Page 27 - Engineering Electromagnetics, 8th Edition

CHAPTER 1 Vector Analysis 9

1.6 THE DOT PRODUCT
We now consider the first of two types of vector multiplication. The second type will
be discussed in the following section.
Given two vectors A and B, the dot product,or scalar product,is defined as the
product of the magnitude of A, the magnitude of B, and the cosine of the smaller
angle between them,

A · B =|A||B| cos θ AB (3)

The dot appears between the two vectors and should be made heavy for emphasis.
The dot, or scalar, product is a scalar, as one of the names implies, and it obeys the
commutative law,

A · B = B · A (4)

for the sign of the angle does not affect the cosine term. The expression A · B is read
“A dot B.”
Perhaps the most common application of the dot product is in mechanics, where
a constant force F applied over a straight displacement L does an amount of work
FL cos θ, which is more easily written F · L.We might anticipate one of the results
of Chapter 4 by pointing out that if the force varies along the path, integration is
necessary to find the total work, and the result becomes

F · dL
Work =
Another example might be taken from magnetic fields. The total flux crossing
a surface of area S is given by BS if the magnetic flux density B is perpendicular
to the surface and uniform over it. We define a vector surface S as having area
for its magnitude and having a direction normal to the surface (avoiding for the
moment the problem of which of the two possible normals to take). The flux crossing
the surface is then B · S. This expression is valid for any direction of the uniform
magnetic flux density. If the flux density is not constant over the surface, the total flux

is = B · dS. Integrals of this general form appear in Chapter 3 when we study
electric flux density.
Finding the angle between two vectors in three-dimensional space is often a
job we would prefer to avoid, and for that reason the definition of the dot product is
usually not used in its basic form. A more helpful result is obtained by considering two
vectors whose rectangular components are given, such as A = A x a x + A y a y + A z a z
and B = B x a x + B y a y + B z a z . The dot product also obeys the distributive law, and,
therefore, A · B yields the sum of nine scalar terms, each involving the dot product
of two unit vectors. Because the angle between two different unit vectors of the
rectangular coordinate system is 90 ,we then have
◦
a x · a y = a y · a x = a x · a z = a z · a x = a y · a z = a z · a y = 0