Solutions to exercises 645
Section 15.5
Section 15.6
Chapter 16
Section 16.2
- (i)b 1 − 1 a,a 1 − 1 b (ii)
(iii) (iv)b 1 + 1 λ(c 1 − 1 d),allλ
Section 16.3
2.(1, 1 ,1)
- (i)(3, 4 ,0) (ii) 5
(iii)(3 25 , 425 ,0)
- (i)(2,− 5 ,1) (ii)
(iii)
- (i)(2, 3 ,1) (ii)
(iii)
6.Both (− 1 , 5 ,−1)
7.(3, 6 ,9),(− 1 ,− 2 ,−3),(1 23 , 223 ,1)
8.(− 1 , 0 ,4) 9.Both (3,− 6 ,12)
- (i)(2, 1 − 1 ,0),
(ii)(1,− 1 ,1),(0, 0 ,0),(− 2 , 2 ,−2)
- (i)(2, 6 ,−6) (ii)(1, 3 ,−3)
13.− 2 i 1 + 1 j
- (i) 4 i (ii)λj (iii)λi
Section 16.4
- 2 i 1 + 16 tj 16.(− 21 sin 12 t, 31 cos 1 t,2)
- (i)v 1 = 1 ai 1 + 1 (a 1 − 1 gt)j,a 1 = 1 −gj
(ii)F 1 = 1 −mgj
(iii)to the right with constant speed v
x1 = 1 a
(iv)vertical up and down under the
influence of gravity
(v)parabolic up and down to the right
27 14 29,+
(,,)214314114
14
(, ,)230530130−
30
1
2
()cd+
1
4
()abcd+++
gy
a
ay
()=
2
22
π
gy
ee
iy
iay iby()=
−
−−
2 π
×−
exp nDtl
22 2
π
Txt
l
n
nx
l
n(),= sin
∑
81
2
33
π
π
odd- (i)v 1 = 1 − 61 sin 13 t 1 i 1 + 161 cos 13 t 1 j 1 + 13 k
a 1 = 1 − 181 cos 13 t 1 i 1 − 1181 sin 13 t 1 j 1
= 1 −9(xi 1 + 1 yj)
(ii)− 9 m(xi 1 + 1 yj)
(iii)simple harmonic in both directions
(iv)circular in xy-plane at constant
angular speed
(v)in z-direction at constant speed
(vi)right-handed circular helix around
z-axis
Section 16.5
19.Both 7 20.Both 10
21.(0, 12 ,4) 23.− 1 24. 0
25.− 3 26. 2 27.π 23 , 2 π 23 ,π 24
- (i) 4 (ii)− 3 (iii) 0
- (i) 2522 (ii)
30.− 6
Section 16.6
- 9 i 1 − 1 j 1 + 13 k,− 9 i 1 + 1 j 1 − 13 k 32. 7 i, 7
33.Both 11 i 1 − 12 j 1 − 1 k 34. 0 35.− 7
36.− 14 j 1 − 121 k,i 1 − 118 j 1 − 19 k 40. 5
- (i)− 7 k (ii)i (iii)− 5 j
- 5 i 1 + 16 j 43.− 3 i 1 + 118 k
- (i)y 1 = 1 ωk (ii)v 1 = 1 −ωyi 1 + 1 ωxj
(iii)l 1 = 1 mω[(x
2
1 + 1 y
2
)k 1 − 1 xzi 1 − 1 yzj]
47.l 1 = 12 m(x
2
1 + 1 y
2
)ωk
Section 16.8
- 4 xi 1 + 16 yj 1 − 12 zk
49.(y 1 + 1 z)i 1 + 1 (x 1 + 1 z)j 1 + 1 (x 1 + 1 y)k
50.−(x
2
1 + 1 y
2
1 + 1 z
2
)
− 322
(xi 1 + 1 yj 1 + 1 zk)
- 3 , 0 52. 0 ,i 1 + 1 j 1 + 1 k 53. 0 , 0
Chapter 17
Section 17.1
1.x 1 = 12 ,y 1 = 13 2. 6 3. 2 4. 1
Section 17.2
5.x 1 = 11 ,y 1 = 12 ,z 1 = 13 6.− 11
V
x
y
z
=− − − +C
2
2
2
2
3
2