Thomas Calculus 13th [Solutions]

944 Chapter 13 Vector-Valued Functions and Motion in Space
(b)
2 2 2 2
v sin t i cos t j sin t k v sin t cos t sin t 1 sin t
sin t i cost j sin t k
T v T(0) j, T i k , T( ) j,
T 3 i k
v
2
1 sin t
2 2 2 2
(c) a cos t i sin t j cos t k;
n i k is
normal to the plane x z 1
n a cost
cos t 0
a is orthogonal to n
a is parallel to the plane;
a(0) i k, a j,
a( ) i k,
a 3 j
2
2
(d)
2
v 1 sin t (See part (b))
(e) L 7.64 (by Mathematica)
L
2 2
1 sin t dt
0
18. (a) r cos 4t i sin 4t j 4tk v 4sin 4t i 4cos 4t
j 4k
2 2 2
v 4sin 4t
4cos 4t
4 32 4 2 Length
(b) r cos t i sin t j t k v 1 sin t i 1 cos t j 1 k
2 2 2 2 2 2 2 2
(c)
2 2
1 1 1
2
1 1 2
v sin t cos t
Length
2 2 2 2 2 4 4 2
2
2
4 2 dt 4 2t
2 2
0 0
4 4
2 2
dt t
0 2 2
0
2 2
2 2 2
r cos t i sin t j tk v sin t i cost j k v sin t cost
1 1 1
2 Length
0
0
2 dt 2t
2 2
2 2
19. PQB QOB t and PQ arc ( AQ)
t since
PQ length of the unwound string length of arc ( AQ);
thus x OB BC OB DP cos t t sin t , and
y PC QB QD sin t t cost
20. r cos t t sin t i sin t t cost j v sin t t cost sin t i cos t t( sin t) cost
j
2 2 2
t cos t i t sin t j v t cost t sin t t t t, t 0
T v t cost i t sin t j costi sin t j
v t t
21. v d x0 t u1 i d y0 t u2 j d z0 t u3 k u1i u2j u3 k u , so
dt dt dt
s( t) t v dt t u d t 1 d t
0 0 0
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