Thomas Calculus 13th [Solutions]

Section 12.2 Vectors 881
66.
2 2 2 2 2 2 2 2 2
( x 0) ( y 0) ( z 0) ( x 0) ( y 4) ( z 0) ( x 3) ( y 0) ( z 0)
( x
2
2) ( y
2
2) ( z
2
3)
2
x
2
y
2
z
2
x
2
y 8y 16
2
z
2
x 6x 9
2
y
2
z
2
x 4x 2
y 4y 2
z 6z
17
2
Solve: x
2
y
2
z
2
x
2
y 8y 16
2
z 0 8y 16 y 2
2 2 2 2 2 2
Solve: x y z x 6x 9 y z 0 6x 9 x 3
2
2 2 2 2 2 2
Solve: x y z x 4x y 4y z 6z 17 0 4x 4y 6z 17 0 4 3
2
4(2) 6z
17
z 1 3 , 2, 1
2 2 2
12.2 VECTORS
1. (a) 3(3), 3( 2) 9, 6
2 2
(b) 9 ( 6) 117 3 13
2. (a) 2( 2), 2(5) 4, 10
2 2
(b) 4 ( 10) 116 2 29
3. (a) 3 ( 2), 2 5 1, 3
(b)
2
1
2
3 10
4. (a) 3 ( 2), 2 5 5, 7
2 2
(b) 5 ( 7) 74
5. (a) 2u
2(3), 2( 2) 6, 4
3v
3( 2), 3(5) 6,15
2u
3v
6 ( 6), 4 15 12, 19
(b)
2 2
12 ( 19) 505 (b)
6. (a) 2u
2(3), 2( 2) 6, 4
5v
5( 2), 5(5) 10, 25
2u
5v
6 ( 10), 4 25 16, 29
2 2
( 16) 29 1097
7. (a)
3 u 3
(3),
3
( 2)
9
,
6
(b)
5 5 5 5 5
4 v 4 4 8
5 5 5 5
( 2), (5) , 4
3 u 4 v 9 8
,
6
4
1
,
14
5 5 5 5 5 5 5
1
2
14
2
197
5 5 5
8. (a)
5 u 5
(3),
5
( 2)
15
,
10
(b)
13 13 13 13 13
12 v 12
( 2),
12
(5)
24
,
60
13 13 13 13 13
5 u 12 v 15 24
,
10 60
3,
70
13 13 13 13 13 13 13
( 3)
2 70
2
6421
13 13
9. 2 1, 1 3 1, 4
10.
2 ( 4)
0,
1 3
0 1,1
2 2
11. 0 2, 0 3 2, 3
12. AB 2 1, 0 ( 1) 1,1 , CD 2 ( 1), 2 3 1, 1 , AB CD 0, 0
13.
2 2 1 3
3 3 2 2
cos , sin , 14. cos
3
, sin
3 1
,
1
4 4 2 2
15. This is the unit vector which makes an angle of 120 90 210 with the positive x-axis;
3
cos 210 , sin 210 ,
1
2 2
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