Page 180 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 7] VECTORS 171

7.29. If A ¼ i þ j, B ¼ 2i 3j þ k, C ¼ 4j 3k, find (a) ðA BÞ C,(b) A ðB CÞ.

j j
i k i k

1 0 ¼ i j 5k. Then ðA BÞ C ¼ 1 1 5 ¼ 23i þ 3j þ 4k:

ðaÞ A B ¼ 1

2 3 1 0 4 3

i j k i j k

B C ¼ 2 3 1 ¼ 5i þ 6j þ 8k. Then A ðB CÞ¼ 11 0 ¼ 8i 8j þ k:

ðbÞ

0 4 3 56 8
It can be proved that, in general, ðA BÞ C 6¼ A ðB CÞ.
DERIVATIVES

2 2
3 2t dr dr d r
7.30. If r ¼ðt þ 2tÞi 3e j þ 2 sin 5tk, find (a) ; ðbÞ ; ðcÞ ; ðdÞ d r at t ¼ 0 and
dt
dt dt 2 2
give a possible physical significance. dt
dr d 3 d 2t d 2 2t
dt dt dt dt
ðaÞ ¼ ðt þ 2tÞi þ ð 3e Þj þ ð2 sin 5tÞk ¼ð3t þ 2Þi þ 6e j þ 10 cos 5tk
At t ¼ 0, dr=dt ¼ 2i þ 6j þ 10k
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
140 ¼ 2 35 at t ¼ 0:
From ðaÞ; 2 2 2 p ffiffiffiffiffiffiffiffi p ffiffiffiffiffi
ðbÞ jdr=dtj¼ ð2Þ þð6Þ þð10Þ ¼
2
d r d d 2 2t 2t
dr
ðcÞ 2 ¼ ¼ fð3t þ 2Þi þ 6e j þ 10 cos 5tkg¼ 6ti 12e j 50 sin 5tk
dt dt dt dt
2
2
At t ¼ 0, d r=dt ¼ 12j.
2
2
ðdÞ From ðcÞ; jd r=dt j¼ 12 at t ¼ 0:
If t represents time, these represent respectively the velocity, magnitude of the velocity, acceleration,
3
and magnitude of the acceleration at t ¼ 0ofa particle moving along the space curve x ¼ t þ 2t,
y ¼ 3e 2t , z ¼ 2 sin 5t.
d dB dA
7.31. Prove that ðA BÞ¼ A þ B, where A and B are differentiable functions of u.
du du du
d ðA þ AÞ ðB þ BÞ A B
Method 1: ðA BÞ¼ lim
du u!0 u
A B þ A B þ A B
¼ lim
u!0 u
B A A dB dA

¼ lim A þ B þ B ¼ A þ B
u!0 u u u du du
Method 2: Let A ¼ A 1 i þ A 2 j þ A 3 k, B þ B 1 i þ B 2 j þ B 3 k. Then
d d
du ðA BÞ¼ du ðA 1 B 1 þ A 2 B 2 þ A 3 B 3 Þ

dB 1 dB 2 dB 3 dA 1 dA 2 dA 3
¼ A 1 þ A 2 þ A 3 þ B 1 þ B 2 þ B 3
du du du du du du
dB dA
B
¼ A þ
du du