A Guide To Dot Products

Source: http://mathinsight.org/media/image/image/dot_product_projection.png

This post is an introduction to dot products. It is assumed that the reader knows about vectors where a vector in is of the form . In addition, the reader should be familiar with the concept of a norm.

Definition of the Dot Product

The dot product or the Euclidean inner product is an algebraic operation which takes two vectors of the same length and returns a scalar number. Think of the dot product as another operation like +, – , and .

Given the vectors and in , the dot product of and denoted by is:

If we have the special case of then we obtain the square of the norm as follows:

From the above result, we can identify that the norm or the length of a vector can be expressed as the square root of the dot product (below).

Properties

When introduced with a new operation, some properties need to be introduced. These may not sound exciting but they are important. Think of it like grammar in languages.

Suppose we have vectors , and in and as a scalar (real number), we have these properties.

[Symmetry Property]

[Distributive Property]

[Homogeneity Property]

[Positivity Property]

The above four properties look scary but they are not too bad. Here are a couple of more below:

.

Geometry and Dot Products

Dot products along with norms can help us find the cosine of an angle . A useful formula is:

where , are vectors in and is the angle between and . The angle is defined to be between 0 and ( in radians or 0 to 90 degrees).

We can find out some out some information about based on the dot product.

If then is an acute angle (0 to less than 90 degrees).

If then is an obtuse angle (More than 90 degrees to less than 180 degrees).

If then is a right angle ( = 90 degrees or ).

A visual aid below summarizes the above cases.

Examples

Example One

The dot product of vectors and is:

Example Two

Given vectors and , the dot product of these vectors in is:\

Example Three

Suppose we are given the vectors , and . What is

We compute first then take the dot product of with \textbf{b}.\

Example Four

Simplify (2) ( + 3).