Page 25 - Engineering Electromagnetics, 8th Edition

CHAPTER 1 Vector Analysis 7

determining its component vectors. Parallelism must, of course, be maintained during
the sliding process.
If we are discussing a force vector F,or indeed any vector other than a
displacement-type vector such as r, the problem arises of providing suitable letters
for the three component vectors. It would not do to call them x, y, and z, for these
are displacements, or directed distances, and are measured in meters (abbreviated m)
or some other unit of length. The problem is most often avoided by using component
scalars, simply called components, F x , F y , and F z . The components are the signed
magnitudes of the component vectors. We may then write F = F x a x + F y a y + F z a z .
The component vectors are F x a x , F y a y , and F z a z .
Any vector B then may be described by B = B x a x + B y a y + B z a z . The magnitude
of B written |B| or simply B,isgivenby

2
2
|B|= B + B + B z 2 (1)
y
x
Each of the three coordinate systems we discuss will have its three fundamental
and mutually perpendicular unit vectors that are used to resolve any vector into its
component vectors. Unit vectors are not limited to this application. It is helpful to
write a unit vector having a specified direction. This is easily done, for a unit vector
in a given direction is merely a vector in that direction divided by its magnitude. A
2
2
2

unit vector in the r direction is r/ x + y + z , and a unit vector in the direction of
the vector B is
B B
=
a B =
2
2
B + B + B 2 z |B| (2)
x
y
EXAMPLE 1.1
Specify the unit vector extending from the origin toward the point G(2, −2, −1).
Solution. We first construct the vector extending from the origin to point G,

G = 2a x − 2a y − a z
We continue by finding the magnitude of G,
2
2
2

|G|= (2) + (−2) + (−1) = 3
and finally expressing the desired unit vector as the quotient,
G
1
2
2
a G = = a x − a y − a z = 0.667a x − 0.667a y − 0.333a z
3
3
3
|G|
A special symbol is desirable for a unit vector so that its character is immediately
apparent. Symbols that have been used are u B , a B , 1 B ,oreven b.We will consistently
use the lowercase a with an appropriate subscript.