Page 28 - Engineering Electromagnetics, 8th Edition

10 ENGINEERING ELECTROMAGNETICS

Figure 1.4 (a) The scalar component of B in the direction of the unit vector a is
B · a.(b) The vector component of B in the direction of the unit vector a is (B · a)a.

The remaining three terms involve the dot product of a unit vector with itself, which
is unity, giving finally

A · B = A x B x + A y B y + A z B z (5)
which is an expression involving no angles.
Avector dotted with itself yields the magnitude squared, or

2 2
A · A = A =|A| (6)
and any unit vector dotted with itself is unity,
a A · a A = 1
One of the most important applications of the dot product is that of finding the
component of a vector in a given direction. Referring to Figure 1.4a,we can obtain
the component (scalar) of B in the direction specified by the unit vector a as

B · a =|B||a| cos θ Ba =|B| cos θ Ba
The sign of the component is positive if 0 ≤ θ Ba ≤ 90 and negative whenever
◦
◦
90 ≤ θ Ba ≤ 180 .
◦
To obtain the component vector of B in the direction of a,we multiply the
component (scalar) by a,as illustrated by Figure 1.4b.For example, the component
of B in the direction of a x is B · a x = B x , and the component vector is B x a x ,or
(B · a x )a x . Hence, the problem of finding the component of a vector in any direction
becomes the problem of finding a unit vector in that direction, and that we can do.
The geometrical term projection is also used with the dot product. Thus, B · a is
the projection of B in the a direction.

EXAMPLE 1.2

In order to illustrate these definitions and operations, consider the vector field G =
ya x −2.5xa y +3a z and the point Q(4, 5, 2). We wish to find: G at Q; the scalar com-
1
ponent of G at Q in the direction of a N = (2a x + a y − 2a z ); the vector component
3
of G at Q in the direction of a N ; and finally, the angle θ Ga between G(r Q ) and a N .