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".

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 of dimension $\mathrm{dim}(Y) + \mathrm{dim}(Z) -\mathrm{dim}(X)$.

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

Indeed much more is true.

Suppose $X$ is a projective manifold $X$ 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".

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".