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The dot product is the most common way to define an inner product between elements of (-dimensional vectors).

Definition
Let and . We define the dot product between and by

Note that some texts use the symbol to denote the dot product between and , preserving the inner-product notation.

The dot product is one of three common types of multiplication compatible with vectors; the other being the cross product and scalar multiplication, the latter belonging to the vector space nature of .

## as an inner-product space

We will now prove that the dot product turns into an inner-product space. There are four statements to prove, namely, given any and any scalar , the following is true:

1. with equality if and only if
Proof.
1. , where we in the second step factored out the
2. . But each (), so , as required. Now suppose that . Then clearly . If , then . Suppose for the sake of contradiction that some . Then so that . But this is a contradiction, so we must have

This completes the proof.

## Euclidean norm and as a metric space

Once we have defined the dot product between elements of Euclidean -space, we may define a map , when applied to is called the norm of .

Definition
If , we define the norm of , denoted by , by

One can show that if and , then is a valid distance between and , and hence turns into a metric space. In fact, this metric space is complete, meaning that every Cauchy sequence of elements in converges to some point in .

## Angles between two elements

The dot product can be used to determine the angle between two elements:

### Orthogonality

Two elements in an inner-product space are said to be orthogonal if and only if their inner-product is 0. In this translates to: and in are orthogonal if and only if . Note that the zero vector is orthogonal to every vector.