Page 29 - Engineering Electromagnetics, 8th Edition

CHAPTER 1 Vector Analysis 11

Solution. Substituting the coordinates of point Q into the expression for G,wehave

G(r Q ) = 5a x − 10a y + 3a z
Next we find the scalar component. Using the dot product, we have
1
1
G · a N = (5a x − 10a y + 3a z ) · (2a x + a y − 2a z ) = (10 − 10 − 6) =−2
3
3
The vector component is obtained by multiplying the scalar component by the unit
vector in the direction of a N ,
1
(G · a N )a N =−(2) (2a x + a y − 2a z ) =−1.333a x − 0.667a y + 1.333a z
3
The angle between G(r Q ) and a N is found from

G · a N =|G| cos θ Ga
√
−2 = 25 + 100 + 9 cos θ Ga
and
−2
θ Ga = cos −1 √ = 99.9 ◦
134

D1.3. 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 ;(b) R AC ;(c) the angle θ BAC at vertex A;(d) the
(vector) projection of R AB on R AC .

◦
Ans. −8a x + 4a y − 6a z ; −9a x + 2a y + 3a z ;53.6 ; −5.94a x + 1.319a y + 1.979a z

1.7 THE CROSS PRODUCT
Given two vectors A and B,wenow define the cross product,or vector product,of A
and B, written with a cross between the two vectors as A × B and read “A cross B.”
The cross product A × B is a vector; the magnitude of A × B is equal to the product
of the magnitudes of A, B, and the sine of the smaller angle between A and B; the
direction of A×B is perpendicular to the plane containing A and B and is along one of
the two possible perpendiculars which is in the direction of advance of a right-handed
screw as A is turned into B. This direction is illustrated in Figure 1.5. Remember that
either vector may be moved about at will, maintaining its direction constant, until
the two vectors have a “common origin.” This determines the plane containing both.
However, in most of our applications we will be concerned with vectors defined at
the same point.
As an equation we can write

A × B = a N |A||B| sin θ AB (7)

where an additional statement, such as that given above, is required to explain the
direction of the unit vector a N . The subscript stands for “normal.”