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.