have been mentioned. Others are distance, 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 itsown distinguishing characteristics, as an example of whichits dimensions in the sense well known to physicists maybe cited. A distance 3, a time 3, a work 3, etc., are verydifferent. The magnitude 3 is, however, a property commonto them all perhaps the only one. Of all scalar quanti-tities pure number is the simplest. It implies nothing butmagnitude. It is the scalar par excellence and consequentlyit is used as the mathematical symbol for all scalars.As examples of vector quantities force, displacement, velocity, and acceleration have been given. Each of these hasother characteristics than those which belong to a vector pureand simple. The concept of vector involves two ideas andtwo alone magnitude of the vector and direction of thevector. But force is more complicated. When it is appliedto a rigid body the line in which it acts must be taken intoconsideration; magnitude and direction alone do not suffice. And in case it is applied to a non-rigid body the pointof application of the force is as important as the magnitude ordirection. Such is frequently true for vector quantities otherthan force. Moreover the question of dimensions is presentas in the case of scalar quantities. The mathematical vector,the stroke, which is the primary object of consideration inthis book, abstracts from all directed quantities their magnitude and direction and nothing but these ; just as the mathematical scalar, pure number, abstracts the magnitude andthat alone. Hence one must be on his guard lest fromanalogy he attribute some properties to the mathematicalvector which do not belong to it ; and he must be even morecareful lest he obtain erroneous results by considering the
  20. 20. 4 VECTOR ANALYSISvector quantities of physics as possessing no properties otherthan those of the mathematical vector. For example it wouldnever do to consider force and its effects as unaltered byshifting it parallel to itself. This warning may not benecessary, yet it may possibly save some confusion.4.] Inasmuch as, taken in its entirety, a vector or strokeis but a single concept, it may appropriately be designated byone letter. Owing however to the fundamental differencebetween scalars and vectors, it is necessary to distinguishcarefully the one from the other. Sometimes, as in mathematical physics, the distinction is furnished by the physicalinterpretation. Thus if n be the index of refraction itmust be scalar ; m, the mass, and , the time, are alsoscalars ;but /, the force, and a, the acceleration, arevectors. When, however, the letters are regarded merelyas symbols with no particular physical significance sometypographical difference must be relied upon to distinguishvectors from scalars. Hence in this book Clarendon type isused for setting up vectors and ordinary type for scalars.This permits the use of the same letter differently printedto represent the vector and its scalar magnitude.1Thus ifC be the electric current in magnitude and direction, C maybe used to represent the magnitude of that current ;if g bethe vector acceleration due to gravity, g may be the scalarvalue of that acceleration ; if v be the velocity of a movingmass, v may be the magnitude of that velocity. The use ofClarendons to denote vectors makes it possible to pass fromdirected quantities to their scalar magnitudes by a merechange in the appearance of a letter without any confusingchange in the letter itself.Definition : Two vectors are said to be equal when they havethe same magnitude and the same direction.1This convention, however, is by no means invariably followed. In someinstances it would prove just as undesirable as it is convenient in others. It ischiefly valuable in the application of vectors to physics.
  21. 21. ADDITION AND SCALAR MULTIPLICATION 5The equality of two vectors A and B is denoted by theusual sign =. Thus A = BEvidently a vector or stroke is not altered by shifting itabout parallel to itself in space. Hence any vector A = PPr(Fig. 1) may be drawn from any assigned point as origin ;for the segment PPfmay be moved parallel to itself untilthe point P falls upon the point and P upon some point T.In this way all vectors in space may be replaced by directedsegments radiating from one fixed point 0. Equal vectorsin space will of course coincide, when placed with their termini 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 scalarsare necessary. These may be chosen in a variety of ways.If r, </>,be polar coordinates in space any vector r drawnwith its origin at the origin of coordinates may be representedby the three scalars r, </>,6 which determine the terminus ofthe vector.r~(r,*,0).Or if #, y9 z be Cartesian coordinates in space a vector r maybe considered as given by the differences of the coordinates a/,y i zfof its terminus and those #, y, z of its origin.r~ (xrx,yry,zrz).If in particular the origin of the vector coincide with theorigin of coordinates, the vector will be represented by thethree coordinates of its terminusr -(* ,*, ,* )When two vectors are equal the three scalars which represent them must be equal respectively each to each. Henceone vector equality implies three scalar equalities.
  22. 22. 6 VECTOR ANALYSISDefinition : A vector A is said to be equal to zero when itsmagnitude A is zero.Such a vector A is called a null or zero vector and is writtenequal to naught in the usual manner. ThusA = if A = 0.All null vectors are regarded as equal to each other withoutany considerations of direction.In fact a null vector from a geometrical standpoint wouldbe represented by a linear segment of length zero that is tosay, by a point. It consequently would have a wholly indeterminate direction or, what amounts to the same thing, none atall. If, however, it be regarded as the limit approached by avector of finite length, it might be considered to have thatdirection which is the limit approached by the direction of thefinite vector, when the length decreases indefinitely and approaches zero as a limit. The justification for disregardingthis direction and looking upon all null vectors as equal isthat when they are added (Art. 8) to other vectors no changeoccurs and when multiplied (Arts. 27, 31) by other vectorsthe product is zero.5.] In extending to vectors the fundamental operationsof algebra and arithmetic, namely, addition, subtraction, andmultiplication, 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 andeasily by looking to physics (for in that science vectors continually present themselves) and by observing how suchquantities are treated there. If then A be a given displacement, force, or velocity, what is two, three, or in general xtimes 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 theequivalent of A and B taken together ? The obvious answersto these questions suggest immediately the desired definitions.
  23. 23. ADDITION AND SCALAR MULTIPLICATION 1Scalar Multiplication6.] Definition: A vector is said to be multiplied by apositive scalar when its magnitude is multiplied by that scalarand its direction is left unalteredThus if v be a velocity of nine knots East by North, 2 timesv is a velocity of twenty-one knots with the direction stillEast 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 akilogram. The direction in both cases is vertically downward.If A be the vector and x the scalar the product of x and A isdenoted as usual byx A or A x.It is, however, more customary to place the scalar multiplierbefore the multiplicand A. This multiplication by a scalaris called scalar multiplication, and it follows the associative lawx (y A) = (x y) A = y (x A)as in ordinary algebra and arithmetic. This statement is immediately obvious when the fact is taken into considerationthat scalar multiplication does not alter direction but merelymultiplies the length.Definition : A unit vector is one whose magnitude isunity.Any vector A may be looked upon as the product of a unitvector a in its direction by the positive scalar A, its magnitude.A = A a = a A.The unit vector a may similarly be written as the product ofA by I/A or as the quotient of A and A.1 Aa = ^ A = -IA A
  24. 24. 8 VECTOR ANALYSIS7.] Definition : The negative sign, prefixed to a vectorreverses 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 thestroke A~B be A, the stroke B A, which is of the same lengthas A but which is in the direction from B to A instead offrom A to 5, will be A. Another illustration of the useof the negative sign may be taken from Newton s third lawof motion. If A denote an "action," A will denote the" reaction." The positive sign, + , may be prefixed to a vector to call particular attention to the fact that the directionhas not been reversed. The two signs + and when usedin connection with scalar multiplication of vectors follow thesame laws of operation as in ordinary algebra. These aresymbolically+ + = + ; +- = -; -+ = -; = +;(ra A) = m ( A).The interpretation is obvious.Addition and Subtraction8.] The addition of two vectors or strokes may be treatedmost simply by regarding them as defining translations inspace (Art. 2), Let S be one vector and T the other. Let Pbe a point of space (Fig. 2). The translation S carries P into P1such that theline PP1is equal to S in magnitude anddirection. The transformation T will thencarry P1into P11the line P P" beingparallel to T and equal to it in magnitude.FIG. 2. Consequently the result of S followed byT is to carry the point P into the pointP". If now Q be any other point in space, S will carry Qinto Q such that Q~Qr= S and T will then carry Qfinto Q"
  25. 25. ADDITION AND SCALAR MULTIPLICATIONsuch that Q Q" = T. Thus S followed by T carries Q into Q".Moreover, the triangle Q QfQ" is equal to PP P". Forthe two sides Q Qfand Q Q", being equal and parallel to Sand T respectively, must be likewise parallel to P P1andP P" respectively which are also parallel to S and T. Hencethe third sides of the triangles must be equal and parallelThat isQ Q" is equal and parallel to PP".As Q is any point in space this is equivalent to saying thatby means of S followed by T all points of space are displacedthe same amount and in the same direction. This displacement is therefore a translation. Consequently the twotranslations S and T are equivalent to a single translation R.Moreoverif S = PP and T = P P", then R = PP".The stroke R is called the resultant or sum of the twostrokes S and T to which it is equivalent. This sum is denoted in the usual manner byR = S + T.From analogy with the sum or resultant of two translationsthe following definition for the addition of any two vectors islaid down.Definition : The sum or resultant of two vectors is foundby placing the origin of the second upon the terminus of thefirst and drawing the vector from the origin of the first to theterminus of the second.9.] Theorem. The order in which two vectors S and T areadded does not affect the sum.S followed by T gives precisely the same result as T followedby 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 Pinto P ". The line PP "is equal and parallel to PP". Con-
  26. 26. 10 VECTOR ANALYSISsequently the points P, P 9Pff,and Pm lie at the vertices ofa parallelogram. Hencepm pn js equal an(J par-allel to PP. Hence Scarries P"finto P". T followed by S therefore carries P into P" through Pwhereas S followed by Tcarries P into P" throughPm. The final result is ineither case the same. This may be designated symbolicallyby writingR = S + T = T + S.It is to be noticed that S = PP1and T = PPm are the two sidesof the parallelogram pprpp" which1have the point P ascommon origin ; and that JL=PP" is the diagonal drawnthrough P. This leads to another very common way ofstating the definition of the sum of two vectors.If two vectors be drawn from the same origin and a parallelogram be constructed upon them as sides, their sum will be thatdiagonal which passes through their common origin.This is the well-known "parallelogram law "according towhich the physical vector quantities force, acceleration, velocity, and angular velocity are compounded. It is important tonote that in case the vectors lie along the same line vectoraddition becomes equivalent to algebraic scalar addition. Thelengths of the two vectors to be added are added if the vectorshave the same direction ; but subtracted if they have opposite directions. In either case the sum has the same directionas that of the greater vector.10.] After the definition of the sum of two vectors hasbeen laid down, the sum of several may be found by addingtogether 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-
  27. 27. ADDITION AND SCALAR MULTIPLICATION 11gle one. The final result is the same as that obtained by placingthe origin of each succeeding vector upon the terminus of thepreceding one and then drawing at once the vector fromthe origin of the first to the terminus of the last. In casethese two points coincide the vectors form a closed polygonand their sum is zero. Interpreted geometrically this statesthat if a number of displacements R, S, T are such that thestrokes R, S, T form the sides of a closed polygon taken inorder, then the effect of carrying out the displacements is nil.Each point of space is brought back to its starting point. Interpreted in mechanics it states that ifany number of forcesact at a point and ifthey form the sides of a closed polygontaken in order, then the resultant force is zero and the pointis in equilibrium under the action of the forces.The order of sequence of the vectors in a sum is of no consequence. This may be shown by proving that any two adjacent vectors may be interchanged without affecting the result.To showLet A = A, B = A B, C = B C, D = D, E = D E.Then_Let now B C1= D. Then C!B C D is a parallelogram andconsequently CfD = C. HenceOJ = A + B + D + C + E,which proves the statement. Since any two adjacent vectorsmay be interchanged, and since the sum may be arranged inany order by successive interchanges of adjacent vectors, theorder in which the vectors occur in the sum is immaterial.11.] Definition : A vector is said to be subtracted when itis added after reversal of direction. Symbolically,A - B = A + (- B).By this means subtraction is reduced to addition and needs
  28. 28. 12 VECTOR ANALYSISno special consideration. There is however an interesting andimportant way of representing the difference of two vectorsgeometrically.Let A = OA, B = 0IT(Fig. 4). Completethe parallelogram of which A and Bare the sides. Then the diagonal~OG = C is the sum A + B of thetwo vectors. Next complete theparallelogram of which A and B= OB are the sides. Then the diagonal 02) = !) will be the sum ofA and the negative of B. But thesegment OD is parallel and equalto BA. Hence BA may be taken as the difference to the twovectors A and B. This leads to the following rule : The difference of two vectors which are drawn from the same origin isthe vector drawn from the terminus of the vector to be subtracted to the terminus of the vector from which it is subtracted. Thus the two diagonals of the parallelogram, whichis constructed upon A and B as sides, give the sum and difference of A and B.12.] In the foregoing paragraphs addition, subtraction, andscalar multiplication of vectors have been defined and interpreted. To make the development of vector algebra mathematically exact and systematic it would now become necessaryto demonstrate that these three fundamental operations followthe same formal laws as in the ordinary scalar algebra, although from the standpoint of the physical and geometricalinterpretation _of vectors this may seem superfluous. Theselaws arem (n A) = n (m A) = (m n} A,(A + B) + C = A+ (B + C),II A + B r, B + A,III a (m + n) A = m A + n A,m (A + B) = m A + m B,III,- (A + B) = - A - B.
  29. 29. ADDITION AND SCALAR MULTIPLICATION 131 is the so-called law of association and commutation ofthe scalar factors in scalar multiplication.I6 is the law of association for vectors in vector addition. Itstates that in adding vectors parentheses may be inserted atany points without altering the result.11 is the commutative law of vector addition.IIIa is the distributive law for scalars in scalar multiplication.III6 is the distributive law for vectors in scalar multiplication.Ill, is the distributive law for the negative sign.The proofs of these laws of operation depend upon thosepropositions in elementary geometry which have to deal withthe first properties of the parallelogram and similar triangles.They will not be given here; but it issuggested that thereader work them out for the sake of fixing the fundamentalideas of addition, subtraction, and scalar multiplication moreclearly in mind. The result of the laws may be summed upin the statement :The laws which govern addition, subtraction, and scalarmultiplication of vectors are identical with those governing theseoperations in ordinary scalar algebra.It is precisely this identity of formal laws which justifiesthe extension of the use of the familiar signs =, +, andof arithmetic to the algebra of vectors and it is also thiswhich ensures the correctness of results obtained by operating with those signs in the usual manner. One caution onlyneed be mentioned. Scalars and vectors are entirely differentsorts of quantity. For this reason they can never be equatedto each other except perhaps in the trivial case where each iszero. For the same reason they are not to be added together.So long as this is borne in mind no difficulty need be anticipated from dealing with vectors much as if they were scalars.Thus from equations in which the vectors enter linearly with
  30. 30. 14 VECTOR ANALYSISscalar coefficients unknown vectors may be eliminated orfound by solution in the same way and with the same limitations as in ordinary algebra; for the eliminations and solutions depend solely on the scalar coefficients of the equationsand not at all on what the variables represent. If forinstanceaA + &B + cC + dD = 0,then A, B, C, or D may be expressed in terms of the otherthreeas D = --:OA + &B + cC).aAnd two vector equations such as3 A+ 4B=Eand 2 A + 3 B = Fyield by the usual processes the solutionsA=3E-4Fand B = 3 F - 2 E.Components of Vectors13.] Definition : Vectors are said to be collinear whenthey are parallel to the same line; coplanar, when parallelto the same plane. Two or more vectors to which no linecan be drawn parallel are said to be non-collinear. Three ormore vectors to which no plane can be drawn parallel aresaid to be non-coplanar. Obviously any two vectors arecoplanar.Any vector b collinear with a may be expressed as theproduct of a and a positive or negative scalar which is theratio of the magnitude of b to that of a. The sign is positivewhen b and a have the same direction ; negative, when theyhave opposite directions. If then OA = a, the vector r drawn
  31. 31. ADDITION AND SCALAR MULTIPLICATION 15from the origin to any point of the line A produced ineither direction isr = x a. (1)If x be a variable scalar parameter this equation may therefore be regarded as the (vector) equation of all points in theline OA. Let now B be any point notupon the line OA or that line producedin either direction (Fig. 5).Let OB = b. The vector b is surelynot of the form x a. Draw through B Flo 5"a line parallel to OA and Let R be anypoint upon it. The vector BE is collinear with a and isconsequently expressible as #a. Hence the vector drawnfrom to R is0~E=0~B + ITRor r = b + #a. (2)This equation may be regarded as the (vector) equation ofall the points in the line which is parallel to a and of whichB is one point.14.] Any vector r coplanar with two non-collinear vectorsa and b may be resolved into two components parallel to aand b respectively. This resolution maybe accomplished by constructing the parallelogram (Fig. 6) of which the sides areparallel to a and b and of which the diagonal is r. Of these components one isx a ; the other, y b. x and y are respectively the scalar ratios (taken with theproper sign) of the lengths of these components to the lengthsof a and b, Hencer = x a + y b (2)is a typical form for any vector coplanar with a and b. Ifseveral vectors r x, r 2, r 3 may be expressed in this form as
  32. 32. 16 VECTOR ANALYSIStheir sum r is thenrl= xla + yl b,r2= #2a + 2/2 b,r3= xza + 2/3b.+ (ft + ft + ft + )This is the well-known theorem that the components of asum of vectors are the sums of the components of thosevectors. If the vector r is zero each of its components mustbe zero. Consequently the one vector equation r = isequivalent to the two scalar equationsy + ft + ft + = (3)15.] Any vector r in space may be resolved into threecomponents parallel to any three given non-coplanar vectors.Let the vectors be a, b,and c. The resolutionmay then be accomplished by constructingthe parallelepiped (Fig.7) of which the edgesare parallel to a, b, andc and of which the diagonal is r. This par-allelopiped may bedrawn easily by passingthree planes parallel respectively to a and b, b and c, c and a through the originof the vector r ; and a similar set of three planes through itsterminus It. These six planes will then be parallel in pairsFIG. 7.
  33. 33. ADDITION AND SCALAR MULTIPLICATION 17and hence form a parallelepiped. That the intersections ofthe planes are lines which are parallel to a, or b, or c isobvious. The three components of r are x a, y b, and zc;where x, y, and z are respectively the scalar ratios (taken withthe proper sign) of the lengths of these components to thelength of a, b, and c. Hencer = # a + 7/b + zc (4)is a typical form for any vector whatsoever in space. Severalvectors r lfr 2,r 3. . .may be expressed in this form asrx= xla + ylb + z l c,r2= #2a + y2b + *2 c1*3= XZa + 2/3bTheir sum r is then1 =rl + r2 + F3 + * = 0*1 + *2 + XZ + a+ (2/i + 2/2+ 3/3 + )!>+ Ol +^2 + ^3+ "O -If the vector r is zero each of its three components is zero.Consequently the one vector equation r = is equivalent tothe three scalar equationsxl + #2 + #3 + - = v2/i + 2/2 + 2/3 + =y r = 0. (5)*i + *2 + % + = /Should the vectors all be coplanar with a and b, all the components parallel to c vanish. In this case therefore the aboveequations reduce to those given before.16.] If two equal vectors are expressed in terms of thesame three non-coplanar vectors, the corresponding scalar coefficients are equal.
  34. 34. 18 VECTOR ANALYSISLet r = r ,r = x9a + y1b + z c,Then x = x , y = yFor r - r = = (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. Inthat case one of the three vectors could be expressed in termsof the other two asc = m a + n b.Then r = #a + y b + s c = (a + m z) a + (y + TIz) b,r = x!a + y1b + 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 directionsa and b (supposed different) are zero.Hence x + mz = xr+ mzry -f n z = yf+ n z1.But this by no means necessitates x, y, z to be equal respectively to x y z1. In a similar manner if a and b were col-linear it is impossible to infer that their coefficients vanishindividually. 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, thecorresponding scalar coefficients are equal. But this is not necessarily 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 expressing vectors in terms of three given non-coplanar ones has beenexplained. The simplest set of three such vectors is the rect-
  35. 35. ADDITION AND SCALAR MULTIPLICATION 19angular system familiar in Solid Cartesian Geometry. Thisrectangular system may however be either of two very distincttypes. In one case (Fig. 8, first part) the Z-axis llies uponthat side of the X Y- plane on which rotation through a rightangle from the X-axis to the F-axis appears counterclockwiseor positive according to the convention adopted in Trigonometry. This relation may be stated in another form. If the Xaxis 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,,kRight-handedFIG. 8.Left-handedment is common in mathematical physics and engineering. Ifa right-handed screw be turned from the Xaxis to the F-axis it will advance along the (positive) Z-axis. Such a system of axes is called right-handed, positive, or counterclockwise.2It is easy to see that the F-axis lies upon that side ofthe ^X-plane on which rotation from the ^-axis to the X-axis is counterclockwise ; and the X-axis, upon that side of1By 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 consistsof the thumb, first finger, and second finger of the right hand. If the thumb andfirst finger be stretched out from the palm perpendicular to each other, and if thesecond 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,secondfinger.
  36. 36. 20 VECTOR ANALYSTSthe F^-plane on which rotation from the F-axis to the Z-axis is counterclockwise. Thus it appears that the relationbetween the three axes is perfectly symmetrical so long as thesame cyclic order XYZXY is observed. If a right-handedscrew is turned from one axis toward the next it advancesalong the third.In the other case (Fig. 8, second part) the ^-axis lies uponthat side of the XF-plane on which rotation through a rightangle from the JT-axis to the F-axis appears clockwise or negative. The F-axis then lies upon that side of the ^X-planeon which rotation from the ^-axis to the X-axis appearsclockwise and a similar statement may be made concerningthe X-axis in its relation to the F^-plane. In this case, too,the relation between the three axes isS3rmmetrical so longas the same cyclic order XYZX Y is preserved but it is justthe opposite of that in the former case. If a fe/Mianded screwis turned from one axis toward the next it advances alongthe third. Hence this system is called left-handed, negative,or clockwise.1The two systems are not superposable. They are symmetric. One is the image of the other as seen in amirror. If the JT- and F-axes of the two different systems besuperimposed, the ^-axes will point in opposite directions.Thus one system may be obtained from the other by reversingthe direction of one of the axes. A littlethought will showthat if two of the axes be reversed in direction the system willnot be altered, but if all three be so reversed it will be.Which of the two systems be used, matters little. But inasmuch as the formulae of geometry and mechanics differslightly in the matter of sign, it is advisable to settle once forall which shall be adopted. In this book the right-handed orcounterclockwise system will be invariably employed.1A left-handed system may be formed by the left hand just as a right-handedone was formed by the right.
  37. 37. ADDITION AND SCALAR MULTIPLICATION 21Definition : The three letters i, j, k will be reserved to denote three vectors of unit length drawn respectively in thedirections of the JT-, T-, and Z- axes of a right-handed rectangular system.In terms of these vectors, any vector may be expressed asr = xi + y] + zk. (6)The coefficients xy y, z are the ordinary Cartesian coordinatesof the terminus of r if its origin be situated at the origin ofcoordinates. The components of r parallel to the X-, F-, and^f-axes are respectivelyx i, y j, z k.The rotations about i from j to k, about j from k to i, andabout k from i to j are all positive.By means of these vectors i, j, k such a correspondence isestablished between vector analysis and the analysis in Cartesian coordinates that it becomes possible to pass at willfrom either one to the other. There isnothing contradictory between them. On the contrary it is often desirableor even necessary to translate the formulae obtained byvector methods into Cartesian coordinates for the sake ofcomparing them with results already known and it isstill more frequently convenient to pass from Cartesiananalysis to vectors both on account of the brevity therebyobtained and because the vector expressions show forth theintrinsic meaning of the formulae.Applications*18.J Problems in plane geometry may frequently be solvedeasily by vector methods. Any two non-collinear vectors inthe plane may be taken as the fundamental ones in terms ofwhich all others in that plane may be expressed. The originmay also be selected at pleasure. Often it is possible to
  38. 38. 22 VECTOR ANALYSISmake such an advantageous choice of the origin and fundamental vectors that the analytic work of solution ismateriallysimplified. The adaptability of the vector method is aboutthe same as that of oblique Cartesian coordinates with different scales upon the two axes.Example 1 : The line which joins one vertex of a parallelogram to the middle point of an opposite side trisects the diagonal (Fig. 9).Let A BCD be the parallelogram, BE the line joining thevertex B to the middle point E of the sideAD, R the point in which this line cuts thediagonal A C. To show A R is one third ofFlG 9AC. Choose A as origin, AB and AD as thetwo fundamental vectors S and T. ThenA C is the sum of S and T. Let AR = R. To showR = 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 AR to A C to be shown equalto.HenceT + 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.
  39. 39. ADDITION AND SCALAR MULTIPLICATION 23From which y = .Inasmuch as x is also -the line j&2? must be trisected asowell as the diagonal A C.Example 2 : If through any point within a triangle linesbe drawn parallel to the sides the sum of the ratios of theselines to their corresponding sides is 2.Let ABC be the triangle, R the point within it. ChooseA as origin, A B and A C as the two fundamental vectors Sand T. LetAR = R = w S + 7i T. (a)m S is the fraction of A B which is cut off by the line throughR parallel to A C. The remainder of A B must be the fraction (1 m) S. Consequently by similar triangles the ratio ofthe 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 Bitself is (1 n ).Next express R in terms of S and T S thethird 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 theline through R parallel to B C. Consequently by similar triangles the ratio of this line to BC itself is (m + n). Addingthe three ratios(1- m) + (1- n) + (m + ri)= 2,and the theorem is proved.Example 3 : If from any point within a parallelogram linesbe drawn parallel to the sides, the diagonals of the parallelograms thus formed intersect upon the diagonal of the givenparallelogram.Let A B CD be a parallelogram, R a point within it, KMand LN two lines through R parallel respectively to AB and
  40. 40. 24 VECTOR ANALYSISAD, the points K, Z, M, N lying upon the sides DA, AS,B C, CD respectively. To show that the diagonals KN andLM of the two parallelograms KRND and LBME meeton A C. Choose A as origin, A B and A D as the two fundamental vectors S and T. LetR = AB = m S 4- ft T,and let P be the point of intersection of KNwith 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 1m~m + n 1Substituting either of these solutions in the expression for P,the result isP^-^-^S + T),which shows that P is collinear with A C.*19.] Problems in three dimensional geometry may besolved in essentially the same manner as those in two dimensions. In this case there are three fundamental vectors interms of which all others can be expressed. The method ofsolution isanalogous to that in the simpler case. Two
  41. 41. ADDITION AND SCALAR MULTIPLICATION 25expressions for the same vector are usually found. The coefficients of the corresponding terms are equated. In this waythe equations between three unknown scalars are obtainedfrom which those scalars may be determined by solution andthen substituted in either of the expressions for the requiredvector. The vector method has the same degree of adaptability as the Cartesian method in which oblique axes withdifferent scales are employed. The following examples likethose in the foregoing section are worked out not so much fortheir intrinsic value as for gaining a familiarity with vectors.Example 1 : Let A B CD be a tetrahedron and P anypoint within it. Join the vertices to P and produce the linesuntil they intersect the opposite faces in A B , C1,Df. ToshowPA PB PC1PDA~AfTTB ~C~Of"Choose A as origin, and the edges A J?, A C, AD as thethree fundamental vectors B, C, D. Let the vector A P beP = A P=IE + raC + 7i D,Also A = A A = A B + BA .The vector BA 1is coplanar with WC = C B and BDD B. Hence it may be expressed in terms of them.A = B + ^ 1 (C-B)+y1 (D~B).Equating coefficients Jclm = x vHence &., =PA _ VZZ7~~&1I + m + nPA* JL-1and"^ 7
  42. 42. 26 VECTOR ANALYSISIn like manner A B = # 2C + y2Dand A B = ^t + B B = B + & 2 (P- B).Hence o;2C + y2D = B + A:2 (ZB + mC + ^D-Band = 1 + *, (J- 1),Hence 2 -i__-andIn the same way it may be shown thatPC .PL1CC* 3DAdding the four ratios the result isid JL vn -4- <w ^ _L 7 J_ w -I- 77 1Example % : To find a line which passes through a givenpoint and cuts two given lines in space.Let the two lines be fixed respectively by two points Aand B, C and D on each. Let be the given point. Chooseit as origin and letC = ~OC, D=d~D.Any point P of A B may be expressed asP= OP= 0~A + xA = A + x (B- A).Any point Q of CD may likewise be writtenIf the points P and Q lie in the same line through 0, P andare collinear That is
  43. 43. ADDITION AND SCALAR MULTIPLICATION 27Before it is possible to equate coefficients one of the fourvectors 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^^THence 1 x = z y /,x = zy m,= z [1 + y (n - 1)J.Hencemx =y =2 =ii-_________I + mSubstituting in P and ftI A+ m B+ mft =Either of these may be taken as defining a line drawn fromand cutting A B and CD.Vector Relations independent of the Origin20.] Example 1 : To divide a line A B in a given ratiom : n (Fig. 10).Choose any arbitrary point asorigin. Let OA = A and OB = B.To find the vector P = ~OP of whichthe terminus P divides AB in theratio m : n.mBFIG. 10.That is, P =B = A-f- 7in A + m Bn(B- A).(7)
  44. 44. 28 VECTOR ANALYSISThe components of P parallel to A and B are in inverse ratioto the segments A P and PB into which the line A B isdivided by the point P. If it should so happen that P dividedthe line AB externally, the ratio A P / PE would be negative, and the signs of m and n would be opposite, but theformula would hold without change if this difference of signin m and n be taken into account.Example 2 : To find the point of intersection of the mediansof a triangle.Choose the origin at random. Let A BC be the giventriangle. Let 0~A = A, ()B = B, and "00 = C. Let Af, ,Cbe respectively the middle points of the sides opposite thevertices A, B, (7. Let M be the point of intersection of themedians and M = M the vector drawn to it. Thenand~< = BAssuming that has been chosen outside of the plane of thetriangle so that A, B, C are non-coplanar, corresponding coefficients may be equated.Hence x = y -9 3Hence M =4 (A + B + C).
  45. 45. ADDITION AND SCALAR MULTIPLICATION 29The vector drawn to the median point of a triangle is equalto one third of the sum of the vectors drawn to the vertices.In the problems of which the solution has just been giventhe origin could be chosen arbitrarily and the result is independent of that choice. Hence it is even possible to disregard the origin entirely and replace the vectors A, B, C, etc.,by their termini A, B, C, etc. Thus the points themselvesbecome the subjects of analysis and the formulae readn A + m Bm + nand M=~(A + B + C).This is typical of a whole class of problems soluble by vectormethods. In fact any purely geometric relation between thedifferent parts of a figure must necessarily be independentof the origin assumed for the analytic demonstration. Insome cases, such as those in Arts. 18, 19, the position of theorigin may be specialized with regard to some crucial pointof the figure so as to facilitate the computation ; but in manyother cases the generality obtained by leaving the origin un-specialized and undetermined leads to a symmetry whichrenders the results just as easy to compute and more easyto remember.Theorem : The necessary and sufficient condition that avector equation represent a relation independent of the originis that the sum of the scalar coefficients of the vectors onone side of the sign of equality is equal to the sum of thecoefficients of the vectors upon the other side. Or if all theterms of a vector equation be transposed to one side leavingzero on the other, the sum of the scalar coefficients mustbe zero.Let the equation written in the latter form be
  46. 46. 30 VECTOR ANALYSISChange the origin from to by adding a constant vectorB = OO1to each of the vectors A, B, C, D---- The equationthen becomesa (A 4- B) + 6 (B + B) + c (C + B) + d (D + R) + - =If this is to be independent of the origin the coefficient of Bmust vanish. HenceThat this condition is fulfilled in the two examples citedis obvious.ifm + nIf M = (A f B + C),m + n m + nl3*21.] The necessary and sufficient condition that twovectors satisfy an equation, in which the sum of the scalarcoefficients is zero, is that the vectors be equal in magnitudeand in direction.First let a A + 6 B =and a + 6 = 0.It is of course assumed that not both the coefficients a and bvanish. If they did the equation would mean nothing. Substitute the value of a obtained from the second equation intothe first.-&A + 6B = 0.Hence A = B.
  47. 47. ADDITION AND SCALAR MULTIPLICATION 31Secondly if A and B are equal in magnitude and directionthe equationA-B =subsists between them. The sum of the coefficients is zero.The necessaiy and sufficient condition that three vectorssatisfy an equation, in which the sum of the scalar coefficientsis zero, is that when drawn from a common origin they terminate in the same straight line.1First let aA + 6B + cC =and a + b + c = 0.Not all the coefficients a, J, c, vanish or the equationswould be meaningless. Let c be a non-vanishing coefficient.Substitute the value of a obtained from the second equationinto the first.orHence the vector which joins the extremities of C and A iscollinear with that which joins the extremities of A and B.Hence those three points -4, -B, C lie on a line. Secondlysuppose three vectors A= OA, B = OB,G= 00 drawn fromthe same origin terminate in a straight line. Then thevectorsAB = B - A and A~C = C - Aare collinear. Hence the equationsubsists. The sum of the coefficients on the two sides isthe same.The necessary and sufficient condition that an equation,in which the sum of the scalar coefficients is zero, subsist1Vectors which have a common origin and terminate in one line are called byHamilton "termino-collinear:
  48. 48. 82 VECTOR ANALYSISbetween four vectors, is that if drawn from a common originthey terminate in one plane.1First let a A + 6B + cC + dV =and a + b + c + d = Q.Let d be a non-vanishing coefficient. Substitute the valueof 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 fourtermini A, B, (7, D of A, B, C, D lie in one plane. Secondlysuppose 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 coplanar vectors. One of them may be expressed in terms ofthe 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 coefficients in this equation is zero.Between any five vectors there exists one equation the sumof whose coefficients is zero.Let A, B, C,D,E be the five given vectors. Form thedifferencesE-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 anequation between them.ft(E- A) + /(E - B) + m (E - C) + n (E- D) = 0.The sum of the coefficients of this equation is zero.1Vectors which have a common origin and terminate in one plane are calledby Hamilton "termino-complanar."
  49. 49. ADDITION AND SCALAR MULTIPLICATION 33*22.] The results of the foregoing section afford simplesolutions of many problems connected solely with the geometric properties of figures. Special theorems, the vectorequations of lines and planes, and geometric nets in two andthree dimensions are taken up in order.Example 1: If a line be drawn parallel to the base of atriangle, the line which joins the opposite vertex to the intersection of the diagonals of thetrapezoid thus formed bisects thebase (Fig. 11).Let ABC be the triangle, EDthe line parallel to the base CB,G the point of intersection of thediagonals EB and DC of the trapezoid CBDE, and Fthe intersection of A G with CB. To showFI(J nthat F bisects CB. Choose theorigin at random. Let the vectors drawn from it to thevarious points of the figure be denoted by the correspondingClarendons as usual. Then since ED is by hypothesis parallel to CB, the equationE - D = n (C- B)holds true. The sum of the coefficients is evidently zero asit should be. Rearrange the terms so that the equationtakes on the formE nC = "D 7i B.The vector E n C is coplanar with E and C. It must cutthe line EC. The equal vector D 7&B is coplanar with Dand B. It must cut the line DB. Consequently the vectorrepresented by either side of this equation must pass throughthe point A. HenceE 7iC = D ?iB = #A.
  50. 50. 34 VECTOR ANALYSISHowever the points E, 0, and A lie upon the same straightline. 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 reasoningE + 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 amultiple of F and such a multiple that the sum of the coefficients 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 coveredconsiderable space because each detail of the reasoning hasbeen given. In reality, however, the actual analysis has consisted of just four equations obtained simply from the first.Example % : To determine the equations of the line andplane.Let the line be fixed by two points A and B upon it. LetP be any point of the line. Choose an arbitrary origin.The vectors A, B, and P terminate in the same line. HenceaA + 6Band a + I + p = 0.,Therefore P =a + b
  51. 51. ADDITION AND SCALAR MULTIPLICATION 35For different points P the scalars a and b have differentvalues. They may be replaced by x and y, which are usedmore generally to represent variables. Thenx + yLet 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. Hence6B + cCand a + b + c+p = Q.aA + 6B + cCTherefore P =-f cAs a, 6, c, vary for different points of the plane, it is morecustomary to write in their stead x, ytz.+ y + zExample 3 ; The line which joins one vertex of a complete quadrilateral to the intersection of two diagonalsdivides the opposite sides harmonically (Fig. 12).Let A, B, C, D be four verticesof a quadrilateral. Let A B meetCD in a fifth vertex E, and ADmeet BC in the sixth vertex F.Let the two diagonals AC and p 12BD intersect in G. To showthat FG intersects A B in a point i" and CD in a point E1such that the lines AB and (7I> are divided internally atE1and 2?" in the same ratio as they are divided externallyby E. That is to show that the cross ratios
  52. 52. 86 VECTOR ANALYSISChoose the origin at random. The four vectors A, B, C, Ddrawn from it to the points A, B, C, D terminate in oneplane. Henceand a + b + e + d = 0.Separate the equations by transposing two terms :Divide : =a + c = (b + d).cC 6B +a + c b + daA + d D __6B + cCa + d b + c(a + C)G (a + d)F cC diIn like manner F =Form:(a + c)- (a + d)"(a + c) (a + d)(a + c)Q (a + rf)F cCorc a c aSeparate 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 ratioc / d. But equation (a) shows that JEffdivides CD in theratio c:d. Hence E and E" divide CD internally andexternally in the same ratio. Which of the two divisions isinternal and which external depends upon the relative signsof c and d. If they have the same sign the internal pointof division is E; if opposite signs, it is E1. In a similar wayE1and 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 composedof points and straight lines obtained in the following manner.Start with a certain number of points all of which lie in one
  53. 53. ADDITION AND SCALAR MULTIPLICATION 37plane. 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 numberof points which may in turn be joined in pairs and so on.The construction may be kept up indefinitely. At each stepthe number of points and lines in the figure increases.Probably the most interesting case of a plane geometric net isthat in which four points are given to commence with.Joining these there are six lines which intersect in threepoints different from the given four. Three new lines maynow 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=(c)and a + b + c + d = Qwhich subsist between the four vectors drawn from an undetermined origin to the four given points. From these it ispossible to obtaina A + 6B cC + dDTjla + b c + dA + cC Z>B + dDa + c b + dA + dJ) &B + cCa + d b + cby splitting the equations into two parts and dividing. Nextfour vectors such as A, D, E, F may be chosen and the equation the sum of whose coefficients is zero may be determined.This would beaA + dV + (a + b) E-f (a + c) P = 0.By treating this equation as (c) was treated new points maybe obtained*
  54. 54. 38 VECTOR ANALYSISa A + dD (a + 6)E + (a + c)FH =1 =a + d 2a + b + caA + (a + ft)E __ <?D+ (a + c)Fa 4- c + d(a + 6) Ec a + b + dEquations between other sets of four vectors selected fromA, B, C, D, E, F, may be found ; and from these more pointsobtained. The process of finding more points goes forwardindefinitely. A fuller account of geometric nets may befound in Hamilton s " Elements of Quaternions," Book I.As regards geometric nets in space just a word may besaid. Five points are given. From these new points may beobtained by finding the intersections of planes passed throughsets of three of the given points with lines connecting theremaining pairs. The construction may then be carried forward with the points thus obtained. The analytic treatmentis similar to that in the case of plane nets. There arefive vectors drawn from an undetermined origin to the givenfive points. Between these vectors there exists an equationthe sum of whose coefficients is zero. This equation may beseparated into parts as before and the new points may thusbe obtained.+ 6B cC + dD +then F =a + b c + d + eA + cC 6B + dV + ea + b b + d + care two of the points and others may be found in the sameway. Nets in space are also discussed by Hamilton, loc. cit.
  55. 55. ADDITION AND SCALAR MULTIPLICATION 39Centers of Gravity*23.] The center of gravity of a system of particles maybe found very easily by vector methods. The two laws ofphysics which will be assumed are the following:1. The center of gravity of two masses (considered assituated at points) lies on the line connecting the two massesand divides it into two segments which are inversely proportional to the masses at the extremities.2. In finding the center of gravity of two systems ofmasses each system may be replaced by a single mass equalin magnitude to the sum of the masses in the system andsituated 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 bywhere 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 thethree points -4, B, C may be found by means of law 2. Themasses a and b may be considered as equivalent to a singlemass a + b situated at the pointa A + &Ba + bThen G = (a + 6)" A + 6B+ c Ca -f- bTT aA-h&B-f-cCHence G =a -f b + c
  56. 56. 40 VECTOR ANALYSISEvidently the center of gravity of any number of massesa, &, c, d, ... situated at the points A, B, C, D, ... maybe found in a similar manner. The result isaA + ftB + cO + rfD + ...^a + b + c + d + ...Theorem 1 : The lines which join the center of gravity of atriangle to the vertices divide it into three triangles whichare proportional to the masses at the opposite vertices (Fig. 13). Let A, B, Cbe the vertices of a triangle weightedwith masses a, &, c. Let G be the center of gravity. Join A, B, C to G andproduce the lines until they intersectthe opposite sides in Af, B C1respectively. To show thatthe areasG 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 comesfrom compounding the first three. It is, however, useful inthe demonstration.ABC AA A G .GA b + c+ 1.HenceGBC~ GA! CTA G~AfABC a + b + cIn a similar mannerandGBC aBCA a + I + cGCA~ bCAB _ a + b + cGAB~~~cHence the proportion is proved.Theorem 2 : The lines which join the center of gravity ofa tetrahedron to the vertices divide the tetrahedron into four
  57. 57. ADDITION AND SCALAR MULTIPLICATION 41tetrahedra which are proportional to the masses at the opposite vertices.Let -4, B, C, D be the vertices of the tetrahedron weightedrespectively with weights a, &, c, d. Let be the center ofgravity. Join A, B, C, D to G and produce the lines untilthey meet the opposite faces in A , B G D . To show thatthe volumesBCDG:CDAG:DABG:ABCG:ABCDBCDABCDGIn like mannerandandABCD dwhich proves the proportion.*24.] By a suitable choice of the three masses, a, J, c located at the vertices A, B, (7, the center of gravity G maybe made to coincide with any given point P of the triangle.If this be not obvious from physical considerations it certainly becomes so in the light of the foregoing theorems.For in order that the center of gravity fall at P, it is onlynecessary to choose the masses a, 6, c proportional to theareas of the triangles PEG, PCA^ and PAB respectively.Thus not merely one set of masses a, &, c may be found, butan infinite number of sets which differ from each other onlyby a common factor of proportionality. These quantities
  58. 58. 42 VECTOR ANALYSISa, 6, c may therefore be looked upon as coordinates of thepoints P inside of the triangle ABC. To each set therecorresponds a definite point P, and to each point P therecorresponds an infinite number of sets of quantities, whichhowever do not differ from one another except for a factorof proportionality.To obtain the points P of the plane ABC which lie outsideof the triangle ABC one may resort to the conception ofnegative weights or masses. The center of gravity of themasses 2 and 1 situated at the points A and B respectivelywould be a point G dividing the line A B externally in theratio 1 : 2. That isAny point of the line A B produced may be represented bya suitable set of masses a, b which differ in sign. Similarlyany point P of the plane ABC may be represented by asuitable set of masses a, 6, c of which one will differ in signfrom the other two if the point P lies outside of the triangleABC. Inasmuch as only the ratios of a, 6, and c are important two of the quantities may always be taken positive.The idea of employing the masses situated at the verticesas coordinates of the center of gravity is due to Mobius andwas published by him in his book entitled " Der barycentrischeCalcul" in 1827. This may be fairly regarded as the startingpoint of modern analytic geometry.The conception of negative masses which have no existencein nature may be avoided by replacing the masses at thevertices by the areas of the triangles GBC, GOA, andGAB to which they are proportional. The coordinates ofa point P would then be three numbers proportional to theareas of the three triangles of which P is the common vertex ;and the sides of a given triangle ABC, the bases. The signof these areas is determined by the following definition.
  59. 59. ADDITION AND SCALAR MULTIPLICATION 43Definition: The area ABC of a triangle is said to bepositive when the vertices A, B, C follow each other in thepositive or counterclockwise direction upon the circle described through them. The area is said to be negative whenthe points follow in the negative or clockwise direction.Cyclic permutation of the letters therefore does not alterthe sign of the area.Interchange of two letters which amounts to a reversal ofthe cyclic order changes the sign.A CB = BA = CBA = -A B C.If P be any point within the triangle the equationPAB+PBC+PCA=ABCmust hold. The same will also hold if P be outside of thetriangle provided the signs of the areas be taken into consideration. The areas or three quantities proportional tothem may be regarded as coordinates of the point P.The extension of the idea of "barycentric"coordinates tospace is immediate. The four points A, B, C, D situated atthe vertices of a tetrahedron are weighted with mass a, J, c, drespectively. The center of gravity G is represented bythese quantities or four others proportional to them. Toobtain points outside of the tetrahedron negative massesmay be employed. Or in the light of theorem 2, page 40,the masses may be replaced by the four tetrahedra whichare proportional to them. Then the idea of negative volumes takes the place of that of negative weights. As thisidea is of considerable importance later, a brief treatment ofit here may not be out of place.Definition : The volume A B CD of a tetrahedron is saidto be positive when the triangle ABC appears positive to
  60. 60. 44 VECTOR ANALYSISthe eye situated at the point D. The volume is negativeif the area of the triangle appear negative.To make the discussion of the signs of the varioustetrahedra perfectly clear it is almost necessary to have asolid modeL A plane drawing is scarcely sufficient. It isdifficult to see from it which triangles appear positive andwhich negative. The following relations will be seen tohold if a model be examined.The interchange of two letters in the tetrahedron ABCDchanges the sign.ACBD = CBAD=BACD=DBCAThe sign of the tetrahedron for any given one of the possible twenty-four arrangements of the letters may be obtainedby reducing that arrangement to the order A B C D bymeans of a number of successive interchanges of two letters.If the number of interchanges is even the sign is the sameas that of A B CD ; if odd, opposite. ThusIf P is any point inside of the tetrahedron A B CD theequationABCP-BCDP+ CDAP-DABP=ABCDholds good. It still is true if P be without the tetrahedronprovided the signs of the volumes be taken into consideration. The equation may be put into a form more symmetrical and more easily remembered by transposing all the termsto one number. ThenThe proportion in theorem 2, page 40, does not hold trueif the signs of the tetrahedra be regarded. It should readBCDG:CDGA:DGAB:GABC:ABCD
  61. 61. ADDITION AND SCALAR MULTIPLICATION 45If the point G- lies inside the tetrahedron a, J, c, d represent quantities proportional to the masses which must belocated at the vertices A,B,C,D respectively if G is to be thecenter of gravity. If G lies outside of the tetrahedron they maystill be regarded as masses some of which are negative orperhaps better merely as four numbers whose ratios determinethe 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 anypoint of the plane A B C is(page 35)aA + yB + zCx + y + zComparing this with the expressionaA + &B + cCa + b + cit will be seen that the quantities x, y, z are in reality nothingmore nor less than the barycentric coordinates of the point Pwith respect to the triangle ABO. In like manner fromequation__#A + yB + 2C + wDx + y + z + wwhich expresses any vector P drawn from an indeterminateorigin in terms of four given vectors A, B, C, D drawn fromthe same origin, it may be seen by comparison with+ &B + c C + rfD=a + b + c + dthat the four quantities x, y, 2, w are precisely the barycentric coordinates of P, the terminus of P, with respect tothe tetrahedron A B CD. Thus the vector methods in whichthe origin is undetermined and the methods of the "Barycentric Calculus"are practically co-extensive.It was mentioned before and it may be well to repeat here
  62. 62. 46 VECTOR ANALYSISthat the origin may be left wholly out of consideration andthe vectors replaced by their termini. The vector equationsthen become point equationsx A + y B 4- zandx + y + zxA + yB + zC + wDw.Atx + y + zThis step brings in the points themselves as the objects ofanalysis and leads still nearer to the "Barycentrische Calcul"of Mobius and the "Ausdehnungslehre"of Grassmann.The Use of Vectors to denote Areas25.] Definition: An area lying in one plane MN andbounded by a continuous curve PQR which nowhere cutsitself is said to appear positive from the point when theletters PQR follow eachother in the counterclockwiseor positive order; negative,when they follow in thenegative or clockwise order(Fig. 14).It is evident that an areacan have no determined signper se, but only in referenceto that direction in which itsboundary is supposed to be traced and to some point outside of its plane. For the area P R Q is negative relative toPQR; and an area viewed from isnegative relative to thesame area viewed from a point O fupon the side of the planeopposite to 0. A circle lying in the XF-plane and describedin the positive trigonometric order appears positive from everypoint on that side of the plane on which the positive axislies, but negative from all points on the side upon which
  63. 63. ADDITION AND SCALAR MULTIPLICATION 47the negative ^-axis lies. For this reason the point of viewand the direction of description of the boundary must be keptclearly in mind.Another method of stating the definition is as follows : Ifa person walking upon a plane traces out a closed curve, thearea enclosed is said to be positive if it lies upon his left-hand side, negative if upon his right. It is clear that if twopersons be considered to trace out together the same curve bywalking upon opposite sides of the plane the area enclosedwill lie upon the right hand of one and the left hand of theother. To one it will consequently appear positive ; to theother, negative. That side of the plane upon which the areaseems positive is called the positive side ; the side uponwhich it appears negative, the negative side. This idea isfamiliar to students of electricity and magnetism. If anelectric current flow around a closed plane curve the lines ofmagnetic force through the circuit pass from the negative tothe positive side of the plane. A positive magnetic poleplaced upon the positive side of the plane will be repelled bythe circuit.A plane area may be looked upon as possessing more thanpositive or negative magnitude. It may be considered topossess direction, namely, the direction of the normal to thepositive side of the plane in which it lies. Hence a planearea is a vector quantity. The following theorems concerningareas when looked upon as vectors are important.Theorem 1 : If a plane area be denoted by a vector whosemagnitude is the numerical value of that area and whosedirection is the normal upon the positive side of the plane,then the orthogonal projection of that area upon a planewill be represented by the component of that vector in thedirection normal to the plane of projection (Fig. 15).Let the area A lie in the plane MN. Let it be projectedorthogonally upon the plane M N . Let MN&nd M*Nrinter-
  64. 64. 48 VECTOR ANALYSISsect in the line I and let the diedral angle between thesetwo planes be x. Consider first a rectangle PQJRS in MNwhose sides, PQ, RS and QR, SP are respectively paralleland perpendicular to the line /. This will project into arectangle P Q R S1in M N . The sides P Qfand JR Swill be equal to PQ and US; but the sides Q1R and S Pwill be equal to QR and SP multiplied by the cosine of #,the angle between the planes. Consequently the rectangleAtFIG. 15.Hence rectangles, of which the sides are respectivelyparallel and perpendicular to I, the line of intersection of thetwo planes, project into rectangles whose sides are likewiserespectively parallel and perpendicular to I and whose area isequal to the area of the original rectangles multiplied by thecosine of the angle between the planes.From this it follows that any area A is projected into anarea which is equal to the given area multiplied by the cosineof the angle between the planes. For any area A may be divided up into a large number of small rectangles by drawing aseries of lines in MNparallel and perpendicular to the line I.
  65. 65. ADDITION AND SCALAR MULTIPLICATION 49Each of these rectangles when projected is multiplied by thecosine of the angle between the planes and hence the totalarea is also multiplied by the cosine of that angle. On theother hand the component A of the vector A, which represents the given area, in the direction normal to the planeMfNfof projection is equal to the total vector A multipliedby the cosine of the angle between its direction which isthe normal to the plane M^and the normal to M Nr. Thisangle is x ; for the angle between the normals to two planesis the same as the angle between the planes. The relationbetween the magnitudes of A and A is thereforeA1= A cos x,which proves the theorem.26.] Definition : Two plane areas regarded as vectors aresaid to be added when the vectors which represent them areadded.A vector area is consequently the sum of its three components obtainable by orthogonal projection upon threemutually perpendicular planes. Moreover in adding twoareas each may be resolved into its three components, thecorresponding components added as scalar quantities, andthese sums compounded as vectors into the resultant area.A generalization of this statement to the case where the threeplanes are not mutually orthogonal and where the projectionis oblique exists.A surface made up of several plane areas may be represented by the vector which is the sum of all the vectorsrepresenting those areas. In case the surface be looked uponas forming the boundary or a portion of the boundary of asolid, those sides of the bounding planes which lie outside ofthe body are conventionally taken to be positive. The vectors which represent the faces of solids are always directedout from the solid, not into it4
  66. 66. 50 VECTOR ANALYSISTheorem 2 : The vector which represents a closed polyhedralsurface is zero.This may be proved by means of certain considerations ofhydrostatics. Suppose the polyhedron drawn in a body offluid assumed to be free from all external forces, gravity included.1The fluid is in equilibrium under its own internalpressures. The portion of the fluid bounded by the closedsurface moves neither one way nor the other. Upon each faceof the surface the fluid exerts a definite force proportionalto the area of the face and normal to it. The resultant of allthese forces must be zero, as the fluid is in equilibrium. Hencethe 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 uponany plane. This consists of a double area. The part of thesurface farthest from the plane projects into positive area ;the part nearest the plane, into negative area. Thus thesurface projects into a certain portion of the plane which iscovered twice, once with positive area and once with negative.These cancel each other. Hence the total projection of aclosed surface upon a plane (if taken with regard to sign) iszero. But by theorem 1 the projection of an area upon aplane is equal to the component of the vector representingthat area in the direction perpendicular to that plane. Hencethe vector which represents a closed surface has no componentalong the line perpendicular to the plane of projection. This,however, was any plane whatsoever. Hence the vector iszero.The theorem has been proved for the case in which theclosed surface consists of planes. In case that surface be1Such a state of affairs is realized to all practical purposes in the case of apolyhedron suspended in the atmosphere and consequently subjected to atmospheric pressure. The force of gravity acts but is counterbalanced by the tensionin the suspending string.
  67. 67. ADDITION AND SCALAR MULTIPLICATION 51curved it may be regarded as the limit of a polyhedral surfacewhose number of faces increases without limit. Hence thevector which represents any closed surface polyhedral orcurved is zero. If the surface be not closed but be curved itmay be represented by a vector just as if it were polyhedral.That vector is the limit lapproached by the vector whichrepresents that polyhedral surface of which the curved surfaceis the limit when the number of faces becomes indefinitelygreat.SUMMARY OF CHAPTER IA vector is a quantity considered as possessing magnitudeand direction. Equal vectors possess the same magnitudeand the same direction. A vector is not altered by shifting itparallel to itself. A null or zero vector is one whose magnitude is zero. To multiply a vector by a positive scalarmultiply its length by that scalar and leave its directionunchanged. To multiply a vector by a negative scalar multiply its length by that scalar and reverse its direction.Vectors add according to the parallelogram law. To subtracta vector reverse its direction and add. Addition, subtraction, and multiplication of vectors by a scalar follow the samelaws as addition, subtraction, and multiplication in ordinaryalgebra. A vector may be resolved into three componentsparallel to any three non-coplanar vectors. This resolutioncan be accomplished in only one way.r = x* + yb + zc. (4)The components of equal vectors, parallel to three givennon-coplanar vectors, are equal, and conversely if the components are equal the vectors are equal. The three unitvectors i, j, k form a right-handed rectangular system. In1 This limit exists and is unique. It is independent of the method in whichthe polyhedral surface approaches the curved surface.
  68. 68. 52 VECTOR ANALYSISterms of them any vector may be expressed by means of theCartesian coordinates #, y, z.r = xi + yj+zk. (6)Applications. The point which divides a line in a givenratio m : n is given by the formula(7)m + nThe necessary and sufficient condition that a vector equationrepresent a relation independent of the origin is that the sumof the scalar coefficients in the equation be zero. Betweenany four vectors there exists an equation with scalar coefficients. If the sum of the coefficients is zero the vectors aretermino-coplanar. If an equation the sum of whose scalarcoefficients is zero exists between three vectors they aretermino-collinear. The center of gravity of a number ofmasses a, &, c situated at the termini of the vectorsA, B, C supposed to be drawn from a common origin isgiven by the formulaA vector may be used to denote an area. If the area isplane the magnitude of the vector is equal to the magnitudeof the area, and the direction of the vector is the direction ofthe normal upon the positive side of the plane. The vectorrepresenting a closed surface is zero.EXERCISES ON CHAPTER I1. Demonstrate the laws stated in Art. 12.2. A triangle may be constructed whose sides are paralleland equal to the medians of any given triangle.
  69. 69. ADDITION AND SCALAR MULTIPLICATION 533. The six points in which the three diagonals of a com*plete quadranglelmeet the pairs of opposite sides lie threeby three upon four straight lines.4. If two triangles are so situated in space that the threepoints of intersection of corresponding sides lie on a line, thenthe lines joining the corresponding vertices pass through acommon point and conversely.5. Given a quadrilateral in space. Find the middle pointof the line which joins the middle points of the diagonals.Find the middle point of the line which joins the middlepoints of two opposite sides. Show that these two points arethe same and coincide with the center of gravity of a systemof equal masses placed at the vertices of the quadrilateral.6. If two opposite sides of a quadrilateral in space bedivided proportionally and if two quadrilaterals be formed byjoining the two points of division, then the centers of gravityof these two quadrilaterals lie on a line with the center ofgravity of the original quadrilateral. By the center of gravityis meant the center of gravity of four equal masses placed atthe vertices. Can this theorem be generalized to the casewhere the masses are not equal ?7. The bisectors of the angles of a triangle meet in apoint.8. If the edges of a hexahedron meet four by four in threepoints, the four diagonals of the hexahedron meet in a point.In the special case in which the hexahedron is a parallelepipedthe three points are at an infinite distance9. Prove that the three straight lines through the middlepoints of the sides of any face of a tetrahedron, each parallelto the straight line connecting a fixed point P with the middle point of the opposite edge of the tetrahedron, meet in a1 A complete quadrangle consists of the six straight lines which may he passedthrough four points no three of which are collinear. The diagonals are the lineswhich join the points of intersection of pairs of sides
  70. 70. 54 VECTOR ANALYSISpoint E and that this point is such that PE passes throughand is bisected by the center of gravity of the tetrahedron.10. Show that without exception there exists one vectorequation with scalar coefficients between any four givenvectors A, B, C, D.11. Discuss the conditions imposed upon three, four, orfive vectors if they satisfy two equations the sum of the coefficients in each of which is zero.
  71. 71. CHAPTER IIDIRECT AND SKEW PRODUCTS OF VECTORSProducts of Two Vectors27.] THE operations of addition, subtraction, and scalarmultiplication have been defined for vectors in the waysuggested by physics and have been employed in a fewapplications. It now becomes necessary to introduce twonew combinations of vectors. These will be called productsbecause they obey the fundamental law of products ; i. e., thedistributive law which states that the product of A into thesum of B and C is equal to the sum of the products of A intoB and A into C.Definition : The direct product of two vectors A and B isthe scalar quantity obtained by multiplying the product ofthe magnitudes of the vectors by the cosine of the angle between them.The direct product is denoted by writing the two vectorswith a dot between them asA-B.This is read A dot B and therefore may often be called thedot product instead of the direct product. It is also calledthe scalar product owing to the fact that its value is scalar. If A be the magnitude of A and B that of B, then bydefinitionA-B = ^cos (A,B). (1)Obviously the direct product follows the commutative lawA-B = B A. (2)
  72. 72. 56 VECTOR ANALYSISIf either vector be multiplied by a scalar the product ismultiplied 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 between them becomes zero or one hundred and eighty degreesand its cosine is therefore equal to unity with the positive ornegative sign. Hence the scalar product of two parallelvectors is numerically equal to the product of their lengths.The sign of the product is positive when the directions of thevectors are the same, negative when they are opposite. Theproduct of a vector by itself is therefore equal to the squareof its lengthA.A=^42.(3)Consequently if the product of a vector by itself vanish thevector is a null vector.In case the two vectors A and B are perpendicular theangle between them becomes plus or minus ninety degreesand the cosine vanishes. Hence the product A B vanishes.Conversely if the scalar product A B vanishes, thenA B cos (A, B) = 0.Hence either A or B or cos (A, B) is zero, and either thevectors are perpendicular or one of them is null. Thus thecondition for the perpendicularity of two vectors, neither ofwhich vanishes, is A B = 0.28.] The scalar products of the three fundamental unitvectors i, j, k are evidentlyii = jj = kk = l, (4)i .j= j. k = k . i = 0.If more generally a and b are any two unit vectors theproducta b = cos (a, b).

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