Page 23 - Engineering Electromagnetics, 8th Edition

CHAPTER 1 Vector Analysis 5

values, which must be used in all other coordinate systems, is to consider the point as
being at the common intersection of three surfaces. These are the planes x = constant,
y = constant, and z = constant, where the constants are the coordinate values of the
point.
Figure 1.2b shows points P and Q whose coordinates are (1, 2, 3) and (2, −2, 1),
respectively. Point P is therefore located at the common point of intersection of the
planes x = 1, y = 2, and z = 3, whereas point Q is located at the intersection of the
planes x = 2, y =−2, and z = 1.
As we encounter other coordinate systems in Sections 1.8 and 1.9, we expect
points to be located at the common intersection of three surfaces, not necessarily
planes, but still mutually perpendicular at the point of intersection.
If we visualize three planes intersecting at the general point P, whose coordinates
are x, y, and z,we may increase each coordinate value by a differential amount and
obtain three slightly displaced planes intersecting at point P , whose coordinates are

x + dx, y + dy, and z + dz. The six planes define a rectangular parallelepiped whose
volume is dv = dxdydz; the surfaces have differential areas dS of dxdy, dydz, and
dzdx. Finally, the distance dL from P to P is the diagonal of the parallelepiped and

has a length of (dx) + (dy) + (dz) . The volume element is shown in Figure 1.2c;
2
2
2
point P is indicated, but point P is located at the only invisible corner.

All this is familiar from trigonometry or solid geometry and as yet involves only
scalar quantities. We will describe vectors in terms of a coordinate system in the next
section.
1.4 VECTOR COMPONENTS
AND UNIT VECTORS
To describe a vector in the rectangular coordinate system, let us first consider a vector r
extending outward from the origin. A logical way to identify this vector is by giving
the three component vectors, lying along the three coordinate axes, whose vector sum
must be the given vector. If the component vectors of the vector r are x, y, and z,
then r = x + y + z. The component vectors are shown in Figure 1.3a. Instead of one
vector, we now have three, but this is a step forward because the three vectors are of
avery simple nature; each is always directed along one of the coordinate axes.
The component vectors have magnitudes that depend on the given vector (such
as r), but they each have a known and constant direction. This suggests the use of unit
vectors having unit magnitude by definition; these are parallel to the coordinate axes
and they point in the direction of increasing coordinate values. We reserve the symbol
a for a unit vector and identify its direction by an appropriate subscript. Thus a x , a y ,
3
and a z are the unit vectors in the rectangular coordinate system. They are directed
along the x, y, and z axes, respectively, as shown in Figure 1.3b.
If the component vector y happens to be two units in magnitude and directed
toward increasing values of y,we should then write y = 2a y .Avector r P pointing

3 The symbols i, j, and k are also commonly used for the unit vectors in rectangular coordinates.