Thomas Calculus 13th [Solutions]

Section 12.3 The Dot Product 887
11.
cos 1 u v 1
3 3 ( 7)(1) (0)( 2)
1 3 7 1 1
u v
cos cos cos 1.77
2
2 2
2
2 2
52 8 26
3 ( 7) 0 3 (1) ( 2)
rad
12.
1 u v 1
(1)( 1) 2 (1) 2 (1)
1 1
1
cos cos cos cos 1 1.83
u v
2
2 2
2 2 2
15
(1) 2 2 ( 1) (1) (1)
5 3
rad
13. AB 3, 1 , BC 1, 3 , and AC 2, 2 . BA 3, 1 , CB 1, 3 , CA 2, 2 .
AB BA 10, BC CB 10, AC CA 2 2,
Angle at
A
cos 1 AB AC cos 1 3(2) 1( 2)
cos 1 1 63.435
AB AC
10 2 2
5
Angle at
1 1 ( 1)( 3) ( 3)( 1) 1
B cos BC BA cos cos 3
10 10
5
53.130 , and
BC BA
Angle at C
cos 1 CB CA cos 1 1( 2) 3(2)
cos 1 1 63.435
CB CA
10 2 2
5
14. AC 2, 4 and BD 4, 2 . AC BD 2(4) 4( 2) 0, so the angle measures are all 90 .
15. (a) cos i v a
j v
, cos b , cos k v c
i v v
j v v
k v v and
2 2 2
2 2 2
2 2 2
cos cos cos a b c a b c v v
1
v v v v v v v
(b) v 1 cos a a, cos b b
v
v
and cos c v
c are the direction cosines of v
16. u 10i 2k is parallel to the pipe in the north direction and v 10j k is parallel to the pipe in the east
direction. The angle between the two pipes is
cos 1 u v cos 1 2
u v
1.55 rad 88.88 .
104 101
17. The sum of two vectors of equal length is always orthogonal to their difference, as we can see from the
2 2
equation v1 v2 v1 v2 v1 v1 v2 v1 v1 v2 v2 v2 v1 v2 0
18.
2 2
CA CB v ( u) ( v u) v v v u u v u u v u 0 because u v , since both equal
the radius of the circle. Therefore, CA and CB are orthogonal.
19. Let u and v be the sides of a rhombus the diagonals are d1
u v and d2
u v
2 2
d1 d2 ( u v) ( u v) u u u v v u v v v u 0 because u v , since a rhombus has
equal sides.
Copyright
2014 Pearson Education, Inc.