Page 42 - Engineering Electromagnetics, 8th Edition

24 ENGINEERING ELECTROMAGNETICS

1.15 Three vectors extending from the origin are given as r 1 = (7, 3, −2),
r 2 = (−2, 7, −3), and r 3 = (0, 2, 3). Find (a)a unit vector perpendicular to
both r 1 and r 2 ;(b)a unit vector perpendicular to the vectors r 1 − r 2 and
r 2 − r 3 ;(c) the area of the triangle defined by r 1 and r 2 ;(d) the area of the
triangle defined by the heads of r 1 , r 2 , and r 3 .
1.16 If A represents a vector one unit long directed due east, B represents a vector
three units long directed due north, and A + B = 2C − D and
2A − B = C + 2D, determine the length and direction of C.

1.17 Point A(−4, 2, 5) and the two vectors, R AM = (20, 18 − 10) and
R AN = (−10, 8, 15), define a triangle. Find (a)a unit vector perpendicular to
the triangle; (b)a unit vector in the plane of the triangle and perpendicular to
R AN ;(c)a unit vector in the plane of the triangle that bisects the interior
angle at A.
1.18 A certain vector field is given as G = (y + 1)a x + xa y .(a) Determine G at
the point (3, −2, 4); (b) obtain a unit vector defining the direction of G at
(3, −2, 4).
2 −1
2
1.19 (a) Express the field D = (x + y ) (xa x + ya y )incylindrical components
and cylindrical variables. (b)Evaluate D at the point where ρ = 2, φ = 0.2π,
and z = 5, expressing the result in cylindrical and rectangular components.
1.20 If the three sides of a triangle are represented by vectors A, B, and C, all
2
directed counterclockwise, show that |C| = (A + B) · (A + B) and expand
the product to obtain the law of cosines.
1.21 Express in cylindrical components: (a) the vector from C(3, 2, −7) to
D(−1, −4, 2); (b)a unit vector at D directed toward C;(c)a unit vector at D
directed toward the origin.
1.22 A sphere of radius a, centered at the origin, rotates about the z axis at angular
velocity rad/s. The rotation direction is clockwise when one is looking in
the positive z direction. (a) Using spherical components, write an expression
for the velocity field, v, that gives the tangential velocity at any point within
the sphere; (b) convert to rectangular components.
1.23 The surfaces ρ = 3,ρ = 5,φ = 100 ,φ = 130 , z = 3, and z = 4.5 define a
◦
◦
closed surface. Find (a) the enclosed volume; (b) the total area of the
enclosing surface; (c) the total length of the twelve edges of the surfaces;
(d) the length of the longest straight line that lies entirely within the volume.
1.24 Two unit vectors, a 1 and a 2 , lie in the xy plane and pass through the origin.
They make angles φ 1 and φ 2 , respectively, with the x axis (a) Express each
vector in rectangular components; (b) take the dot product and verify the
trigonometric identity, cos(φ 1 − φ 2 ) = cos φ 1 cos φ 2 + sin φ 1 sin φ 2 ;(c) take
the cross product and verify the trigonometric identity
sin(φ 2 − φ 1 ) = sin φ 2 cos φ 1 − cos φ 2 sin φ 1 .