VECTOR ANALYSIS FINALS.docx - 117 Find the distance vector...

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117. Find the distance vector between P 1 (1, 2, 3) and P 2 (1, 2, 3) in Cartesian coordinates.
118.Find the angle q between vectors A and B of Example 3.1 using the cross product between them. | x z x
119. Find the angle that vector B of Example 3-1 makes with the z-axis. °
120. Vectors A and B lie in the y-z plane and both have the same magnitude of 2 (Fig). Determine (a) A B and (b) A B.
a. A B = AB cos (90 ° + 30 ° ) = 2 x 2 x cos 120 ° = - 2 b. A = y 2 B = -y 2 cos 60 ° + z 2 cos 30 ° = -y + z 1.73 A x B = y 2 (-y + z 1.73 ) = x 3.46 121. A circular cylinder of radius r = 5 cm is concentric with the z-axis and extends between z = -3 cm and z = 3 cm. Use Eq. (3.44) to find the cylinder’s volume. r r 3
122.Point P = (2 3 ; p=3; 2) is given in cylindrical coordinates. Express P in spherical coordinates.
= R = r 2 + z 2 = ( 2 3 ) 2 +(− 2 ) 2 = 4 = π 3 ( unchanged ) ¿ tan 1 r 2 = tan 1 2 3 2 =− 60 ° = π 3 2 π 3 123.Transform Vector A = x ' ( x + y ) + y ' ( y x ) + z' z
124.Given V = x 2 y + xy 2 + xz 2 , (a) find the gradient of V, and (b) evaluate it at (1; -1; 2). 4
125.Find the directional derivative of V = r z 2 cos2 along the direction A = r 2 z and evaluate it at ( 1, π 2 , 2 ) . z

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