Elementary Vector Geometry

seymour

schuster)

B) A)

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u)

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B)

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ELEMENTARY

GEOMETRY)

VECTOR

SEYMOUR

SCHUSTER)

DOVER PUBLICATIONS,INC. Mineola,

New

York)))

Bibliographical Note Dover

This

the work

first published in 2008, by John Wiley published

edition,

originally

of Congress

Library

is an

unabridged

& Sons,

vector geometry /

Elementary

p. em. Originally Includes

ISBN-13:

published:

of 1962.)

Cataloging-in-Publication Data

Seymour.

Schuster,

republication

Inc., New York, in

Seymour

New York:

John

Schuster.

Wiley

- Dovered.

& Sons,

Inc., 1962.

index.

978-0-486-46672-9

ISBN-lO: 0-486-46672-8

1.Vector

QA433.S38

analysis.

2. Geometry.

I. Title.

2008

515' .63-dc22)

2007050541) Manufactured

Dover

Publications,

in the United States of America Inc., 31 East 2nd Street, Mineola, N.Y.

11501)))

to

my

parents)))

preface)

This short presented at a

is

work

held at lectures

outgrowth

of

pattern

teachers.

geon1etry

mathematics

in

events

of

the material covered is

material

essential

rather,

but,

that were

of lectures

Institute

Foundation

Science

Carletol1 College ill the summer were to serve the purposeof

backgrounds

that

the

National

The

1959.

of

the

\"enriching\"

However,

no longer

the

indicates

education

enrichment

for every

knowledge

teacher.

It

ago that linear algebra'was a course for beginning graduate students and vector analyclass course by taken as an upper sis was typically The students. and engineering mathematics, physics, was

just

a few

years

last decade has brought mathematics vectors quite

is

\037

revolution

in

undergraduate

today the knowledgeof earlier stage. Indeed,it is to study vectors and freshmen

and education, a much at acquired for

usual

matrices,

particularly

college as

more, the studies and

applied

to geometry.

recommendatio11s made

v)))

Furtherby

the)

PREFACE)

VI)

Commission Group,

Study

Programin

on Mathematics, the School M\037thematics on the Undergraduate and the Committee all in the direction Mathematics of point

getting some of these

into the c011cepts

school

high

I have curriculum. 1011g felt that vector techniques find ,viII their should way into the high school n otas of the mathecurriculum-perhaps integral part matical training of all studentsbut, at the as least, work to excite and challenge superior students. and

an

very

On

a

level,

elementary

very

marily development mathematical tool in greater insight into the

theorems

proofs proofs,

a

knowledge

prerequisite.

oped slowly-more so on vectors. Simple

vector

natural

vector any

are develstandard works

algebra

of the

as

explanations,

well

as

two

(in

geometry

developedasa

gain a

are used. Beyond this, some and three dimensions) is

illustrations,

analytic

is to

attempting

by

of in

than

geometric

numerous

aim

proofs in contrast to the synthetic of which the reader brings as a

elements

The

vector The

geometry.

and analytic

deals prialgebra as a

this textbook of

the

with

of the

outgrowth

vector treatment. to assist in other

In addition, the vectorapproach is used areas of elementary mathematics: algebra,trigonometry In short, and and higher geometry. (plane spherical), it was felt that whenever vectorshave been employed in facilitating they would aid in gaining and/or insight to develop and proofs. I have tried very computations this small little machinery but to go a long way with himself amount. the reader will. find Accordingly, with such topics as linear inequalities,convexity, dealing

linear because

involutes,

programming,

As for I

and

projective

theorems.

prerequisites, they are not listed do not claim to have given a logical

development

of

of

Loosely

geometry.

assumed that the and concepts

formally

(axiomatic)

reader

Ellclidean

is

familiar geometry

speaki11g, I have the definitions and with the bare)))

\\vith

VII)

PREFACE)

area

and

parallelism,

angle,

For example, the notionsof are assumed. It is further

of trigonometry.

essentials

assumedthat the knows sine, cosine, and tangent

functions

as ratios

sides

the

of

resliits

from

of a

(in

the

right triangle).

and

geometry

of the

definitions

the

reader

naive

I haye

trigonometry,

sense,

In regard to indeed

taken very little for of. geometric granted. Samples information that are calledupon are: formulas for the area of a parallelogram and volume of a parallelopiped\037 the fact that two points determinea unique and the line, result that opposite sides of a parallelogram are equal. 3 I use the)aw of cosinesfor motivatio11, In Chapter but who has not seen it before will be consoledby the reader a

shortly

given

proof

thereafter.

It is entirely possibleto give a vector of development Euclidean from \"scratch.\" In fact, some geometry believe that a first course in geometry should people Others believe that the coordinate beginwith vectors. method should be given at the and still others olltset, of have faith in a combinatio11

coordinate

development,

beenadopted several in rewriting the by

high

approaches.

The

in various forms, has recently of the current groups interested school

curriculum.

mathematics

the reader interested in seeing how a strict vector would do the job, I strongly recomme11d the \"Geometric excellent Vector and the Analysis paper Conceptof Vector Space\" by Professor Walter Prenowitz. For

approach

This fine expositionconstitutesoneofthe chapters of the Yearboolc Third of the of National Council Twenty-

Teachersof Sincere who

of

came comfortably

l\\lathematics.

thanks to

and

Carleton learning

are appreciation i11 the SlImmer mathematics

due to the teachers of 1959 in the hope in cool

Minnesota

the strains but who, instead, laboredal1dperspiredunder of vector geometry and the 96% humidity. For reading the and for their valuable suggestiol1SI am))) n1a11uscript

PREFACE)

VIII)

grateful especially School in Lincoln of

the

University

the

of

Chicago

to Mr. Saul Birnbaum of the New New York City, ProfessorRoy Dubisch of Washington, Professor J. M. Sachs Technical College, and my Carleton

colleague,Professor

B.

William

special

thanks

courageously Science

National

1961.

This

tribute

substantially

Jr.

Houston,

Also,

who go to ProfessorDick Hall, used the text in mimeograph form at a Wick

Institute

Foundation

enabled

experience by

pointing

out

Sllmmer of Hall to conerrors ill judg-

in the

Professor ID.y

ment and typography.) SEYMOUR

N orthfield,

January,

Minnesota 1962)))

SCHUSTER)

contents)

1

Chapter

Fundamentalproperties4.

\302\267

of

5. Auxiliary

vectors. of

Uniqueness

\037

Chapter

IN

combinations

point technique.6.

COORDINATE

SYSTEMS

40)

systems and orientation.8.

7. Rectangular and

vectors

Linear

representations.)

VECTORS

Basis

of vector. 3.

2. Definition

Introduction.

1.

1)

OPERATIONS)

ELEMENTARY

applications

\302\267 9.

The

complex

plane.)

Chapter 10.

3

INNER

Definition.

12.

Components.

14.

Work.)

60)

PRODUCTS)

11. Properties of inner product \302\267 13. Inner product formulas.

IX)))

CONTENTS)

x)

ANALYTIC GEOMETRY

4.

Chapter

15. Our pointof view A,nalytic

Distance

\302\267

16.

geometry a

from

The

to

point

line

straight

21. method of proof. 20. Circles. 24.

line in

straight

two lines

plane

\302\267 28.

Chapter 5 29.

a point to a plane three dimensions \302\267 26. Angle

\302\267 25.

\302\267 27.

a line

\302\267

The

be-

a

line with

and a plane.) 135)

PRODUCTS)

a

from

to

point

a plane

\302\267 32.

Dis-

cross

lines. 33. Triple

two

between

scalar product.

30. Triple

products.

31. Distance tance

of a

Intersection

between

Angle

CROSS

Cross

by points on it

from

Distance

tween

a plane

Det\037rmining

17.

22.

Spheres.

\302\267 23.

Planes

\302\267

line continued. 18. a line. 19. Analytic

the

of

76)

products.)

Chapter

151)

TRIGONOMETRY)

6

34. Plane

trigonometry .,35. Spherical

trigonometry.)

Cl\037apter

7

36. Loci defined booby traps. 38. Linear

more parametric

by

inequalitjes.

37.

A

few

Segments and convexity.39. 40.

programming.

general

160)

GEOMETRY)

MORE

Theorems

arising

in

geometries. 41. Applicationsof

equations

to

locus

problems.

42.

Rigid motions.) APPENDIX

204

ANSWERS

fJ06

INDEX)

\03711)))

elem.entary

operations)

1.

INTRODUCTION

The

history

of the

of mathematical ideas

development

indicatesthat abstractconcepts arise

from

generally

in some

problemsof

counting,

roots

Arithmetic stemmedfrom

\"practical\" problem.

arose

geometry

from

problems

of

surveying land in Egypt, and calculusdevelopedprincito solve the problems of motion. from the efforts pally goes quite beyond the point of however, Mathematics, the that initiate the particular solving merely problems a for is with mathematics concerned building study, deductive science that is general and abstract,that may science have a wide range of application. By a deductive that a we mean, logical development beginswith briefly, of a set of assumptions a basic framework consisting and a set of terms used in or (calledaxioms postulates)

stating the assumptions. the of the assumptions are then the All

theorems

science,

which is

logical of

consequences deductive

the

concerned with abstractions or idealiza1)))

ELEMENTARY

2)

GEOMETRY)

VECTOR

tions of concepts from the original rather than problem the original problem itself. For example, the study is based on a set of assumptions of geometry that deal with lines and POil1tS than rather with fences essentially Line is an abstract concept; and fenceposts. an idealization of the fence, and it admits to all sortsof other of light, the edge of a board, ray.

with

it is

a interpretations:

the path of a

some

under

molecule

circumstances,

still others. Thus geometry,with finds application in a variety surveying,

a host of

and

in

origins

its of

problems.

The

however, goes on-and far

mathematician,

pure

oncehe a mathematical beyond. Because free to exercisehis imagination making

study,

begins

he is

logical

by

deductiollS

and developing theories from the realities of the motivating is a reality mathematician there theorems)

(proving

that qllite apart the problem. within his deductivescience. drawing metry for illustration, we can point to the are

li\"'or

from

Again

dimensional

fOllr

of

geo-

developments

n-dimensional

geometry-even

world is spite of the fact that our physical non-Euclidean or to the of invention dimel1sional,

geometry-in three

geometries,that contradictEuclid'sParallelPostulate for

(which,

2000

over

mathematicaltruth). were

consequences the

beyond

consideration

was accepted

years, Such

as absolute

by mathematicians

creations

of strong imagination and quite of any elementary problem in

the physical

world.

Vector

in

physical

is

analysis problems.

also It

a subject that was developed

has its roots primarily

handle problemsin physics, problems chanics but, later, problems in various other

in

initially

physical

twentieth of

a vector

science.

Developments

have resulted in a consequently,

centuries and,

of

th\037

branches

nineteenth

to meof

and

in the abstractconcept wide range of interpre-

tationsofthisabstractconcept.

The

result

is that

vec-)))

OPERATIONS)

ELEMENTA.RY

tors

now

to name

3)

a prominent role in a variety a few: just physicalchemistry,fluid-flow play

studies, theory,

and

psychology,

economics,

theory,

electro-magnetic

of

electrocardiography.

with illustrations filled in Geometry books aI,vays fact that the circle of are abstract point,line, spite that do 110texistin physicalreality in spite concepts of the fact that beginning students are apprisedof the are

and

and

the at of abstract subject their course. The reasonis

Abstract

simple.

quite

is difficult;

reasoning

beginning of

the very

nature

students therefore need-or

leastassisted by-the

of

help

some

real

model

are at

(or inter-

a pretation) of the abstract concepts.Consequently, with a sharp pencil is a convenient dot marked of point, and a sharppencildrawn for the concept model a ruler leaves a n1arkthat isusedasa of the along edge of line. for the Such pencil marks are a model concept until they get to feel at for convenience beginl1ers great and in the home subject begin to feel that there is a in itself. Later in their mathematical reality geometry

small

studies students

other

encounter

abstract

concepts,

but

models to by this time they can, and do, use geometric assist in still more abstract reasoning. This them is precisely of what occurs in the pattern development the concept of vector can of vectors. Although study a geometric be made abstract, model (directed line segthat assists the beginner in development) is the crutch ing steadylegsinthe field that is new to him. that It is the author'sview steady legs in abstract and that vector algebra are developedslowly reasoning in the model (now geometry)for some extended period should be done preliminary to engagingin the abstract

study.

this

Therefore

with a geometricstudy

entire of

vectors

textbook (i.e.,

co\037cerns itself the application

vectors to geometry),in contrastto the general study

of vectors.

Let us

begin.)))

abstract

of

2. DEFINITION Earlier

we

OF VECTOR out

pointed

from

originally

GEOMETRY)

VECTOR

ELEMENTARY

4)

physics.

the idea of vector came let us considerTherefore,

that

from the physicist's point of view-the statement of the television announcer who, beforegiving his final \"Good and the wind night,\" states, \"The temperature is 110W 37\302\260 is' 12 miles per hour in a northeasterly In direction.\" this simple weather announcementwe observe examples different types of quantities in the sense of two distinctly that the first (temperature) requires only a single num-

units, of course-for its description, (wind velocity) requires two facts, quantity

ber-with

whereas

second

the

magnitude

These

dire-ction.

and

the quantities encounteredin Hence, quantities

called tude

examples

are typical

elementary

physics.

of

classification is made: that are singled out and possess only magnitude whereas that both quantities possess scalars, magniare called vectors. and direction the

simple

following

In additionto of scalar examples quantities are mass, length, area, in addition volume; and, to velocity, examples of vectors are force,acceleration, temperature,

and

a11delectricalintensity.

Just as the

for his means

trained which he can A convenient the

geometry-to by

of

reasoning.

is a directedline

a

desires

mathematician

general concepts, so

segment

model

geometric

doesthe physicist.

For

has a reality

physicist-also

\"visualize\" and be aidedin his geometric model for a vector (/)

because

this

possesses

both magnitude (length) and direction,simultaneously. which suits the needs of physicists, is also This model, our for for it is our aim to quite satisfactory purposes, mearlS of vectors). study (by geometry Hence, for our

mathematical development,

we

make

the

for-

following

mal definition.

Definition. use

boldface

A

vector

is

a directed

type to indicate a

line segment.

vector. The

W

e shall

symbol1AI)))

OPERATIONS)

ELEMENTARY

5)

Q

Terminus

or

endpoint)

c) p

Origi n)

.0)

(b))

(a))

FIGURE

that the

event

we

PQ,

P

being

of

PQ

to designate

used

be

will

(see

the length of vector A.

In

vector we speak of isthe directedsegment its

emphasize origin

and

Figure

1a).

the

1)

the \037

nature by writi11gPQ, Q being the terminus or endpoint Another useful conventionwhen vector

\037

--7

\037

a vectors OB, OC, and OD with several and common 0 is to call these vectors D, origin with concerned a is if discussion That respectively. is, from a single point, we may several vectors emanating

to

referring

B, C,

them

designate

merely

by

their

individual

endpoints

(see Figure1b).

The one is called a unit vector. of zero length (with direction), any conalthough peculiar, is actually a great apparently vector zero venience. We refer to such a vector as the The to time. and shall point to its usefulnessfromtime of the direction notation for the zero vector is o. The 4. zero vector is discussed further in Secti011 A

of

vector

notion

Scalars,

being

bered scalars they

length

vector

of a

merely

will

magnitudes,

be real-num-

mathematics (In more may be elements of the complexnumbers;indeed, be from any number field. Our needs, may

quantities.

advanced

how-)))

GEOMETRY)

VECTOR

ELEMENTARY

6)

not require such generalityand will therefore be scalars to the real by restricting numbers.) are designated by lower-case Latin letters: a, b, c,

ever, do satisfied They

or

by

3.

FUNDAMENTAL

numerals.)

Our desire end we must tors equal.

Definition. A

=

(ill)

vectors

Two

and only if I

B) if

(i) (ii)

PROPERTmS

is to build an algebraof first present a criterion

A

is

A

and

the same sense of the length of A equals \\BI, i.e., B

=

possess

It cannot be emphasized toostrongly

even

be equal

space.

a

A

vector

if

may

do

they

hold:

not

direction; and the that

vectors

the same

possess

of B.)

length

may

position in

our definition indicates that relocated provided that we move it to its original position and parallel of fact,

matter

be

a position

to

rigidly

that (see

a

As

conditions

three

following

vec-

two

to B;2

parallel

IAt

calling

called equal (written

B are

A and the

for

to this

and

vectors,

its length or sense of direction It may therefore be relocated in a posi2a). Figure with its in space that we choose. origin anywhere are vectors this freedom, they are termed given not

do

we

change

tion When

free. 1 The

phrase

is actually

tion

is,

(a) If

(b)

If 2

We

\"parallel in

high

where Section to

itself.

in

this

on the

A

=

conditions use or

the on

\"if and only if\" points a double implication, conditions B, then

(i), (ii), and (iii) word parallel in the same line.\"

up the

(i),

(ii)

,

more

Although

and

A =

then

hold, the

fact that the

defini-

or logicalequivalence. general this

(iii) hold;

That

and

B. sense to mean is not given

usage

it is quite common in analytic school geometry, geometry, two lines possessingequal slopesare called parallel (see a vector is equal and Thus a line is parallel to itself, 16). The latter would not be tru\037 if we didn't use \"parallel\" vectors sense. Figure 2b exhibits two equal generalized

same

line.)))

OPERATIONS)

ELEMENTARY

7)

\"

Y

\"

,

\"

y

, \"-

\"

Y)

,

\ '\\

,

(b))

(a))

FIGURE

,)

2)

study of geometry this libertyto make displaceis highly advantageous. In the applicato other this freedom is not sciences tionsof vectors for it is to restrict vectors to necessary always granted, in For the of rigid mechanics some example, degree. that a vector be confined to a bodies it isoftenrequired line;that is,it may be moved rigidly but only in the line This line is referred to as its line of it lay originally. In Figure 3a we show three vectors, F, G, and action. three forces of the same magnitude, which H, represent and haviIlg the same sense of) lines on parallel acting In the

of vectors

ments

/ //

/

/

z/)

(b))

(a))

FIGURE

3)))

(a))

(b))

4)

FIGURE

direction. our

effect

F represents

definition.

ofthe bar of

and

and

G

represents

F and

bar.

the

therefore

would

They

are

GEOMETRY)

VECTOR

ELEMENTARY

8)

be equal accordingto

a pulling force a pushing force

G would have considered

therefore

at the center at the center

the same

mechanical

mechanically equal.

a effect However,H appliedat theendofthebar of turning motion, which is quite different from the F = G. Thus H is not equal to the otherforces. it would be natural to insist that forces such studies different lines of action be unequal. Thisjustifies having would

effect

In

two

inthe the stipulation of of permitting a vectorto be theory

of rigid

mechanics

displaced

only

along

bodies, its

line

of action.

If the fieldof applicationwere the theory of elasticity, still would be necessaryto restrict(force) vectors both of the more. Figure 3b shows two forces J and K, of line same magnitude and directed along the same the K has on a soft material. acting plasticlike action, of effect of stretching the mass, whereas J has the effect This illustration indicates why, in the it. compressing then it

theory

of

elasticity,

two

vectors

applied

at

different)))

OPERATIONS)

ELEMENTARY

considered

not

are

points

9)

restrictedto its

position;

original

displace it. Such vectorsare

bound.

called

emphasis, once again: in accordance with our

with state, are free,

We

book

this

In this field a vectoris there is no freedom to

equal.

all

in

vectors

of

definition

equality. any two vectors (Figure 4a). for vector B so that its origin of A. Now we construct a third the terminus at placed A + B, whose origin coincides with the called vector, A with the terof and whose terminus coincides origin

Let

Addition.

B be

and

A

is

a location

select

can

We

minus of B.

The construction of

nal sense

(ii)

B =

+

A

Addition

(A

Part (ii)

the

indi-

both are the same diagothe same parallelogram, and they possessthe same Hence we have the followingresult. of direction. =

A

Theorem 1. (i) that is

A clearly

B +

of

4b)

(Figure

cates that B +

of

the of

definition

A +

B, for

Addition

of

is

vectors

commutative;

B + A. of

vectors

+

B)

+ C

theorem addition.

that is

is associative;

=

A

+

is easily

(B +

C).

established 5 illustrates

Figure

Co> \037

Co \037

\037 \037

\037 \037

\037)

B+C

\037)

A)

A)

FIGURE

6)))

by using the proof.)

GEOMETRY)

VECTOR

ELEMENTARY

10)

the However, the reader is advisedto phrase proof from deduction elementarygeometryindependent logical of

as a

a figure.)

EXERCISE

1. Give an elementarygeometry vectors areaddedto equal vectors, As indicated in Section1, the toricallYr notably

from

an It

force.

but nonethelessexcellent,Dutch experimented

(1548-1620),

replace the covered by the

two

by

sums

the

of a

notion

to characterize

attempt is interesting

a single

theorem: If equal are equal vectors.

of the

proof

his-

physical quantities,

to note that the little-known, in

forces

one, called the

the resultant was actually the diagonal of a parallelogram,of which

sented the two original forces (Figure6). of formulation of the principles of addition complete statics.

an effort He

resultant.

that

a

Stevin

Simon

scientist, two

with

used extensivelyin developing rium-the beginningof modern

vector arose,

force

the This

forces,

theory (It is for

to dis-

represented

sides repreto his led

which he of equilib-

this reason that the parallelogramsof Figure 6 are sometimes referred to as parallelogramsof forces.) the many other accomAmong of are his: work on hydrostatics,which Stevin (1) plishments of) laid for the reclamation of the below-sea-Ievel land plans

Force Fl)

Force

Fl)

Force

FIGURE

6)))

Fl)

and (2) developmentof

Holland, with

the

first

to

11)

OPERATIONS)

ELEMENTARY

a give entitled

chapter

systematic \"Stevin

for numbers

notation

decimal

He was the

for computation.

methods

consequent

treatment of decimals. (Seethe on Decimal Fractions\" in A Source

by D. E. Smith, or A History oj MatheJ. F. Scott, 1960.) of vector addition is consistentwith our Thus definition the desires of the physicist who is interestedin applying techthe of vector analysis to his problems. (The of student niques a critical science should constantly maintain attitude applied toward the mathematical definitions, care to see whether taking or not they accurateiy reflect situations.) given physical Before continuing, it should be mentioned that Galileo (15641642), quite independently, cameto the sameconclusionas did Simon two scientists discovered how vectors Thus Stevin. two centuries prior to the inven\"should\" add, approximately tion of vector algebra and vector analysis in the nineteenth in Mathematics,

Book

matics, by

century.

Our

can

addition

vectors:

n

of

Of course,

(ii) of

of

definition

sum

the

Al +

this can be doneby

Theorem 1) and applying the

However, as

simply

grouping the

pairs

definition

(note

part

repeatedly.

process might be described that its origin is at the so that its origin is at the Aa

geometric

follows: of

terminus

now be extended to find \302\267 \302\267 \302\267 + An. 2 + Ag +

A

AI;

move move

A 2 so

of A 2 ; continue this process until An is placed its origin at the terminus of An-I. The sum Al + \302\267 \302\267 \302\267 A is then the vector whose + + An g + origin

terminus

with A

2

coincides cides the with

with

the

origin

terminus

of Al

and whose

terminus coin-

of An.

What would be the sum of the vectors that form a closed polygon with arrowstakingusallthe way around? to find the answer before reading on.) Consider, (Try for A + B + C + D + E + F 'of Figure 7. example, This is the vector whose the coincides with origin origin of A and whose terminus with the terminus of coincides F as))) after are placed \"origin to terminus\" the vectors

FIGURE

sum

We then

length.

A

This answer

+

our

D+ E+ F=

C +

B +

query

for

a polygon

o.)

of n sides, so the

zero vector.

is: The

Multiplicationof a by convenient to introduce

a scalar.

vector

In arithmetic as

multiplication

of addition. 4 + 4 + 4.

the

zero

write

holds

argument to

the origil1 and terminus of same point, al1dthe vector is of

are the

vector

7)

Hence

above.

described

GEOMETRY)

VECTOR

ELEMENTARY

12)

For example, 3 X 4 may be

\037\037 rt,\037)

FIGURE

8)))

\037

of

thought

Similarly, we can-at least to begin

x

it is

extension

an

with-)

OPERATIONS)

ELEMENTARY

13)

think of multiplying a vector by a of vector addition. An illustration A +. A. that 2A should represent of

we

addition,

parallel to

the

however,

A +

vector

know

and

A

the

having

of A +

length

is

A

twice

is

8)

(Figure

From our definition

same sense A

extension might be

as an

scalar

a vector

actually

of direction as A; the

length

of A.

A by A, the result of multiplying 2 is a vectorparallelto A having the same sense of direction as A but with twice the length of A.

Therefore,

=

2A

if

A +

the scalar

Before proceedingto the general case of multiplying vector by a scalar, let us considerthe questionof would

be appropriate

f\037r

ble demand might

for the

In

parallel of

if

general,

A

X =

+

to A, (b)

direction

this

stipulate

moment,

opposite

+

A

and

see where (a)

(c) X must

=

IAI

(-A)

= 0;

0, we know that

\\xl, and to that of

A reasona-

of -A.

a definition

be that

A

(Figure

a what

9).

so let us,

it takes us. X

must

be

have a sense Thus-A

precisely the properties a, b, and c menin the previous sentence. (Alternatively, if tioned A + X = 0, then A followed by X can be thought of as a closed in which the origin of X is at the terpolygon minus of A.) Consequently,our definition should (and that a will) stipulate multiplying by negative scalarhas the effect of changing the sense of direction of a vector. We are now ready to present a definition for the multia of a vector scalar.) by plication should

have

FIGURE

9)))

VECTOR

ELEMENTARY

14)

GEOMETRY)

;1A)

FIGURE 10)

3

vector parallel to A with magnitude In \\ = the times of A. In symbols, InAI magnitude Inl iAI. to have the same sense of Further, if n > 0, nA is defined direction as A; and if n < 0, nA is defined to have a sense direction to that of A; finally, if n = 0, nA is of opposite to be the zero vector (which follows from the first defined Definition.

sentence

nA

of our n =

Figure definition). 3, n = -3, and n

for

init.ion

Theorem

2.

is a

m(nA)

(ii) (iii)

illustrate, a

=

nA 3

2A,

The

sy-mbol

(m + n)A

definition

=

=

(mn)A. mA

nA.

+

B) = mA + consider m = 5 and meA +

vector twicethe

Inl refers

as follows:If n > 0, then rfhe

i-.

If m and n are scalars,then (i)

To

10 illtlstratesthe def-

=

asserts that the absolute

n

=

n

of

length

to the absolute value = n; and if Inl

mB.

- 2.

A btlt

directed

of n,

which is

< 0,

then

value

of

always non-negative; e.g., 131 = 3, 1-31 = 3, and = shall need the fact that of \\mnl Imllnl, the truth he clear from definition.))) the

a 101

Then

Inl

defined =

number =

which

O.

-n.

is We

should

15)

OPERATIONS)

ELEMENTARY

(i) states:

oppositely to A.

5(-2A) = (5)(-2)A

=

(-2))A =

(5 +

states:

(ii)

3A =

5A

+

5A

+

-lOA.

or

(-2)A

(- 2)A.

(iii) states: 5(A + B) = 5A + 5B. Proof. (i) By examining the length of ber of (i), our definitionof multiplication a scalar yields) 1 m(nA)

I

=

=

ImllnAI

that the directions that

of

same

the

have

the definition (ii)

=

they are parallel follows multiplies of A. The reader

That length. that both are

a vector

by

= ImnllAl

ImllnllAI

that the vectors of

1 proves

Equation

mem-

left

the

of

I

are

(i)

(1)

(mn)AI.

in

equal

fact

from

the

is left

to check

sense.

Use

(Hint.

nA..)

+ n = 0, both sidesof (ii) point in the same direction \037s A. If m

vectors

represent

m +

If

n <

0,

that point in the direcboth sidesof represent vectors tion oppositeto that of their The A. comparison lengths is left to the reader. (Hint. Usethe (ii)

of

definition

of

nA.)

Let us

(iii)

We consider the

nonzero. A

+

B

suppose that

(see

--7

then P R

=

and

Bare

B.

and

nOllparallel

trianglePQR,-?

which

11), by having A

Figure A +

A

=

PQ,

defines

-?

B =

QR,

P' Q'

R'

triangle , -? \037 = = = mA mA where + mB. P'Q', mB Q'R', then P'R' is similar to triPQR SincemAilAandmBilB.triangle

-?

We

construct

-?

angle

is meA + B), and we have = mA + mB. + B) meA The reader should consider two questions cOllcerning of (iii). The first is: What of the direction proof

result

the

P'Q'R'.

Thus

P'R'

the

that

of)))

GEOMETRY)

VECTOR

ELEMENTARY

16)

p')

p) A)

mA)

11)

FIGURE

B) as comparedto that of mA + mB? The and with the proof explicitly concerned itself, lengths of the two sense parallelism vectors, but it didnotdiscuss of The direction. second is: Does the proof question

m.(A +

break

if

down

A\\.IB?

Since

NOTE.

part

(i)

change of parenthesis is legitimate,we be no confusion if we eliminate For entirely and write mnA. example,) = (3

3(2A)

=

\302\267

2)A

As il1 elementary

Subtraction.

that traction is operation we define subtractionof

is

an

2A

the

=

parenthesis

6A.)

arithmetic, where subthe inverse of addition, of' vector the inverse real number, we write the expresses equation and in terms of addition as

vectors

More precisely,if

addition.

\302\267

3

there

know

now

would

a

is a

a - a = a + (-a) O. This fact that subtractioll is of addition. that subtraction is the -a is the that the realilumber maticians say =

defined

Mathe-

inverse

a relative

is the

to the

operation inverse

additive

of

of

a

inverseof -a). carry defining subtraction.

Definition. to

mean

-lB.)))

A

-

B =

A

+

or

additioll

(similarly, these

We

a

that

states

2

Theorem

of

(-B)

ideas

simply

a is

that

of

- a

the additive

over to

where

inverse

-B

vectors in

is

written

ELEMENTARY OPERATIONS)

17)

of subtraction can take in 12. Note that the any preeented Figure A of the is equal to the sum diagonal parallelogram B (A - B) and also to the sum (A - B) + B. Such who is for the algebraic checking advised beginner the

Geometrically

operation

forms

the

of

+

is

having difficulty in finding the correctorientationfor the difference A-B. With these few tools of addivector and subtraction we can begin applying vectorsto tion geometry.

elementary

We shall use our vector operationsto work an states exercise, one equivalent to the theorem which elementary that the diagonals of a _parallelogram Let each other. bisect not on one line. Call M the mid0, B, and C be threepoints We shall prove that BC (see Figure 13a). point of segment 1.

EXAMPLE

--+--+

--+

OM

cussed

=

(OB \037

+ OC).

on page

In accordance with

5, we shallwrite

\037

=

B

OB,

convention

the --7

C = OCand M

Then) \037

M

= B

+ BM)

M

= C

- MC = C \037

and

\037 BM

\037

(since

Bill

\037

= MC).)

B)

-B)

B)

FIGURE 12)))

dis\037

= OM.

GEOMETRY)

VECTOR

ELEMENTARY

18)

B)

B)

D)

o)

o)

c)

c)

(a))

(b))

13)

FIGURE

we get)

Adding,

2M=B+C

M =

or)

If we considerour figure

(Figure13b),

then

o to the midpoint

to

the

result

of

diagonal

\037

for

=

IODI

we

conclude

may

eachother.

of a

t\\VO sides

any

one half of

that the line joining the triangle is parallel to the third

Prove

2.

EXAMPLE

be part of a parallelogram OBDC states that the line joining vertex BC is one half the diagonal OD,

In equivalent (and more usual) language CI. that the diagonals of a parallelogram bisect

+

\\B

it.

In triangle PQR (seeFigure of

PQ

A

=

and

\037

Then

=

.C

A

these

that

(Note

\037

B = PkI

= NR,

vector

Adding

and D

to

of

that

C

equality

=

A - B, we of vectors)

the segment

and

N be

C+

midpoints

\037

\037 C

J.lfQ, A

-

follow from

=

MN,

D

to

equal

and

D =

\037

QR.

- B = o. about

vectors

summing

quadrilateral JfNRQ, respectively.) both sides of the last equation, we get C +

Since

=

- Band equations

MNP

triangle

111 and

let

14)

midpointsof

Call)

respectively.

PR,

\037

PN

+ C).)

(B \037

A

- B = D.

(by the proves simultaneously thatNM\\lRQ have

2C

= D, which

NM equals one half

of

the

base

definition

RQ.)))

and

OPERATIONS)

ELEMENTARY

19) p)

R:)

14)

FIGURE

EXERCISES

the easily rememberedSHORTCUTLEMMA: \037 AB + BC = AC. (This lemma has also been appropriately the Bypass Lemma by Professor D. W. Hall.) named 2. Reproduce Figure 15 on another sheet. Then construct and labelthe vectorsC - A, B - C, B + C, and -B - C. 1. Establish \037

\037

\037

3. four

Show

that

arbitrarily

PQ + chosen

\037

TlS =

\037

where

2MN,

points

P,

and where

FIGURE

15)))

Q, R, \037f

and

and S N

are

are the)

midpoints of PR and QS, respectively. around the polygon NMRS and 4.

Draw

5.

\037

\037

point,

+ XR.

current. The 5 mph N. and give a the resultant. is

of the forces. (vectors) acting is the zero vector, solve the following

in equilibrium

a body

mid-

\037

force resulting the direction of

the

.of

geometricconstructionthat shows 6. Using the fact that the sum on

\037

X Q

+

XP

upon by the wind and mph E and the current velocity

the magnitude

Compute

of A.B, Q the

CA. If X is any

acted

is

is 8

velocity

\037

XB + XC =

NMPQ.)

midpoint

midpoint of

\037

sailboat

A

wind

P the

with

R the

and

XA +

that

show

ABC,

triangle BC,

the

Sum

(Hint.

vectors

point of

GEOMETRY)

VECTOR

ELEMENTARY.

20)

problem.

of 100 lb hangs by a wire and is pushed by a horiA weight zontal force until the wire makes an angle of 1r/4 (or 45\302\260) the vertical. with Find the magnitude of the horizontal force and the tension in the wire.

7.

8. If

the

that

Show

A,

D are

any four points \037 \037 AB + AD + CB + \037

that

plane),

prove

and Q

are the

of

a

quadri-

of a parallelogram.

and

C,

B,

sides

of consecutive

midpoints

are vertices

lateral

of

midpoints

and

A.C

in a

necessarily

(not \037

\037

=

CD

4PQ

where

(How does

BD.

relateto Exercise

P

this

2?)

Using associative (A

-

properties -

B) and

(A

10. Establish the

nology: the 0'

vector

0' =

4.

of subtraction, the

definition

the

9.

zero

of addition, B) +

B, do actually

that has the

reduce

to

equal

A.

the termithus justifying Consider the possibility of a properties of O. Then prove that

uniquenessof

vector.

and

commutative

show that the sums B +

0,

(Hint.

0.))

LINEAR

COMBINATIONS

OF

VECTORS

we have learned to add and subtract vectors, we can combine multiply vectors by scalars, to enrich and vectors these operations to generate new our algebra of vectors. For example,if we are given A)))

Now that and also to

21)

OPERATIONS)

ELEMENTARY

can perform our various operationsto get B, B, 2A - 3B, 5A + 6B, etc. Such combinationsof A and B are called linear combinations of A and B. The set of aillillear combinationsof A and B could be written {xA + yB Ix and y real}. 4 The defil1ition of linear combinations is now extended in the folB, we

and

A

+

A

If

way:

lowing

Xl, X2,

.

X3,

.

xiA

AI, ,X n

.

+

I

A 2, A 3 , . . . are n scalars,

X2A 2

+

X3

and

n vectors

the vector \302\267 \302\267 \302\267

+

A g

are

, An

+

xnAn

of AI, A 2, A 3 , . . to be a linearcombination of section is devoted to the study The present of certain sets of vectors, and the combinatiolls contained herein are perhaps the most difficult Thus we shall proceedslowly. el1tire book. The is cautioned to study the definitionsand to take

is said

.

,An. linear

ideas the

in

reader them

literally!

Iloted

We

that

X

tiplying

=

earlier that -A, that is,

Since

-1.

A by

could, in a sense,say

conversely,that

,ve began with

A

the equationA X is the vector X is

that

is

A

dependent

Y

=

A =

and)

X =

obtained

0 implies by

from

derived

mul-

A, we

depe11dent upon A; UpOll X. Similarly,

or, if

we could write

3Y = 0,

+

be

can

X

+

--lA

-

3Y.)

Y is shown to depel1d 011A and A to depel1d on A a11d to state that Y. It might be preferable simply that to observe Yare dependent. It is almost trivial if the scalar such dependency would be impossibleto sho,v Hellce we exclude coefficients of A and Y were both zero. Thus

this case

in

consideratiol1

from

making

the

following

definItion. 4

The

defining

z

satisfying

synlbolism sets in the the

{I

},

following

condition

borrowed \"ray:

or sentence

from {zIS(z)}

S (z).)))

set theory, is useful in the set of all represents

VECTOR

ELEMENTARY

22)

Two vectors

Definition.

dependent if and

not both

zero,

if

only

aA +

so that

A

there

are called two scalars

B

and

GEOMETRY)

exists

bB = o.

Remark:The studentshould

recognize

the

linearly

a and b, fact

that

of words are statements of logical definitions equivalence and its (see footnote 1, page 6); i.e., boththe statement hold. converse Using our present definitionto illustrate this explicitly, we would say that the definition states: dependent implies that and both of them zero, so that b, not aA aA + bB = 0 holds, + bB = 0; and (2) if a relation a and to not both b zero with equal (i.e., at leastonebeing A and B are linearly dependent. then nonzero), is equivalent to saying that Algebraicallyour definition B and being (1) there existscalarsa A

linearly

two vectors are linearly dependentif and only if (see \"if and only if\ footnote 1 for explanation of the phrase one of them is a scalar multiple of the other (show this!). A geometric would be the following:Two interpretation if and only if they are vectorsare linearly dependent

parallel. The

can

reader

verify

these

interpretations

of the form formally by constructing a general that preceded our definition. One point, however, must be mentioned; this concerns the presenceof 0 as argument fine

one

of

that 0 10 +

vectors

the and

any

under vector

consideratio11. A are linearly

First,

we note

dependent, for

OA 0 and for 0, which is the definingcondition be linearly depe11de11t nonzero (1 is the required would say that Our scalar). geometric i11terpretation o isparalleltoA, where A may be any vector. This may howto the beginning student of vectors; appear strange it is a of conmatter convenience to retain the ever, great vention that the zero vector is parallel to every vector! as having no direcInstead of regarding the zero vector tion (as would to some), we regard 0 as having appeal and direction, Vectors direction. any specify magnitude so we chooseto say that 0 has any or all directio11s simul-)))

A to

=

OPERATIONS)

ELEMENTARY

the zero vectoristhe only

Of course,

taneously.

23) vector

a property, for any other vector (line segment has a u11iquedirection. Later,when of nonzero length) work we with perpendicular vectors, we shall have occasion regard zero vector as perpendicular the to every such

with

to

vector.

zero

has

vector

In the we

ent, we

the idea

with

consistent

is

this

Again,

that the

direction.

any

event that two vectors arenot linearly dependcall them linearly independent. Summarizing,

that:)

say

of vectors,

A pair

(1)

is a

one

which

of

is a

zero vector,

linearly dependentsetof vectors. A

(2)

(3)

of

pair

vectors is a linearly

nonzero

parallel

dependentset. A

of

pair

a linearly

vectors is

nonparallel

nonzero,

independent

set.)

merely the generalizing would leave us

contradict fllrthermore, of linear

and, theory

extend Xn,

so

where not all

If,

that

A 2,

. . if

example,

of

the

are

x's

the

zero).

implies

one

definition.

following

. .

. , An a set

exist

X2 A 2

+

of

is called

n vectors

of scalars Xl,

Xl

. , An three

\302\267 \302\267 \302\267

+

(i.e. when

zero

X2A 2 = is

X2 = said

+

XnAn

+

on the other +

xlAl

AI,

A 2,

vectors

.

X2,

.

.

,

that)

xlAI

equal

the

with

Definition. A set AI, linearly dependentif there

spirit

of

mathematics

of

with a rather meager We therefore proceed to

dependence.

notions

Ollr

to pairs

attention

our

Confining

would

hand,

the

\302\267 \302\267 \302\267

+

\302\267 . = \302\267

to be \037

= 0,

at least one doesnot equation

xnAn

Xn =

= 0

0, . then

linearly independent.

vectors, AB, cases must following

\037

set

the

For

\037

CD, and EF,

occur: (a) It

are

is

given,

possible)))

three scalars a,

to find

\037

\037

not

is

It

to

possible

The

shall

we

of

a

geometric

3.

any

vector

third

bination of A

are

the they

It dealing

segments

be

can 16).

Figure mille

form

step ill the

A

C;

and

B

and

which

are

is parallel

B, can

be

then

independent,

linearly

to (or

expressed

as

in)

the

plane

com-

a linear

B.

be should be recalled that vectors may they possess a commonorigin, that is, with free vectors. Therefore, even when A, B, and C might be in space, representing to positions in the same plane (see moved A and B with a commonorigin deterFor

so that

arranged we

If

by A and

Proof.

o.

may be rather

theorem.

Theorem determined

to be linearly dependto be linearly independent. a = 0, b = 0, and c =

an intermediate

provide

In

scalars.

said

to n vectors

vectors

two

from

jump

steep,so

if

hold

will

that)

(*))

such

three

find

the three vectorsare In case (b) they are said

In both cases(*)

such

\037

case (a)

ent.

all zero,

not

c,

+ cEF = o.)

bC D

+

aAB

(b)

and

b,

GEOMETRY)

VECTOR

ELEMENTARY

24)

a

alld

pla11e;

C (being

parallel to this plane) may

then be displacedsothat it is

actually

the

in

plalle

A and B.)

'B) \037)

A)

B

\037

)) A)

FIGURE

16)))

of

OPERATIONS)

ELEMENTARY

I

C

I

yBII

25)

B xA

\037----

A)

0

/

and

0

oX <

A

> a

y

/x >o

is a

if C

nOllzerovector, there isa parallelogram

C alld

diagonal

y>O

17)

FIGURE

Now,

and

(b))

(a))

with

A \037

I

A)

'\\vith

B.

A and

along

edges

An

in explicit constructioll of this parallelogramis given 17 and call be as described follows:Call0 the Figure common of the vectors A, B, and C, and call A, origin and C the respective B, endpoints of the three vectors. Constructa line \302\243through C parallel to B, alld call D

theintersectionof

the

\302\243 with

line

of actiol1

of A

the

(when

\037

of A

origin

of

multiple

is 0). A; let

\037

=

DC

yB.

OD =

us say

xA.

In the

trivially

\037

DC,

let

B;

us

say

Then)

C = xA

+

(2))

yB,)

of C to be a linear combination the event that C is the zero vector, true, for)

shows

which

some

is

l\037\"urthermore,

is some multipleof

to B,

parallel

being

Then OD, being parallelto A, \037

o

=

A

B.

and

is

theorem

OA + OB,)

(3))

and our proof is complete.

Toassociate more

the

concept

Corollary.

strollgly

of lillear Any

linearly depertdent.)))

three

the

idea

3 '\\vith

we state the

dependence, vectors

of Theorem

in

the

same

plane. are

Proof. Equati