Page 41 - Engineering Electromagnetics, 8th Edition

CHAPTER 1 Vector Analysis 23

1.4 A circle, centered at the origin with a radius of 2 units, lies in the xy plane.
Determine the unit vector in rectangular components that lies in the xy plane,
√
is tangent to the circle at (− 3,1, 0), and is in the general direction of
increasing values of y.
2
1.5 Avector field is specified as G = 24xya x + 12(x + 2)a y + 18z a z .Given
2
two points, P(1, 2, −1) and Q(−2, 1, 3), find (a) G at P;(b)a unit vector in
the direction of G at Q;(c)a unit vector directed from Q toward P;(d) the
equation of the surface on which |G|= 60.
1.6 Find the acute angle between the two vectors A = 2a x + a y + 3a z and
B = a x − 3a y + 2a z by using the definition of (a) the dot product; (b) the
cross product.
1.7 Given the vector field E = 4zy cos 2xa x + 2zy sin 2xa y + y sin 2xa z for
2
2
the region |x|, |y|, and |z| less than 2, find (a) the surfaces on which
E y = 0; (b) the region in which E y = E z ;(c) the region in which E = 0.
1.8 Demonstrate the ambiguity that results when the cross product is used to
find the angle between two vectors by finding the angle between
A = 3a x − 2a y + 4a z and B = 2a x + a y − 2a z . Does this ambiguity exist
when the dot product is used?
2
2
1.9 A field is given as G = [25/(x + y )](xa x + ya y ). Find (a)a unit vector
in the direction of G at P(3, 4, −2); (b) the angle between G and a x at P;
(c) the value of the following double integral on the plane y = 7.

4 2
G · a y dzdx
0 0
1.10 By expressing diagonals as vectors and using the definition of the dot
product, find the smaller angle between any two diagonals of a cube, where
each diagonal connects diametrically opposite corners and passes through the
center of the cube.
1.11 Given the points M(0.1, −0.2, −0.1), N(−0.2, 0.1, 0.3), and P(0.4, 0, 0.1),
find (a) the vector R MN ;(b) the dot product R MN · R MP ;(c) the scalar
projection of R MN on R MP ;(d) the angle between R MN and R MP .
1.12 Write an expression in rectangular components for the vector that extends
from (x 1 , y 1 , z 1 )to(x 2 , y 2 , z 2 ) and determine the magnitude of this vector.
1.13 Find (a) the vector component of F = 10a x − 6a y + 5a z that is parallel to
G = 0.1a x + 0.2a y + 0.3a z ;(b) the vector component of F that is
perpendicular to G;(c) the vector component of G that is perpendicular
to F.
1.14 Given that A + B + C = 0, where the three vectors represent line segments
and extend from a common origin, must the three vectors be coplanar? If
A + B + C + D = 0, are the four vectors coplanar?