Suppose a body moves under the influence of a single mass producing a gravitational field. The equation of motion of the body is
Taking the cross product of this equation with
gives
Now
since angular mometum is conserved.
Hence
a constant vector. We can write
Using the identity
gives
Integrating this equation gives
and taking the dot product withgives
where
is the angle between
and
Now use the identity
to give
Hence