Page 21 - Engineering Electromagnetics, 8th Edition

CHAPTER 1 Vector Analysis 3

Figure 1.1 Two vectors may be added graphically either by drawing
both vectors from a common origin and completing the parallelogram or
by beginning the second vector from the head of the first and completing
the triangle; either method is easily extended to three or more vectors.

The rule for the subtraction of vectors follows easily from that for addition, for
we may always express A−B as A+(−B); the sign, or direction, of the second vector
is reversed, and this vector is then added to the first by the rule for vector addition.
Vectors may be multiplied by scalars. The magnitude of the vector changes, but
its direction does not when the scalar is positive, although it reverses direction when
multiplied by a negative scalar. Multiplication of a vector by a scalar also obeys the
associative and distributive laws of algebra, leading to
(r + s)(A + B) = r(A + B) + s(A + B) = rA + rB + sA + sB
Division of a vector by a scalar is merely multiplication by the reciprocal of that
scalar. The multiplication of a vector by a vector is discussed in Sections 1.6 and 1.7.
Twovectors are said to be equal if their difference is zero, or A = B if A − B = 0.
In our use of vector fields we shall always add and subtract vectors that are defined
at the same point. For example, the total magnetic field about a small horseshoe mag-
net will be shown to be the sum of the fields produced by the earth and the permanent
magnet; the total field at any point is the sum of the individual fields at that point.
If we are not considering a vector field,we may add or subtract vectors that are
not defined at the same point. For example, the sum of the gravitational force acting
on a 150 lb f (pound-force) man at the North Pole and that acting on a 175 lb f person
at the South Pole may be obtained by shifting each force vector to the South Pole
before addition. The result is a force of 25 lb f directed toward the center of the earth
at the South Pole; if we wanted to be difficult, we could just as well describe the force
as 25 lb f directed away from the center of the earth (or “upward”) at the North Pole. 2

1.3 THE RECTANGULAR
COORDINATE SYSTEM
To describe a vector accurately, some specific lengths, directions, angles, projections,
or components must be given. There are three simple methods of doing this, and
about eight or ten other methods that are useful in very special cases. We are going

2 Students have argued that the force might be described at the equator as being in a “northerly”
direction. They are right, but enough is enough.