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We prove that the number of directions contained in a set of the form $A\times B \subset AG(2,p)$, where $p$ is prime, is at least $|A||B| -\min\{|A|,|B|\} + 2$. Here $A$ and $B$ are subsets of $GF(p)$ each with atleast two elements and $|A||B| <p$. This bound is tight for an infinite classof examples. Our main tool is the use of the R\'edei polynomial withSz\H{o}nyi's extension. As an application of our main result, we obtain anupper bound on the clique number of a Paley graph, matching the current bestbound obtained recently by Hanson and Petridis.

On the directions determined by a Cartesian product in an affine Galois plane