Page 30 - Engineering Electromagnetics, 8th Edition

12 ENGINEERING ELECTROMAGNETICS

Figure 1.5 The direction of A × B is in the
direction of advance of a right-handed screw
as A is turned into B.

Reversing the order of the vectors A and B results in a unit vector in the opposite
direction, and we see that the cross product is not commutative, for B×A =−(A×B).
If the definition of the cross product is applied to the unit vectors a x and a y ,we find
a x × a y = a z , for each vector has unit magnitude, the two vectors are perpendicular,
and the rotation of a x into a y indicates the positive z direction by the definition of a
right-handed coordinate system. In a similar way, a y × a z = a x and a z × a x = a y .
Note the alphabetic symmetry. As long as the three vectors a x , a y , and a z are written
in order (and assuming that a x follows a z , like three elephants in a circle holding tails,
so that we could also write a y , a z , a x or a z , a x , a y ), then the cross and equal sign may
be placed in either of the two vacant spaces. As a matter of fact, it is now simpler to
define a right-handed rectangular coordinate system by saying that a x × a y = a z .
A simple example of the use of the cross product may be taken from geometry
or trigonometry. To find the area of a parallelogram, the product of the lengths of
two adjacent sides is multiplied by the sine of the angle between them. Using vector
notation for the two sides, we then may express the (scalar) area as the magnitude of
A × B,or |A × B|.
The cross product may be used to replace the right-hand rule familiar to all
electrical engineers. Consider the force on a straight conductor of length L, where
the direction assigned to L corresponds to the direction of the steady current I, and
a uniform magnetic field of flux density B is present. Using vector notation, we may
write the result neatly as F = IL × B. This relationship will be obtained later in
Chapter 9.
The evaluation of a cross product by means of its definition turns out to be more
work than the evaluation of the dot product from its definition, for not only must
we find the angle between the vectors, but we must also find an expression for the