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1.2

VECTORS IN THREE-DIMENSIONAL

SPACE

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z axis

xy-plane xz-plane

yz-plane

y axis

x axis

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

1

2 3

4

5

-1

-2

-3

-4

-5

1

2 3

4 5

-2 -3

-4 -5

x

y

z

4 3 2 , , P

12 3 , ,Q 3 4 5 , , R

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Distance and midpoint points: 1111 z ,y , x P

2 2 2 2 z ,y , x P

2 12 2 12 2 12 zzyy x x

Distance: or 2 1 P P 2 1 P , Pd

Midpoint: 2 2 2

2 12 12 12 1

zz ,

yy ,

x x M P P

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Example 1. Determine the distance between the given points and the

midpoint of the segment joining them.

2 3 4 2 , , P

4 2 3 1 , , PSolution:

2 1 P P 2 12 2 12 2 12 zzyy x x

2 2 2 4 2 2 3 3 4 2 2 2 6 5 7

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Solution (continued)

2 3 4 2 , , P4 2 3 1 , , P

2 1 P P 2 2 2 6 5 7 36 25 49 110

2 2 2

2 12 12 1

2 1zz ,yy , x x M P P

2 2 4

2

3 2

2

4 3 2 1 , , M P P

12

1

2

12 1 , , M

P P

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2 3 4 2 , , P

4 2 3 1 , , P

1 2 3 4 5

1

2 3

4

5

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

-2

-3

-4

-5

1

2 3

4 5

-2 -3

-4 -5

x

y

z

12

1

2

12 1 , , M P P

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Vector in 3D

A vector in three-dimensional space is an ordered triple of real numbers . a, band b are called components

of the vector.

c ,b ,a

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On the space

Position representation of

c ,b ,a

initial point: the origin

terminal point:

0 0 0 , ,

c ,b ,a

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Magnitude and direction

length of any of its representation

Magnitude of vector A

: A

Direction angles of a non-zero vector A:

smallest radian measure measured from the positive side of each axis

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x

y

z

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MUST!!!

Initial point: Terminal point:

VECTOR COMPONENTS:

t t t z ,y , x iii z ,y , x

it it it zz ,yy , x x

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Magnitude and direction

Consider vector .c ,b ,a A 2 2 2

cba A

If , and are the direction angles,

Aa

cos Ab

cos Ac

cos

12 2 2 coscoscoswhere

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Example 2.

Determine the components of the vector with initial point at

and terminal point at . Also, determine the

magnitude and the cosines of

the direction angles.

3 5 4 , ,5 4 2 , ,

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Solution:

2 12 , , A

A 2 2 2 2 12 4 14 9 3

it it it zz ,yy , x x 3 5 5 4 4 2 , ,

2 12 , ,

initial: terminal: 3 5 4 , , 5 4 2 , ,

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Solution (continued) 2 12 , , A 3 A

Aacos

Abcos

Accos

3

2 cos

3

1cos

3

2 cos

3 2 cos Arc

3 1cos Arc

3 2 cos Arc

rad .8410 rad .9111 rad .8410

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1 2 3 4 5

1

2 3

4

5

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

-2

-3

12

3 4 5

-2 -3

-4 -5

x

y

z

initial:

terminal:

3 5 4 , ,5 4 2

, ,2 12 , , A

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Operations on vectors in 3D

3 2 1 b ,b ,bB3 2 1 a ,a ,a ASUM:

3 3 2 2 11 ba ,ba ,baB A

NEGATIVE: 3 2 1 a ,a ,a ADIFFERENCE:

3 3 2 2 11 ba ,ba ,baB ASCALAR PRODUCT:

3 2 1 ca,ca,cacA

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Unit vectors 0 0 1 , ,i : unit vector in the

direction of positive x-axis

0 10 , , j : unit vector in the direction of positive y-axis

10 0 , ,k : unit vector in the direction of positive

z-axis

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Unit vector

ckbjai A

Given .c ,b ,a A

Ac ,

Ab ,

AaU A

cos ,cos ,cos

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Example.

Solution:

3 Consider and . Determine the unit vector in the direction of .

3 2 2 , , A0 2 4 , ,B

B A 2 3

B A 2 3 0 2 4 2 3 2 2 3 , , , ,

0 4 8 9 6 6 , , , , 0 9 4 6 8 6 , ,9 2 14 , ,

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Solution (continued)

B A2 3 9 2 14 , ,

B AU 2 3 B AB A

2 3

2 3

2 2 2 9 2 14 9 2 14 , ,

18 4 196

9 2 14 , ,

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Solution (continued)

B AU 2 3 18 4 196 9 2 14 , ,

218

9 2 14 , ,

218

9

218

2

218

14 , ,

610 14 0 95 0 . ,. ,.

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END