Elementary Vector Geometry

seymour

schuster)

A)

-

---

_N

---)

u)

A')

B)

---L)

B)

-...--

'\\.. \037)

-)

c')))

ELEMENTARY

VECTOR

GEOMETRY)

SEYMOUR

SCHUSTER)

DOVER

_{PUBLICATIONS,}

INC.

Bibliographical

Note

This

Dover

edition,

first

published

in

2008,

is

an

unabridged

republication

of

the

work

originally

published

by

John

Wiley

&

Sons,

Inc., New

York,

in

1962.)

Library

of

Congress

Cataloging-in-Publication

Data

Schuster,

Seymour.

Elementary

vector

geometry /

Seymour

Schuster.

-

Dover

ed.

p.

em.

Originally

published:

New

York:

John

Wiley

&

Sons,

Inc.,

1962.

Includes

index.

ISBN-13:

978-0-486-46672-9

ISBN-lO:

0-486-46672-8

1.

Vector

analysis.

2.

Geometry.

I.

Title.

QA433.S38

2008

515'

.63-dc22)

2007050541)

Manufactured

in

the

United

States

of

America

preface)

This short

work

is

the

_outgrowth

of

lectures

that

were

presented

at a

National

Science

Foundation

Institute

held

at

Carletol1

_College

ill

the

summer

of

1959.

The

lectures

were

to

serve

the

_purposeof

\"enriching\"

the

backgrounds

of

geon1etry

teachers.

However,

the

pattern

of

events

in

mathematics

education

indicates

that

the

material

covered

is no longer

enrichment

material

_but,

_rather,

essential

_knowledge

for

every

teacher.

It

was

_just

a

few

_years

_ago

that

linear

_algebra'

was

a

course

for

_beginning

_graduate

students

and

vector

analy-sis

was

typically

taken

as

an

upper

class

course

by

mathematics,

physics,

and

engineering

students.

The

last

decade has brought

\037

revolution

in

undergraduate

mathematics

_education,

and

_today

the

knowledge

of

vectors

is

_acquired

at

a

much

earlier

_stage.

_Indeed,

it

is

quite

usual

for

college

freshmen

to

study

vectors

and

matrices,

particularly

as

applied

to

geometry.

Further-more,

the

studies and

recommendatio11s

made

by

the)

VI)

PREFACE)

Commission

on

_Mathematics,

the

School

M\037thematics

Study

Group,

and

the

Committee

on

the

Undergraduate

Programin

Mathematics

all

_point

in

the

direction

of

getting

some of

these

c011cepts

into the

high

school

curriculum.

I

have

_1011g

felt

that

vector

_techniques

should

and

,viII

find

their

_way

into

the

_high

school

curriculum-perhapsnotas

an

_integral

_part

of

the

mathe-matical

training of

all

students

but,at the

very

least,

as

work

to

excite

and

_challenge

_superior

students.

On

a

very

elementary

level,

this

textbook

deals

pri-marily

with

the

_development

of

vector

_algebra

as

a

mathematical

tool in

geometry.

The

aim

is

to

gain

a

greater

insight into

the

theorems

_by

_attempting

vector

proofs

and

analytic

proofs in

contrast

to the

synthetic

proofs,

a

knowledge

of

which

the

reader

brings

as a

prerequisite.

The

elements

of

vector

_algebra

are

devel-oped

slowly-more

so

than

in

_any

of

the

standard

works

on

vectors.

_Simple

geometric

explanations,

as

well

as

numerous

_{illustrations,}

are

used.

_Beyond

_this,

some

analytic

geometry

(in

two

and

three

dimensions)

is

developedasa

natural

_outgrowth

of

the

vector

treatment.

In

_addition,

the

vector

approach

is

used

to

assist

in

other

areas of

elementary

mathematics:

algebra,

trigonometry

(plane

and

spherical),

and

higher

geometry.

In

short,

vectors

have

been

_employed

whenever

it

was

felt

that

they

would

aid in

gaining

insight

and/or

in

facilitating

computations

and

proofs.

I

have

tried

to

_develop

_very

little

_machinery

but

_{to go}

a long

way

with

this

small

amount.

_Accordingly,

the

reader

will.

find

himself

dealing

with

such

topics

as

linear

inequalities,convexity,

linear

programming,

involutes,

and

projective

theorems.

As

for

_{prerequisites, they}

are not

listed

formally

because

I

do

not

claim

to

have

_given

a

_logical

(axiomatic)

development

of

geometry.

Loosely

speaki11g,

I

have

assumed

that the

reader

is

familiar

\\vith

the

definitions

PREFACE)

VII)

essentials

of

_{trigonometry.}

For

_example,

the

notions

of

angle,

parallelism,

and

area

are

assumed.

It

is further

assumedthat the

reader

knows

the

definitions

of

the

sine,

cosine,

and

tangent

functions

_(in

the

naive

_sense,

as

ratios

of

the

sides

of

a

_right

_triangle).

In

_regard

to

resliits

from

_geometry

and

_{trigonometry,}

I

_haye

indeed

taken

very

little

for

granted.

Samples

of.

geometric

information

that

are calledupon

are:

formulas

for

the

area

of

a

_{parallelogram}

and

volume

of

a

_{parallelopiped\037}

the

fact

that

two

_points

determine

a

unique

line,

and

the

result

that

_opposite

sides of

a

_{parallelogram}

are

_equal.

In

_Chapter

3

I

use

_{the)aw of}

cosinesfor

_motivatio11,

but

the

reader

who

has

not

seen

it

before

will

be

consoled

by

a

proof

given

shortly

thereafter.

It

is

_entirely

_possible

to

give

a

vector

development

of

Euclidean

_geometry

from

\"scratch.\"

_{In fact,}

some

people

believe

that

a

first

course in

geometry

should

begin

with

vectors.

Others

believe

that

the

coordinate

method

should

be

given

at

the

olltset,

and

still

others

have

faith in

a combinatio11

of

_approaches.

The

coordinate

_development,

in

various

_forms,

has

_recently

been

adopted

by

several

of

the

current

groups

interested

in

_rewriting

the

high

school

mathematics

curriculum.

For

the

reader

interested in

_seeing

how

a

strict

vector

approach

would

do

the

job,

I

strongly

recomme11d

the

excellent

_paper

\"Geometric

Vector

_Analysis

and

the

Concept

of

Vector

_Space\"

_by

Professor

Walter

Prenowitz.

This fine

_exposition

constitutesoneofthe

chapters

of

the

Twenty-

Third

Yearboolc

of

the

National

Council of

Teachersof

l\\lathematics.

Sincere

thanks

and

_appreciation

are

due

to

the

teachers

who

came

to

Carleton

i11

the

SlImmer

of

1959 in

the

_hope

of

comfortably

learning

mathematics

in

cool

Minnesota

but

_who,

_instead,

laboredal1dperspired

under

the

strains

of

vector

_geometry

and

the

_{96% humidity.}

For

reading

VIII)

PREFACE)

especially

grateful

to

Mr.

Saul

Birnbaum

of

the

New

Lincoln

School

in

New

_{York City,}

Professor

Roy

Dubisch

of

the

_University

of

_Washington,

Professor

J.

M.

Sachs

of

the

_Chicago

Technical

_College,

and

_my

Carleton

colleague,

Professor

William

B.

_Houston,

Jr.

_Also,

special

thanks

go

to

ProfessorDick

Wick

_Hall,

who

courageously

used

the

text

in

mimeograph

form at a

National

Science

Foundation

Institute

in

the

Sllmmer

of

1961.

This

_experience

enabled

Professor

Hall

to

con-tribute

substantially

by

pointing

out

ID.y

errors

ill

judg-ment

and

typography.)

SEYMOUR

SCHUSTER)

N

_orthfield,

Minnesota

contents)

Chapter

1

ELEMENTARY

OPERATIONS)

1)

1.

Introduction.

2.

Definition

of

vector.

3.

Fundamental

_properties

\302\267

_4.

_Linear

_combinations

of

vectors.

5.

_Auxiliary

_point

_technique.

6.

Uniqueness

of

representations.)

Chapter

\037

VECTORS

IN

COORDINATE

SYSTEMS

₄₀₎

7.

_Rectangular

_{systems and}

orientation.

8.

Basis

vectors

and

_applications

\302\267

9.

The

_complex

plane.)

Chapter

3

INNER

PRODUCTS)

60)

10.

Definition.

11.

_Properties

of

inner

_product

\302\267

12.

_Components.

13.

Inner

_product

formulas.

14.

Work.)

x)

CONTENTS)

Chapter

4.

ANALYTIC

GEOMETRY

15.

Our

pointof

view \302\267

16.

The

_straight

line

\302\267

_17.

A,nalytic

geometry

of

the

line

continued.

18.

Distance

from

a

_point

to

a

line.

19.

_Analytic

method of

proof.

20.

Circles.

21.

Spheres.

22.

Planes

\302\267

_23.

Det\037rmining

a

plane

by

points

on

it

\302\267

24.

Distance

from

a

_point

to

a

_plane

\302\267

_25.

_The

straight

line

in

three

dimensions

\302\267

26.

_Angle

be-tween

two

lines

\302\267

_27.

_Intersection

_of

_a

_line

_with

a

plane

\302\267

_28.

Angle

between

a

line

and

a

plane.)

Chapter

5

CROSS

PRODUCTS)

29.

Cross

_products.

30.

Triple

scalar

product.

31. Distance

from

a

_point

to

a

_plane

\302\267

_32.

Dis-tance

between

two

lines.

33.

_Triple

cross

products.)

Chapter

6

TRIGONOMETRY)

34.

Plane

_trigonometry

.,35.

Spherical

trigonometry.)

Cl\037apter

7

MORE

GEOMETRY)

36.

Loci

defined

by

inequalitjes.

37.

A

few

booby

traps.

38.

Segments and

convexity.

39.

Linear

_programming.

40.

Theorems

_arising

in

more

_general

_geometries.

41.

_Applications

of

parametric

equations

to

locus

problems.

42.

Rigid

motions.)

APPENDIX

ANSWERS

INDEX)

76)

135)

151)

160)

204

fJ06 \03711)))

elem.entary

operations)

1.

INTRODUCTION

The

_history

of

the

_development

of

mathematical

ideas

indicatesthat abstract

_concepts

arise

generally

from

roots

in

some

_{\"practical\"}

_problem.

Arithmetic

_stemmedfrom

problemsof

counting,

geometry

arose

from

problems

of

surveying

land

in Egypt, and

calculusdeveloped

princi-pally

from

the

efforts

to

solve

the

problems

of

motion.

Mathematics,

however,

goes

quite

beyond

the

point of

solving

merely

the

problems

that

initiate

the

particular

study, for

mathematics

is

concerned

with

building

a

deductive

science

that

is

_{general and}

_{abstract,that}

may

have

a

wide

_range

of

_application.

_By

a

deductive

science

we

_mean,

_briefly,

a

_logical

_development

that

begins

with

a

basic

framework

consisting

of

a

set of

assumptions

(called

axioms

or

_postulates)

and

a

set

of

terms

used in

stating the

assumptions.

All

the

_logical

_consequences

of

the

_assumptions

are

then

the

theorems

of

the

deductive

science,

which

is

concerned with

abstractions

or

2)

ELEMENTARY

VECTOR

GEOMETRY)

tions

of

_concepts

from

the

original

problem

rather

than

with

the

_original

_problem

itself.

For

example,

the

study

of

_geometry

is

based

on

a

set of

assumptions

that

deal

essentially

with

lines

and

POil1tS

rather

than

with

fences

and

_fenceposts.

Line

is an

abstract

_concept;

it

is

an

idealization

of

the

_fence,

and

it

admits

to

all

sortsof

other

_{interpretations:}

a

ray.

of

light,

the

edge

of

a

board,

the

_{path of a}

molecule

under

some

_{circumstances,}

and

a

host of

still

others. Thus

geometry,

with

its

origins

in

surveying,

finds

application

in

a

variety

of

problems.

The

_pure

_{mathematician,}

_however,

_goes

on-and far

beyond.

Because

oncehe

begins

a

mathematical

study,

he

is

free

to

exercisehis

imagination

by

making

logical

deductiollS

(proving

theorems)

and

developing

theories

that

are

_qllite

_apart

from

the

realities

of

the

motivating

problem.

li\"'or

the

mathematician

there

is

a

reality

within

his

deductive

science.

Again

drawing

from

geo-metry

for

illustration,

we

can

point

to the

developments

of

fOllr

dimensional

_{geometry-even}

n-dimensional

geometry-in

spite of

the

fact that our

physical

world

is

three

_{dimel1sional,}

or

to

the

invention of

non-Euclidean

geometries,that contradict

Euclid'sParallel

Postulate

(which,

for

over

2000

years,

was

accepted

as

absolute

mathematical

_truth).

Such

creations

_by

mathematicians

were

_consequences

of

_strong

_{imagination and}

_quite

beyond

the

consideration

of

any

elementary

problem

in

the

physical

world.

Vector

analysis

is

also

a

subject

that

has

its

roots

in

physical

problems.

It

was

developed

primarily

to

handle

_{problemsin physics,}

initially

problems

in

me-chanics

_but,

_later,

_problems

in various

other

branches

of

physical

science.

Developments

of

th\037

nineteenth

and

twentieth

centuries

have

resulted

in

_{the abstractconcept}

of

a

vector

and,

consequently,

in

a

wide

range of

ELEMENTA.RY

OPERATIONS)

3)

tors

now

_play

a

_prominent

role

in

a

variety

of

studies,

to

name

_just

a

few:

_physical

_chemistry,

fluid-flow

_theory,

electro-magnetic

theory,

economics,

psychology,

and

electrocardiography.

Geometry

books

are

_aI,vays

filled

with

illustrations

in

spite of

the

fact

that

point,

line,

and

circle

are

abstract

concepts

that

do

110t

existin physical

reality

and

in

_spite

of

the

fact

that

beginning

students

are apprisedof the

abstract

nature

of

the

_subject

at

the

_very

_{beginning of}

their

course.

The reason

is

quite

simple.

Abstract

reasoning

is

difficult;

students

therefore

need-or

are at

least

assisted

_by-the

help

of

some

real

model

(or

inter-pretation) of

the

abstract

concepts.

Consequently,

a

small

dot

marked

with

a

_sharp

_pencil

is a

convenient

model

for

the

_concept

of

_point,

and

a

_sharppencil

drawn

along

the

edge

of

a

ruler

leaves

a

n1arkthat

isusedas a

model

for

the

_concept

of

line.

Such

_pencil

marks

are a

great

convenience

for

beginl1ers

until

they

get

to

feel

at

home

in

the

_subject

and

_begin

to

feel

that

there

is a

reality

in

geometry

itself.

Later

in

their

mathematical

studies

students

encounter

other

abstract

_concepts,

but

by

this

time they

can, and

do, use

geometric

models

to

assist

them

in

still

more

abstract

_reasoning.

This

pattern

of

development

is

precisely

what

occurs

in

the

study

of

vectors.

Although

the

concept

of

vector

can

be

made

_abstract,

a

_geometric

model

_(directed

line

seg-ment)

is the

crutch

that

assists

the

_beginner

in

develop-ing

steady

legsinthe

field

that

is

new

to

him.

It

is

the

author's

view

that

_steady

_legs

in

abstract

vector

_algebra

_{are developed}

slowly

and

that

reasoning

in

the

model

_(now

_geometry)

for

some

extended

_period

should

be

done

_preliminary

to

_engagingin

the abstract

study.

Therefore

this

entire

textbook

co\037cerns

itself

with

a

_geometric

study

of

vectors

(i.e.,

the

application

of

vectors

to

_{geometry),in contrast}

to the

general

abstract

4)

ELEMENTARY

VECTOR

GEOMETRY)

2.

DEFINITION

OF

VECTOR

Earlier

we

_pointed

out

that

the

idea of

vector

came

originally

from

physics.

Therefore,

let

us

consider-from

the

_physicist's

_point

of

view-the

statement

of

the

television

announcer

_who,

before

giving

his

final

\"Good

night,\"

states,

\"The

temperature is

110W 37\302\260

and

the

wind

is'

12

miles

_per

hour

in

a

_{northeasterly}

direction.\"

In

this

_simple

weather

announcement

we

observe

_examples

of

two

_distinctly

different

_{types of}

_quantitiesin

the sense

that the

first

_{(temperature)}

_requires

_only

a

_single

num-ber-with

_{units, of}

course-for

its

_description,

whereas

the

second

_quantity

_(wind

_velocity)

_requires

two

facts,

magnitude

and

dire-ction.

These

_examples

are

_typical

of

the

_quantities

encountered in

elementary

physics.

Hence,

the

following

simple

classification

is

made:

quantities

that

possess

only

magnitude

are

singled

out

and

called

_scalars,

whereas

_quantities

that

possess

both

magni-tude

and

direction

are

called

vectors.

In

addition

to

temperature,

examples

of

scalar

quanti-ties

are

mass,

length,

area,

and

_volume;

_and,

in

addition

to

_velocity,

_{examples of}

vectors

are force,acceleration,

a11delectricalintensity.

Just as the

mathematician

desires

a

_geometric

model

for

his

_general

_concepts,

so

doesthe

_physicist.

For

geometry-to

the

trained

physicist-also

has

a

reality

by

means

of

which

he

can

\"visualize\"

and

be

aided

in

his

reasoning.

A

convenient

geometric

model for

a

vector

is a

directedline

segment

(/)

because

this

possesses

both

_magnitude

_{(length) and}

_direction,

simultaneously.

This

_model,

which

suits

the

needs of

physicists,

is

also

quite

satisfactory

for

our

purposes,

for

it

is

our

aim

to

study

geometry

(by

mearlS

of

vectors).

Hence, for

our

mathematical

_development,

we

make

the

_following

for-mal

definition.

Definition.

A

vector

is

a

directed

line

_segment.

W

e

shall

ELEMENTARY

OPERATIONS)

5)

Q

Terminus

or

endpoint)

p

Origi

n)

.0)

c)

(a))

(b))

FIGURE

1)

will

be

used

to

_designate

the

_{length of}

vector

A.

In

the

event

that

the

vector we

_speakof

is the

directed

segment

\037

PQ,

we

emphasize

its

vector

nature

by

writi11g

PQ,

P

being

the

origin

and

Q

being

the

terminus

or

endpoint

of

PQ

(see

Figure

1a).

Another

useful

convention

when

\037

--7

\037

referring

to

several

vectors

OB,

OC,

and

OD

with

a

common

_origin

0

is

to

call

these

vectors

_{B, C,}

and

_D,

respectively.

That

is, if

a

discussion

is

concerned

with

several

vectors

_emanating

from

a

_single

point, we

may

designate

them

merely

by

their

individual

endpoints

(see

Figure

1b).

A

vector

of

_length

one

is

called

a

unit

vector.

The

notion

of

a

vector

of

zero

_length

(with

any

direction),

although

apparently

peculiar,

is

actually

a

great

con-venience.

We

refer

to

such

a

vector

as

the

zero

vector

and

shall

_point

to

its

usefulness

from

time

to

time.

The

notation

for

the

zero

vector

is

o.

The

direction

of

the

zero

vector

is

discussed

further in

Secti011

4.

Scalars,

being

merely

magnitudes,

will

be

real-num-bered

_quantities.

_(In

more

advanced

mathematics

scalars

_may

be

elements

of

the

_complex

numbers;

indeed,

6)

ELEMENTARY

VECTOR

GEOMETRY)

ever,

do

not

require such

generality

and

will

therefore

be

satisfied

_by

_restricting

scalars

to

the

real

_numbers.)

They

are

designated

by

lower-case

Latin

letters:

a, b, c,

or

by

numerals.)

3.

FUNDAMENTAL

PROPERTmS

Our

desire

is

to build an

_algebra

of

_vectors,

and

to

this

end

we

must first

_present

a

criterion

for

calling

two

vec-tors

_equal.

Definition.

Two

vectors

A

and

B

are

called

_equal

(written

A

=

_B)

_if

and

_only

if I

the

_following

three

conditions

hold:

(i)

A

is

_parallel

to

_B;2

(ii)

A

and

B

_possess

the

same

_{sense of}

_direction;

and

(ill)

IAt

=

\\BI,

i.e., the

length

of

A

_equals

the

_length

_of

B.)

It

cannot

be

_emphasized

too

strongly

that

vectors

may

be

_equal

even

if

_they

do

not

_possess

the

same

_{position in}

space.

As

a

matter

of

_fact,

our

definition

indicates

that

a

vector

A

_may

be

relocated

_provided

that

we

move

it

rigidly

to

a

position

parallel

to

its

original

position

and

that

we

do

not

_change

its

_length

or

sense of

direction

(see

Figure

2a).

It

may

therefore

be

relocated

in

a

posi-tion

with

its

_origin

_anywhere

in

_space

that

we

choose.

When

vectors

are

_given

this

_freedom,

_they

are

termed

free.

1

The

phrase

\"if

and

only

if\"

_points

_up

the

fact that

the

defini-tion

is

_actually

a

double

_implication,

or logical

equivalence.

That

is,

(a)

If

A

=

_B,

then

conditions

_(i),

(ii)

,

and

(iii)

hold;

and

(b)

If

conditions

(i),

(ii),

and

(iii)

hold,

then

A

=

B.

2

_We

_use

_the

_word

parallel

in

the

more

_general

sense

to

mean

\"parallel

or

on

the

same

line.\"

Although

this

usage

is

not

given

in

high

school

geometry,

it is quite

common

in

analytic

geometry,

where

two

lines

possessingequal

slopesare called

parallel

(see

Section

_16).

Thus

a line

is parallel to

itself,

and

a

vector

is equal

to

itself.

The

latter

would not

be

_tru\037

if

we

didn't

use

\"parallel\"

in

this

_generalized

sense.

_{Figure 2b}

exhibits

two

equal

vectors

ELEMENTARY

OPERATIONS)

Y

y

Y)

(a))

7)

\"

\"

,

\"

_,

\"-\"

,

\

(b)) '\\

,

,)

FIGURE

2)

In

the

study of

geometry

this

liberty

to

make

displace-ments

of

vectors

is

_highly

_{advantageous.}

In

the

applica-tions

of

vectors

to

other

sciences

this

freedom

is

not

always

granted,

for

it

is

necessary

to

restrict

vectors

to

some

_degree.

For

_example,

in

the

mechanics of

rigid

bodies

it

isoften

required

that

a

vector

be

confined

to

a

line;that

is,

it

may

be

moved

rigidly

but

only

in

the

line

it

lay

originally.

This

line

is

referred

to

as

its

line

of

action.

In

_Figure

3a

we

show

three

_vectors,

_F,

_G,

and

H,

which

represent

three

forces

of

the

same

magnitude,

acting

on

parallel

lines

and

haviIlg

the

same

sense

of)

/

/

/

/

/

z/)

(a))

(b))

FIGURE

3)))

8)

ELEMENTARY

VECTOR

GEOMETRY)

(a))

(b))

FIGURE

4)

direction.

_They

would

therefore

be

_equal

_according

to

our

definition.

F

_represents

a

_pulling

force

at

the

center

ofthe bar

and

G

_represents

a

_pushing

force

at

the

center

of

the

bar.

F

and

G

would

have

the

same

mechanical

effect

and

are

therefore

considered

_mechanically

_equal.

However,

H

appliedat theendofthebar

would

effect

a

turning

motion,

which

is

quite different

from

the

effect

of

F

=

G.

Thus

H

is

not

_equal

to the other

forces.

In

such

studies

it

would

be

natural

to

insist

that

two

forces

having

different

lines

of

action

be

unequal.

Thisjustifies

the

_stipulation

inthe

theory

of

mechanics

of

rigid

bodies,

of

_permitting

a

vector

to

be

displaced

only

along

its

line

of

action.

If

the fieldof

_application

were

the

_theory

of

_elasticity,

then

it

would

be

_necessary

to restrict

(force)

vectors

still

more.

Figure

3b

shows

two

forces

J

and

_K,

both

of

the

same

_magnitude

and

directed

_{along the}

same

line

of

action,

acting

on

a

soft

plasticlike

material. K

has

the

effect

of

_stretching

the

_mass,

whereas

_J

has the

effect

of

compressing

it.

This

illustration

indicates

why,

in

the

ELEMENTARY

OPERATIONS)

9)

points

are

not

considered

equal.

In

this field

a

vector

is

restricted

to

its

original

position;

there

is

no

freedom

to

displace

it. Such

vectors

are

called

bound.

We

_state,

with

emphasis,

once

again:

all

vectors

in

this

book

are

_free,

in

accordance

with

our

_definition

_of

equality.

Addition.

Let

A

and

B

be

_any

two

vectors

_(Figure

_4a).

We

can

select

a

location

for

vector

B

so

that

_itsorigin

is

placed

at

the

terminus

of

A.

Now

we

construct

a

third

vector,

called

A

+

B,

whose

origin

coincides with

the

origin of

A

and

whose

terminus

coincides

with

the

ter-minus of

B.

The construction

(Figure

4b)

of

B

+

A

clearly

indi-cates

that

B

+

A

=

A

+

_B,

for

both

are

the

same

diago-nal

of

the

same

parallelogram,

and they

possess

the

same

sense

of

direction.

Hence

we

have

the

_following

result.

Theorem

_{1. (i)}

Addition

_of

vectors

is

commutative;

that

is

A

+

B

=

B

+ A.

(ii)

Addition

of

vectors

is

associative;

that

is

(A

+

B)

+

C

=

A

+

(B

+

C).

Part

_(ii)

of

the

theorem

is

_easily

established

_by

_using

the

definition

of

addition.

_Figure

5

illustrates

_{the proof.)}

Co> \037 \037 \037 \037)

Co

\037 \037 \037

B+C

\037)

A)

A)

FIGURE

6)))

10)

ELEMENTARY

VECTOR

GEOMETRY)

However,

the

reader

is

advised

to

phrase

the

proof

as

a

logical

deduction

from

elementary

geometry

independent

of

a

_figure.)

EXERCISE

1.

Give an

_elementary

geometry

proof

of

the

theorem:

If

equal

vectors

areaddedto

equal

vectors,

the

sums

are

equal

vectors.

As

indicated in

Section

_1,

the

notion

of

a

vector

_arose,

his-toricallYr

from

an

attempt

to

characterize

physical

quantities,

notably

force.

It

is

interesting

to

note that the little-known,

but

nonetheless

_excellent,

Dutch

_scientist,

Simon

Stevin

(1548-1620),

experimented

with

two

forces

in

an

effort

to

replace

the

two

by

a

single

one,

called

the

resultant.

He

dis-covered

that

the

_{resultant was actually the}

force

_represented

by

the

diagonal

of

a

parallelogram,

of

which

the

sides

repre-sented

_{the two original}

forces

_(Figure

_6).

This

led

to

his

formulation

of

the

principles

of

addition

of

_forces,

which

he

used

extensivelyin

developing

a

complete

theory

of

equilib-rium-the

_beginning

of

modern

statics.

_(It

is

for

this

reason

that the parallelograms

of

_Figure

6

are

sometimes

referred

to

as parallelograms

of

forces.)

Among

the

many

other

accom-plishments

of

Stevin

are

his:

₍₁₎

work

on

_{hydrostatics,}

which

laid

_plans

for

the

reclamation

of the

below-sea-Ievel

land

of)

Force Fl)

Force

Fl)

Force

_Fl)

ELEMENTARY

OPERATIONS)

11)

Holland,

and

(2)

development

of

decimal

notation

for

numbers

with

the

_consequent

methods

for

_computation.

He was the

first

to

_give

a

_systematic

treatment

of

decimals.

_(See

the

chapter

entitled

\"Stevin

on

Decimal

Fractions\" in

A

Source

Book

in

_Mathematics,

_by

D.

E. Smith, or

A

_History

_oj

Mathe-matics,

by

J.

F.

Scott,

1960.)

Thus

our

definition

of

vector

addition

is

consistent

with

the

desires

of

the

_{physicist who}

is

interestedin

applying

the

tech-niques

of

vector

analysis to

his

problems.

(The

student

of

applied

science

should

constantly

maintain

a

critical

attitude

toward

the

mathematical

definitions,

taking

care

to

see

whether

or not they

accurateiy

reflect

_given

_physical

_situations.)

Before

_{continuing, it}

should

be

mentioned

that

Galileo

(1564-1642),

quite

independently,

cameto the sameconclusionas

did

Simon

Stevin.

Thus

two

scientists

discovered

how

vectors

\"should\"

_add,

_{approximately}

two

centuries

prior to the

inven-tion

of

vector

algebra

and

vector

analysisin the nineteenth

century.

Our

definition

of

addition

can

now

be

extended

to

find

the

sum

of

n

vectors:

Al

+

A

2

+

A

g

+

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

+

An.

Of

_course,

this

can

be

done

by

grouping

pairs

(note

part

(ii)

of

Theorem

1) and

applying

the

definition

repeatedly.

However,

the

geometric

process

might

be described

simply

as

follows:

move

A 2

so

that

its

origin

is at the

terminus

of

AI;

move

Aa

so

that

its

origin

is at the

terminus

of

_{A 2 ;}

continue

this

_process

until

An

is

placed

with

its

_origin

at

the

terminus

of

An-I.

The

sum

Al

+

A

₂

₊

A

_g

₊

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

+

An

is

then

the

vector

whose origin

coincides

with

the

_origin

of

Al

and

whose

terminus

coin-cides

with

the

terminus

of

An.

What

would

be

the sum

of

the

vectors

that

form

a

closed

_{polygon with}

arrows

_{takingusallthe}

way

around?

(Try

to

find

the

answer

before

reading

on.)

Consider,

for

_example,

A

₊

B

₊

C

+

D

+

E

+

F

'of

_Figure

7.

This

is

the

vector

whose

_origin

coincides

with

the

_origin

of

A

and

whose

terminus

coincides

with

the

terminus

of

12)

ELEMENTARY

VECTOR

GEOMETRY)

FIGURE

7)

described

above.

Hence

the

_origil1

and

terminus

of

the

sum

vector

are

the

same

_point,

al1d

the

vector

is

of

zero

length.

We

then

write

A

+

B

+

C

+

D

+ E

+ F =

o.)

This

_argument

holds

for

a

_polygon

of

n

_sides,

so the

answer

to

our

_query

is:

The

zero

vector.

Multiplicationof

a

vector

_by

a

scalar.

In

arithmetic

it

is

convenient

to introduce

multiplication

as

an

extension

of

addition.

For

_{example, 3 X 4 may}

be

thought

of

4

₊

4

₊

4.

_{Similarly, we}

can-at

least

to

begin

with-)

\037

x

\037\037 rt,\037)

ELEMENTARY

OPERATIONS)

13)

think

of

_multiplying

a

vector

by

a

scalar

as

an

extension

of

vector

addition.

An

illustration

_(Figure

₈₎

_might

be

that

2A

should

_represent

A

_+.

A.

From

our

definition

of

_addition,

we

know

vector

A

₊

A

is

_actually

a

vector

parallel

to

A

and

_having

the

same

sense

of

direction

as

A;

however,

the

length

of A

+

A

is

twice

the

_length

of A.

Therefore,

if

2A

=

A

+

_A,

the

result

of

_multiplying

A

_by

the

scalar

2

is

a

_{vectorparallel}

to

A

_having

the

same

sense

of

direction

as

A

but

with

twice

the

_length

of A.

Before

_proceeding

to the

general

case

of

multiplying

a

vector

_by

a

_scalar,

let us considerthe question

of

what

would

be

_appropriate

f\037r

a

definition

of

-A.

A

reasona-ble

demand

_might

be that

A

+

(-A)

=

0;

so

let

us,

for

the

moment,

stipulate

this

and

see

where

it

takes

us.

In

general,

if

A

+

X

=

0,

we

know

that

(a)

X

must

be

parallel

to A,

(b)

IAI

=

\\xl,

and

(c) X

must

have

a

sense

of

direction

_opposite

to

that

of

A

_(Figure

_9).

Thus-A

should

have

_precisely

the

_properties

_{a, b,}

and

c

men-tioned

in

the

_previous

sentence.

_{(Alternatively,}

if

A

₊

X

=

_0,

then

A

followed

_by

X

can

be

_thought

of

as

a

closed

polygon

in

which

the

origin of

X

is

at

the

ter-minus

of

_A.)

_{Consequently,}

our

definition

should

(and

will)

stipulate

that

multiplying

by

a negative

scalarhas

the

effect

of

changing

the

sense

of

direction

of

a

vector.

We

are

now

_ready

to

_present

a

definition

for

the

multi-plication

of

a

vector

by

a

scalar.)

14)

ELEMENTARY

VECTOR

GEOMETRY)

;1A)

FIGURE

10)

Definition.

3

nA

is

a

vector

_parallel

to

A

with

_magnitude

_In

\\

times

the

_magnitude

_of

A.

In

_symbols,

_InAI

=

_Inl

_iAI.

Further,

if

n

>

0,

nA

is

_defined

to

have the

same

_{sense of}

direction

as

A;

and

if

n

<

0,

nA

is

_defined

to

have

a

sense

of

direction

opposite

to

that

of

A;

finally,

if n

=

0,

nA

is

defined

to

be the

zero

vector

(which follows from

the

first

sentence

_of

our

_definition).

_Figure

10

illtlstrates

the

def-init.ion

for

n

=

_3,

_n

=

_-3,

and

n

=

_i-.

Theorem

2.

_{If m}

and n

are

_scalars,

then

(i)

m(nA)

=

(mn)A.

(ii)

(m

+ n)A

=

mA

+

nA.

(iii)

meA

+

B)

=

mA

+

mB.

To

_illustrate,

consider m

= 5

and

n

=

-

2.

Then

nA

=

-

2A,

a

vector

twicethe

_length

of

A

btlt

directed

3

_The

sy-mbol

Inl

refers

to

the absolute

value

of

_n,

which

is

defined

as follows:If n

_{> 0,}

then

_Inl

=

_n;

and

if

n

<

_0,

then

Inl

=

-n.

rfhe

definition

asserts that

the absolute

value

of

a

number

is

always

non-negative;

e.g.,

131

=

3,

1-31

=

3,

and

101

=

O.

We

shall

need

the

fact

that

_\\mnl

=

_Imllnl,

the

truth

of

which

should

he

clear

from

the

definition.)))

ELEMENTARY

OPERATIONS)

15)

oppositely

to A.

(i)

states:

5(

-2A) =

(5)(

-2)A

=

_-lOA.

(ii)

states:

(5

+

(-2))A

=

5A

+

(-2)A

or

3A

=

5A

+

(-

2)A.

(iii)

states: 5(A

+ B)

=

5A

+

5B.

Proof.

(i)

By

examining

the length

of

the

left

mem-ber

of

_(i),

our

definition

of

_{multiplication}

of

a

vector

_by

a

scalar

yields)

1

m(nA)

I

=

ImllnAI

=

ImllnllAI

=

ImnllAl

=

I

(mn)AI.

(1)

Equation

1

proves

that

the

vectors of

(i)

are

equal

in

length.

That

they

are

parallel

follows

from

the

fact

that

both

are

_multiplies

of

A.

The

reader

is

left

to

check

that

the

directions

have

the

same

sense.

(Hint.

Use

the

definition

of

_nA..)

(ii)

If

m

+

n

=

0,

both

sides

of

_(ii)

_represent

vectors

that

_point

in

the

same

direction

\037s

A.

If

m

+

n

<

0,

both

sidesof

(ii)

represent

vectors

that

point

in

the

direc-tion

_opposite

to that

of

A.

The

_comparison

of

their

lengths

is

left

to the reader.

(Hint. Usethe

definition

of

_nA.)

(iii)

Let

us

suppose

that

A

and

Bare

_nOllparallel

and

nonzero.

We

consider

the

_triangle

_PQR,

which

defines

-?

-?

A

+

B

(see

Figure

11),

by

having

A

=

_PQ,

B

=

QR,

--7

then

P

R

=

A

₊

B

.

We

construct

_triangle

P'

_Q'

_{R' ,}

-?

-?

\037

where

mA

=

_P'Q',

mB

=

_Q'R',

then

P'R' =

mA

+

mB.

SincemAilAandmBilB.triangle

_-?

PQR

is

similar

to

tri-angle

P'Q'R'.

Thus

P'R'

is meA

+

B),

and

we

have

the

result

that

_meA

+

B)

=

mA

+

mB.

The

reader

should

consider two

_questions

cOllcerning