  We prove that if a subset of a $d$-dimensional vector space over a finite
field with $q$ elements has more than $q^{d-1}$ elements, then it determines
all the possible directions. If a set has more than $q^k$ elements, it
determines a $k$-dimensional set of directions. We prove stronger results for
sets that are sufficiently random. This result is best possible as the example
of a $k$-dimensional hyperplane shows. We can view this question as an Erd\H os
type problem where a sufficiently large subset of a vector space determines a
large number of configurations of a given type. For discrete subsets of ${\Bbb
R}^d$, this question has been previously studied by Pach, Pinchasi and Sharir.
