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Vectors in the Plane and 3D Space: A Comprehensive Guide with Examples, Lecture notes of Mathematics

A comprehensive guide to vectors in the plane and 3d space, covering fundamental concepts, operations, and applications. It explains vector addition, subtraction, scalar multiplication, dot product, cross product, and their geometric interpretations. The document also explores lines and planes in 3d space, including parametric and symmetric equations, distance calculations, and cylindrical surfaces. It includes numerous examples and illustrations to enhance understanding.

Typology: Lecture notes

2024/2025

Uploaded on 03/13/2025

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Sections11.111.211.3
Vectors in thePlane
vector in aplane is adirected linesegment
betweentwo
points
points Ppi Pa Qq92
vector PT is thevector staring at Pandending atQ
PQ qpgapa Avector hasdirection andlength
Q
xP25Q1,2
EIiiii
quivalentVectors
Two vec tors are equivalentif theyhave the samedirectionandlength
info 1direction
2length
omponent
form
the componentform of avector is its xycoordinates
whenplaced
withthestarting
point oftheorigin
PT P2,5 Q1,2
PQ 31333isthecomponent
form
ofPQ
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff

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Sections11.

Vectors in the

Plane

vector in a plane

is a

directed

line segment

betweentwo

points

points

P

pi

Pa

Q

q

vector

PT

is the

vector

staring

at P

and ending

atQ

PQ

q

p

ga

pa

A

vector has

direction

and

length

Q

x P

5 Q

1,

E Iiiii

quivalent

Vectors

Two

vectors

are equivalentif they

have

the

same

direction and

length

info 1

direction

2

length

omponentform

the

component

form

of a

vector is

its x

y

coordinates

when

placed

with the starting

point

of the

origin

PT P

Q

PQ

31

3 3 isthe

component

form

of PQ

Length

qfayu.fi

EIfV

v

Va

in

componetform

then

the length

of w̅ is

denoted

by

In

eiyf.IE i

isii

isoewovecters

1

Addition

itw̅ V tw

Vatwa

at

ouromens

ex.i.ly

e

s

Subtraction

Vw̅ v

W

Va

Wa

E

Isee

direction

Scalar

Multiplication

Let

R

and J v

Ja

be a vector

CJ CV C

V

changes direction

if

c is negative

Ex

V 1 1

if c o

CJ

25

2,

2gÉejft E

E

and

for the real

numbers

Addition

is

commutative

It

w̅ w̅

to

Associative

utw̅

I I Wti

Scalaris

commutative

and

associative

Let

a b c ER

ab

J

ba i

b eV a

b

A

Tectoffs

silly

a ve

Ing iÑÉ

ual

E

D

act if celR andw̅ a

vector

he kill Id

till

roof l cill

Eternal

Futura

Ic

Fta

c

11011

Dimen

sionalVectg Standard

Basis

Vectors

3D

Vectors

in

3D

still

determined

by

V

by

length

and

direction

ull

Tv

lengthfor

2D

ength

for

3D

Vectors

mail.FI

E

ii

ts

Find the length

of the

vectorPT

where

p

a

14

go.ee

erforiginIIPQT

MH35 MaH

T

ector Algebra

Remains the same in all

dimensions

rite v

VaV

4 41,

43

Utu u.tv

U2V2Ust

V

LetCCR

thenCv Cv

Cva

cuz

Ex Findunit

vector

1

U

3,

Citi 6,

f

0

to

1 6 6 1 2 3

ati

tw

u

utw̅

cdk Cda

x P 5

2,3 Q

0,

3 PQ

5,

Find a

sphere

where

the

center

is IIPQ

11

5 3636 V

the

midpoint

of PT

and with

line

segment

being

a

diameter

r

v

Midpoint

5

2

1,

Final

r.mg

9cf

aEPzhs

istance

between

points

Xo

yo

20

and x

g

z

distance

EHg

y.DE

osphefo

ygntgedatCxo

go

zo

of

radius r

is the set

of

points

of constant distance

equation

for

sphere

centered

at

Xo

yo

20

of

radius

r is

x

xo

y yo

2

20 r

ndthe midpoint

between

P

and

Q

Midpoint

is

O

IQ

P

PQ

7 p pj

Pi

P

P

Pi

92

Pa

Ps

PII

PII

PII

brigin

f

If

here

s

career's.tn

thatbeup.IE

x Let w have

intial

poin

2

1,3 and

terminal

point

is

ñ 3i

4j

K

parallel

to w

Find w in component

form

PQ 4

1

3 6

the

Eat

taw

then

6,

130

4 c

355

its

Yes

they

are

parallel

Project u

onto

v

a

projule

Hullcoso

HullYff

1

II

proju

Y

cos Iprojuall

11411

example

Forcedistan

W

F

d

IT

Puffit

EffHow

much work

did

the

farmer

do

to

close

the door

pros

F

III

T

F IF

1

1 IF Sin

f

50

cost

sin 601

E

F.EC

E'stt

oprojyF

i

25

sW

bsft

Cross

Product

of

Let 4

41,42 43

v u

Va

V

then the cross

product

of u and v

to be

a

tl

5

it

I

11

u

xv

uavz vsuz.li

a

v

v

u

Jj

a

v

viualk

Ex 4

a

u u

talk

Troperties

of Cross

Product

axv vxu

2 ax vta uxutuxw

3sccans

casva.cat

www.t

fittj

L

itam

uxv is perpendicular

to u and v with

0

respect

to

the

right

hand

rule

ataxv

a xu 0

nuxu

I

xj

E.TT

remember

as

i 1 0,

J

Corio

ph

creadia

Geometry of

Cross

Product

lemmata

d

it is

the

angle

between u v

oof

HullHull

sin Hall

2110112

f

Hall

Hull

Hul12Y

5

triffid 14

2

12

caravans cars

uiussacau.ua

Lx

Suppose

that

a force of 50lb is imparted

to a lever

that

makes

a 60

angle

with the x axis

How muchtorque is

there if the force is

directed

straight

down

able

control

15

État.si

oj

5T

5Bj

E

so

151

riple

Product

et u v

w be

three vectors

in 1R

he

triple

product is

u

v w a uxw

v wxu w

uxv

Y Y

0243

4203

42

v w

Uzw Uz

u

wa

w

v

Thm The

triple

product

gives

the

absolute

vale of

the

parallelepiped

was.ie

iiijiii

volume

Area of

base.n

like

angle

between w

and uxv

X

Find

the

volume of

the

parallelepiped

u

1

v

w 0,

volume

u

uxw

1

8

0

0

3

Lines

and

Planes

on to

describe

a line

in

3D

space

Point

on

line

pi Pa

Ps

Direction vector

v v

V2 V

Q

parametric

form

pi

pitty

Patva

Pottus

origin

P

Pictor

pathe

line

I

IE

Symmetric

Equations

for a line

FFTTeline

through

D 1

0,

Cartesian

solve for

t and

get

equations in

direction v

1,

only

involving x

y

2

YEI.EE

nsnespt

Y

Pi

YPa

2

9

170

with

t o

6

Example

Take

p

0,

Q

2,

IF If

v

x

p

v.ly

Pa

O

find

the line throughthese

two points

qq.gg

v

y

pa

v

z

p

PT

Q

P

Each

of these

describes

a plane

Y

Therefore

this

gives

theequation for

a line as

the

intersection of two

planes

Pi Pa Ps

is a point

in the

plane

PQ

ñ

is equivalent

PQ.in 0

Ty

IYgtey.lt

pI

Leto

p

sk

n

cEf

naly

paltns

2

ps

is

the

normal

origin

vector

to

plane

The white

the

equations

for a plane

9

on

plane

n x

p

thaly

pa

this

z

ps

a

point

P in the

plane

Parametric

form

Two

vectors

u v

insidethe plane

2 the

normal vector ñ

which are not parallel to

each other

Let

p

be

a

point

in the plane

t s

Pitatsu

t

s

ER

x̅ x

y

z

parameters

eatin

is

aao

Cylindrical

Surfaces

Ruled Surfaces

These are

surfaces

made

up

of lines

The equations

involve

only

two

of the

Cartesian

coordinates and

one of the

coordinates

is

missing

Ex x

ty

In

the x

y

plane this

is

a

circle of a radius

1 centered

at

the

origin

Notice that in

any

2 2

this

gives

the

circle

m

y

f

x

2

92

y

1

Y

1

Ellipsoid

He

I

x

ty

221

2ty

foreach

V

One sheet

hyper

beliad

In To

at the

cur

Ex

x

y

2

22

92

22 21

FEY the

ya

plane

the

tin

radius is K

e

get

circles