| 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:
- with equality if and only if
- , where we in the second step factored out the
- . 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 .
| 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:
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.