  We generalize Ehrhart's idea of counting lattice points in dilated rational
polytopes: Given a rational simplex, that is, an n-dimensional polytope with
n+1 rational vertices, we use its description as the intersection of n+1
halfspaces, which determine the facets of the simplex. Instead of just a single
dilation factor, we allow different dilation factors for each of these facets.
We give an elementary proof that the lattice point counts in the interior and
closure of such a "vector-dilated" simplex are quasipolynomials satisfying an
Ehrhart-type reciprocity law. This generalizes the classical reciprocity law
for rational polytopes. As an example, we derive a lattice point count formula
for a rectangular rational triangle, which enables us to compute the number of
lattice points inside any rational polygon.
