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.