Dot product
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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 innerproduct 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 innerproduct space
We will now prove that the dot product turns into an innerproduct space. There are four statements to prove, namely, given any and any scalar , the following is true:
 with equality if and only if
 Proof.
 , 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 .
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 innerproduct space are said to be orthogonal if and only if their innerproduct is 0. In this translates to: and in are orthogonal if and only if . Note that the zero vector is orthogonal to every vector.