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
between two vectors
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.