Page 31 - Engineering Electromagnetics, 8th Edition

CHAPTER 1 Vector Analysis 13

unit vector a N . This work may be avoided by using rectangular components for the
two vectors A and B and expanding the cross product as a sum of nine simpler cross
products, each involving two unit vectors,

A × B = A x B x a x × a x + A x B y a x × a y + A x B z a x × a z
+ A y B x a y × a x + A y B y a y × a y + A y B z a y × a z
+ A z B x a z × a x + A z B y a z × a y + A z B z a z × a z
We have already found that a x × a y = a z , a y × a z = a x , and a z × a x = a y . The
three remaining terms are zero, for the cross product of any vector with itself is zero,
since the included angle is zero. These results may be combined to give

A × B = (A y B z − A z B y )a x + (A z B x − A x B z )a y + (A x B y − A y B x )a z (8)
or written as a determinant in a more easily remembered form,

a x a y a z

A × B = A x A y A z (9)

B x B y B z
Thus, if A = 2a x − 3a y + a z and B =−4a x − 2a y + 5a z , we have

a x a y a z

A × B = 2 −3 1

−4 −2 5

= [(−3)(5) − (1(−2)]a x − [(2)(5) − (1)(−4)]a y + [(2)(−2) − (−3)(−4)]a z
=−13a x − 14a y − 16a z

D1.4. The three vertices of a triangle are located at A(6, −1, 2), B(−2, 3, −4),
and C(−3, 1, 5). Find: (a) R AB × R AC ;(b) the area of the triangle; (c)a unit
vector perpendicular to the plane in which the triangle is located.

Ans. 24a x + 78a y + 20a z ; 42.0; 0.286a x + 0.928a y + 0.238a z

1.8 OTHER COORDINATE SYSTEMS:
CIRCULAR CYLINDRICAL COORDINATES
The rectangular coordinate system is generally the one in which students prefer to
work every problem. This often means a lot more work, because many problems
possess a type of symmetry that pleads for a more logical treatment. It is easier to
do now, once and for all, the work required to become familiar with cylindrical and
spherical coordinates, instead of applying an equal or greater effort to every problem
involving cylindrical or spherical symmetry later. With this in mind, we will take a
careful and unhurried look at cylindrical and spherical coordinates.