Page 188 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 7] VECTORS 179

7.64. Find the shortest distance from the point ð3; 2; 1Þ to the plane determine by ð1; 1; 0Þ; ð3; 1; 1Þ; ð 1; 0; 2Þ.
Ans.2

TRIPLE PRODUCTS
7.65. If A ¼ 2i þ j 3k; B ¼ i 2j þ k; C ¼ i þ j 4, find (a) A ðB CÞ,(b) C ðA BÞ,(c) A ðB CÞ,
(d) ðA BÞ C. Ans. (a) 20, (b) 20, (c)8i 19j k; ðdÞ 25i 15j 10k
7.66. Prove that ðaÞ A ðB CÞ¼ B ðC AÞ¼ C ðA BÞ
A ðB CÞ¼ BðA CÞ CðA BÞ.
ðbÞ
7.67. Find an equation for the plane passing through ð2; 1; 2Þ; ð 1; 2; 3Þ; ð4; 1; 0Þ.
Ans. 2x þ y 3z ¼ 9
7.68. Find the volume of the tetrahedron with vertices at ð2; 1; 1Þ; ð1; 1; 2Þ; ð0; 1; 1Þ; ð1; 2; 1Þ.
Ans. 4
3
7.69. Prove that ðA BÞ ðC DÞþðB CÞ ðA DÞþ ðC AÞ ðB DÞ¼ 0:

DERIVATIVES
t
t
t
7.70. A particle moves along the space curve r ¼ e cos t i þ e sin t j þ e k. Find the magnitude of the
ffiffiffi ffiffiffi
p t p t
(a)velocity and (b)acceleration at any time t. Ans.(a) 3 e ; ðbÞ 5 e
d dB dA
7.71. Prove that ðA BÞ¼ A þ B where A and B are differentiable functions of u.
du du du
2
3
7.72. Find a unit vector tangent to the space curve x ¼ t; y ¼ t ; z ¼ t at the point where t ¼ 1.
ffiffiffiffiffi
p
Ans. ði þ 2j þ 3kÞ= 14
7.73. If r ¼ a cos !t þ b sin !t, where a and b are any constant non-collinear vectors and ! is a constant scalar,
2
dr d r 2
prove that (a) r ¼ ¼ !ða bÞ; ðbÞ; þ ! r ¼ 0.
dr dt 2
@ 2
3
2
7.74. If A ¼ x i yj þ xzk, B ¼ yi þ xj xyzk and C ¼ i yj þ x zk, find (a) ðA BÞ and
(b) d½A ðB Cފ at the point ð1; 1; 2Þ: Ans.(a) 4i þ 8j; ðbÞ 8 dx @x @y

2
@ B 2 p ffiffiffi
2 2
2
2
7.75. If R ¼ x yi 2y zj þ xy z k, find @ R at the point ð2; 1; 2Þ. Ans.16 5
@x 2 2
@y
GRADIENT, DIVERGENCE, AND CURL
7.76. If U; V; A; B have continuous partial derivatives prove that:
(a) rðU þ VÞ¼ rU þrV; ðbÞr ðA þ BÞ¼r A þr B; ðcÞr ðA þ BÞ¼r A þr B.
2
2
2
7.77. If ¼ xy þ yz þ zx and A ¼ x yi þ y zj þ z xk, find (a) A r ; ðbÞ r A; and (c) ðr Þ A at the
point ð3; 1; 2Þ. Ans: ðaÞ 25; ðbÞ 2; ðcÞ 56i 30j þ 47k
2
7.78. Show that r ðr rÞ¼ 0 where r ¼ xi þ yj þ zk and r ¼jrj.
7.79. Prove: (a) r ðUAÞ¼ðrUÞ A þ Uðr AÞ; ðbÞr ðA BÞ¼ B ðr AÞ A ðr BÞ.
7.80. Prove that curl grad u ¼ 0, stating appropriate conditions on U.
2 4
2
7.81. Find a unit normal to the surface x y 2xz þ 2y z ¼ 10 at the point ð2; 1; 1Þ.
p ffiffiffiffiffi
Ans: ð3i þ 4j 6kÞ= 61
2
7.82. If A ¼ 3xz i yzj þðx þ 2zÞk, find curl curl A. Ans: 6xi þð6z 1Þk
2
7.83. (a)Prove that r ðr AÞ¼ r A þrðr AÞ.(b)Verify the result in (a)if A is given as in Problem 7.82.