A

vector space V along with an

inner product I is said to be a

Euclidean vector space. The inner product I allows for a

geometry on V, by the

definition of

angle and

distance between two

vectors as follows:

d(a,b) := I(a-b,a-b)

theta(a,b) := I(a,b)/( I(a,a)*I(b,b) )

where d:V*V -> R is said to be the distance between two vectors and theta:V*V->(0,1) is taken to be the

cosine of the angle between the two vectors.

With these definitions of distance and angle, along with the definition of a "

point" as an

element of V, the

axioms of pure geometry (

Euclid's first four axioms) may be verified.