Indeed much more is true. 

Suppose $X$ is an irreducible algebraic manifold admitting a transitive action
of a linear algebraic group $G$. If $Y$ and $Z$ are irreducible subvarieties of $X$ then
for a general $g \in G$ the intersection of $gY$ (the translate of $Y$ by $g$) and $Z$
is empty or equidimensional with dimension $\mathrm{dim}(Y) + \mathrm{dim}(Z) -\mathrm{dim}(X)$.

See Kleiman's "[The transversality of a general translate][1]".


  [1]: http://www.numdam.org/item?id=CM_1974__28_3_287_0