  Given a postulated set of points, an algebraic system of axioms is proposed
for an "arrow space'". An arrow is defined to be an ordered set of two points
<T, H>, named respectively Tail and Head. The set of arrows is an arrow space.
The arrow space is axiomatically endowed with an arrow space "pre-inner
product" which is analogous to the inner product of a Euclidean vector space.
Using this arrow space pre-inner product, various properties of the arrow space
are derived and contrasted with the properties of a Euclidean vector space. The
axioms of a vector space and its associated inner product are derived as
theorems that follow from the axioms of an arrow space since vectors are
rigorously shown to be equivalence classes of arrows. Applications of using an
arrow space to solve geometric problems in affine geometry are provided.
