Thomas Calculus 13th [Solutions]

954 Chapter 13 Vector-Valued Functions and Motion in Space
8.
2 2
r cos t i sin t j tk v sin t i cost j k v sin t cos t 1 2
1 sin 1 cos 1 d T
T v t i t j k 1 cos t i 1 sin t j
v 2 2 2 dt 2 2
d T
dt
d T
1 2 1 2 1
dt
cos t sin t N cos t i sin t j ; thus T(0)
1 j 1 k and N(0)
2 2 2
d T
2 2
dt
i j k
B(0) 0 1 1 1 j 1 k , the normal to the osculating plane; r(0) i P 1, 0, 0 lies on the
2 2 2 2
1 0 0
osculating plane 0 x 1 1 y 0 1 z 0 0 y z 0 is the osculating plane; T is normal to the
2 2
normal plane 0 x 1 1 y 0 1 z 0 0 y z 0 is the normal plane; N is normal to the
2 2
rectifying plane 1 x 1 0 y 0 0 z 0 0 x 1 is the rectifying plane.
i
9. By Exercise 9 in Section 13.4, T 3 cost
i 3 sin t j 4 k and N
5 5 5
sin t i cos t j so that
i j k
B T N 3 cost 3 sin t 4 4 cos t i 4 sin t j 3 k . Also v
5 5 5 5 5 5
3cos t i 3sin t j 4k
sin t cost
0
i j k
a 3sin t i 3cost j
d a
dt
3cos t i 3sin t j and v a 3cost
3sin t 4
3sin t 3cost
0
12cos t i 12sin t j 9 k | v
2
a |
2
12cost 2
12sin t
2
( 9) 225. Thus
3cost
3sin t 4
3sin t 3cost
0
3cost
3sin t 0 4
2
9sin t
2
9cos t
36 4
225 225 225 25
10. By Exercise 10 in Section 13.4, T cos t i sin t j and N sin t i cos t j ; thus
i j k
B T N cos t sin t 0
2
cos t
2
sin t k k . Also v t cost i t sin t j
sin t cos t 0
a t sin t cost i t cos t sin t j
d a
dt
t cos t sin t sin t i t sin t cos t cost
j
i j k
t cost 2sin t i 2cost t sin t j . Thus v a t cost t sin t 0
t sin t cost t cost sin t 0
t cos t t cost sin t t sin t t sin t cos t k
2
t k v
2 2
2
a t
4
t .
t cost t sin t 0
cost t sin t sin t cost
0
2sin t t cost 2cost sin t 0
Thus 0 0
4 4
t
t
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