Begin2.DVI

The physical interpretation of projection can be assigned to the dot product as

is illustrated in figure 6-6. In this figure A
and B
are nonzero vectors with ˆeAand

ˆeBunit vectors in the directions of A
and B,
respectively. The figure 6-6 illustrates

the physical interpretation of the following equations:

ˆeB·A
=|A
|cos θ=Projection of A
onto direction of ˆeB

ˆeA·B
=|B
|cos θ=Projection of B
onto direction of ˆeA.

In general, the dot product of a nonzero vector A
with a unit vector ˆe is given

by A
·ˆe=ˆe·A
=|A
||ˆe|cos θand represents the projection of the given vector onto

the direction of the unit vector. The dot product of a vector with a unit vector

is a basic fundamental concept which arises in a variety of science and engineering

applications.

Figure 6-6. Projection of one vector onto another.

Observe that if the dot product of two vectors is zero, A
·B
=|A
||B
|cos θ= 0,

then this implies that either A
=
0 , B
=
0 ,or θ=π 2. If A
and B
are both nonzero

vectors and their dot product is zero , then the angle between these vectors, when

their origins coincide, must be θ=π 2. One can then say the vector A
is perpendicular

to the vector B
or one can state that the projection of B
on A
is zero. If A
and B
are

nonzero vectors and A
·B
= 0 , then the vectors A
and B
are said to be orthogonal

vectors.

Direction Cosines Associated With Vectors

Let A
be a nonzero vector having its origin at the origin of a rectangular cartesian

coordinate system. The dot products