The Inner Product

The Inner Product (A.K.A. dot product)

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.

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