For geometric vectors, such as seen in elementary physics:
the inner product operation is performed on a pair of vectors, and is denoted by a pair of angled brackets: a· b is read as “the dot product of vectors a and b”. (Note that vectors in this document will be denoted by boldface type, and that |v| will represent the length or magnitude of some vector v.) For any two vectors a and b with an angle of q radians between them,
Finding
the dot product of two vectors
if a = a1 i + a2 j and b = b1 i + b2
j then
a· b = a1b1 + a2b2.
One multiplies the respective components together and adds them up. An alternative
formula is a· b = |a| |b| cos(q).
If the two vectors are
perpendicular , meaning that q = 90 degrees
or π/2 radians, then a· b = 0 and the vectors are referred to
as being orthogonal. This can be interpreted as showing that they are
completely independent of one another.
If the dot product is positive
then the angle q is less then 90 degrees
and the each vector has a component in the direction of the other. If the dot
product is negative then the angle
is greater than 90 degrees and one vector has a component in the opposite direction of the other. Thus
the simple sign of the dot product gives information about the geometric
relationship of the two vectors.
For signals:
any two real signals a(t)
and b(t), the dot product (also called the inner product)
of the signals over the interval [t1,
t2] is
.
Recalling from
elementary calculus that integration is really just the limit of an addition
process, we can write this integral as
and we see that, as
before, we are multiplying components together and adding them up. There are
just more of them.
Like its vector
counterpart, the dot product in the above equation is a way of measuring of the
“relatedness” of two signals. As a result—
just like the vector dot product—we can make assumptions about the relationship
between the two signals a(t) and b(t), given the sign of
their dot product. A positive dot
product means that two signals have a lot in common—they are related in a way very
similar to two vectors pointing in the same direction. Likewise, a negative dot product means that
the signals are related in a negative
way, much like vectors pointing in opposing directions.