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8/8/2019 MATH 28 UNIT 1.2
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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