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UNIT 5

VECTORS, L INESand PLANES in

SPACE

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OBJECTIVES

By the end of the unit, you must be

able to: enumerate and apply properties of

vectors in the plane and in space;

perform and interpret vector

operations;

find the equations of a line and

equation of a plane in space; and

identify and sketch cylinders and

quadric surfaces.

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5.1

VECTORS IN 2Dand IN 3D

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NOTION

VECTOR

SCALAR

quant i ty that has both

magn i tude and d irect ion

quant i ty that on ly hasmagni tude

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Examples

SCALARS

MATH 37 GRADE

speedlength

t imetemperature

densi ty

mass

energy

SCALARS VECTORS

magnet ic f ield

veloci tydisplacement

accelerat ionforce

elec tr ic f ieldmomentum

LIFE

VECTORS

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L IFE IS NOT JUST

ABOUT MAGNITUDE.

IT NEEDS DIRECTION.

JUST FOR FUN

-someth ing I overheard

MAGNITUDEDIRECTION

LIFE

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Geometr ic representat ion

magni tude

direct ion

in i t ial po int

term inal po int

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Example.

Cons ider a vecto r w i th

ini t ial po int at and

term inal po int at .

A42 ,

65,Determ ine its magn i tude and

direct ion.

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Solut ion

1 2 3 4 5

1

2

34

5

-1-2-3-4-5-1

-2

-3

-4

-5

65,

42 ,

magni tude

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Solut ion (cont inued)

magn itude of A

6542 ,,,d

246225 2

102

7

149

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Solut ion (cont inued)

1 2 3 4 5

1

2

3

4

5

-1-2-3-4-5-1

-2

-3

-4

-542 ,

65,

10

7

tan 710

7

10tanArc

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Solut ion (cont inued)

Hence, vecto r has a

magn i tude of and inthe d irect ion o f .

A149

7

10tanArc

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Representations . . .

A vecto r has several representat ions

on the plane depend ing on the ini t ial

and term inal poin t .

Posi t ion Representat ion

in i t ial po int at the or ig in

direct ion is measu red from theposi t ive x-axis

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Example. The fol low ing are

d i f feren t representat ions o f one

vector.

In i t ial po int Term inal po int

23, 11 ,

15 , 41

,51, 25,

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1 2 3 4 5

1

2

3

4

5

-1-2-3-4-5-1

-2

-3

-4

-5

23,

11 ,

51,

25,

41 , 34 ,

15 ,

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Vecto r in the p lane

A vectoris an ordered pair of

real numbers . aand bare called as componentso fthe vec to r.

b,a

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Representat ion

Posi t ion representat ion o f

in i t ial poin t :

term inal po int :

b,a

00, b,a

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Example 1.

Determ ine the components o f

the vecto r w i th ini t ial po int at

and term inal po int at

.

42, 65,Solut ion:

itit yy,xx 4625 ,

107

,

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1 2 3 4 5

1

2

3

4

5

-1-2-3-4-5-1

-2

-3

-4

-542,

65,107,

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Example 2.

Determ ine the components o f

the vecto r w i th ini t ial po int at

and term inal po int at

.

23,

11 ,Solut ion:

itit yy,xx 2131 ,

34,

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1 2 3 4 5

1

2

3

4

5

-1-2-3-4-5-1

-2

-3

-4

-5

23

,

11 ,34,

34 ,

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Equali ty o f vecto rs

Two vecto rs are equal i f theirmagni tudes and d irect ions are

equal.

Vectors and are

equali f and on ly i f and.

b,a

ca d,c

db

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Magn i tude and direct ion

Consider vecto r A.

The magni tudeo f A, , is theleng th o f any of i ts

representat ions.

A

The d irect ion angle o f A, , is

the measu re o f the angle form edby the vector w i th the posi t ivex-axis.

A

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Magn i tude and direct ion

Consider vector .b,aA 22 baA

a

btan A

Also, .AA sinA,cosAA

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Example.

Solut ion:

Determ ine the magni tude and

direct ion of vectors

and .

44,A 31,B

44,A 22 44 A 32 24

1

4

4 A

tan 1tanArcA

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1 2 3 4 5

1

2

3

4

5

-1-2-3-4-5-1

-2

-3

-4

-5

44,A A

Solut ion (cont inued)

4

1 tanArc

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4

Solut ion (cont inued)

14

4 Atan 1tanArcA

4

3 A

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Solut ion (cont inued)

31

3 Btan 3tanArcB

31,B 22 31 B 4 2

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1 2

1

2

-1-2

-1

-2

B

Solut ion (cont inued)

3tanArcB

31,B

3

B

Also, .3

5 B

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Example.

Solut ion:

Determ ine the components o f

the vector w i th a magn i tude of

6 un i ts in the d irect ion o f .

3

5

AA sinA,cosA

3

5

3

566

sin,cos

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Solut ion (cont inued)

2

36

2

16 ,

AA sinA,cosA 3

5

3

566

sin,cos

333,

5

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1 2 3 4 5

1

2

3

4

5

-1-2-3-4-5-1

-2

-3

-4

-5

Solut ion (cont inued)

3

5

333,6 un i ts

hor izontal componentv

ert

ica

lc

ompone

nt

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Unit vecto r

A un i t vecto r has a

magni tude of 1.

01,i : un i t vector in thedi rect ion of pos i tive

x-axis

10,j : un i t vector in thedi rect ion of pos i tive

y-axis

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Unit vecto r

bjaiA or

Given .b,aA

1001 ,b,aA

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Unit vecto r

Unit vecto r in the d irect ion o f A:

Given .b,aA

A

b,

A

aUA

AA sin,cos

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Example.

Determ ine a un i t vecto r in the

direct ion o f .512 ,Solut ion:

Let .512 ,A

AU

A 22 512 25144

169 13 135

13

12 ,

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I l lustrat ion

2 4 6 8 10-2-2

2

4

6

-4-6-8-10

-4

-6

8

10

512 ,A

AU 135

13

12 ,

-8

-10

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Example.

Determ ine a uni t vecto r in the

direct ion of the vecto r w i th amagn i tude of 10 in the direct ion

of .6

Solut ion:

BU

Let be the g iven vecto r.B

BB sin,cos

66

sin,cos

2

1

2

3,

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VECTORS IN 3D

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The 3D space

The set o f al l ordered tr ipleso f real numbers is called as

the three-dimensional

number space.

3

R z,y,x|z,y,x R

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z axis

xy-planexz-plane

yz-plane

y axis

x axis

z

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-1-2-3-4-5 1 2 3 4 5

1

2

3

4

5

-1

-2

-3

-4

-5

12

34

5

-2-3

-4

-5

x

y

z 432 ,,P

123

,,Q

345 ,,R

Di t d id i t

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Distance and m idpo int

points : 1111 z,y,xP2222 z,y,xP

2

12

2

12

2

12 zzyyxx

Distance: or21PP 21 P,Pd

Di t d id i t

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Distance and m idpo int

Midpoint :

222

212121

21

zz,

yy,

xxM

PP

points : 1111 z,y,xP2222 z,y,xP

E l D t i th d i t

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Example. Determ ine the d istance

between the given points and the

m idpo int of the segment jo in ingthem.

2342 ,,P4231 ,,P

Solut ion:

21PP 2

122

122

12 zzyyxx

222 422334

110362549

S l t i ( t i d)

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Solut ion (cont inued)

2342 ,,P4231 ,,P

222

212121

21

zz,

yy,

xxM PP

2

24

2

32

2

43

21 ,,M PP

12

1

2

1

21 ,,M PP

z

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2342 ,,P

4231 ,,P

1 2 3 4 5

1

2

3

4

5

-1-2-3-4-5-1

-2

-3

-4

-5

12

34

5

-2-3

-4

-5

x

y

z

1

2

1

2

1

21 ,,M PP

V t i 3D

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Vector in 3D

A vecto r in three-d imensional

space is an ordered tr ip le o f

real numbers . a, band bare called components

o f the vecto r.

c,b,a

O th

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On the space

Posi t ion representat ion o f

c,b,a

in i t ial po int : the or ig in

term inal poin t :

000 ,,

c,b,a

M it d d di t i

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Magn i tude and direct ion

Magn i tude o f vector A: A

Direc t ion ang les o f a non-zero

vector A:

smal les t rad ian measu re

measu red from the posi t ive sideo f each axis

z

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x

y

z

MUST!!!

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MUST!!!

In i t ial po int :Term inal po int :

VECTOR COMPONENTS:

ttt z,y,xiii z,y,x

ititit zz,yy,xx

Magn i tude and direct ion

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Magn i tude and direct ion

Consider vector .c,b,aA

222 cbaA

If , and are the d irect ionangles,

A

a

cos

A

b

cos

A

c

cos

1222 coscoscoswhere

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Unit vectors

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Unit vectors

001 ,,i : un i t vecto r in thedi rect ion of pos i tive

x-axis010

,,j : un i t vector in thedi rect ion of pos i tivey-axis

100 ,,k : un i t vecto r in thedi rect ion of pos i tive

z-axis Unit vecto r

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Unit vecto r

ckbjaiA

Given .

c,b,aA

Ac,

Ab,

AaUA

cos,cos,cos

E l

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Consider vector w i th in i t ial

poin t at and term inal

po int at . 542 ,, 314 ,,

Example.

A

Determ ine the fol low ing :i. the components of the vector ;

i i . i ts magn i tude and direct ional

cos ines; andii i . the un i t vecto r in the same

direct ion

Solut ion:

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Solut ion:

542 ,,: tP 314 ,,:Pi

components of the vector : 256 ,,magni tude: 65

direct ion cos ines:

65

6cos

65

5cos

65

2cos

un i t vecto r in the

same direct ion : 65

2

65

5

65

6,,

z

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1 2 3 4 5

1

2

3

4

5

-1-2-3-4-5-1

-2

-3

-4

-5

12

34

5

-2-3

-4

-5

x

y

256 ,,

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END