Intersection of a locally complete intersection using Tor

Question

I just started learning about some basic intersection theory, after an introduction to algebraic geometry. I heard a lecture in which the following was left as an exercise and I cannot understand it so I was hoping for a hint. Suppose that $Z,Z' $ are irreducible subvarieties of of a smooth projective scheme $ Y$ such that $dim(Z)+dim(Z')=dim(Y)$. If $Z,Z'$ is a locally complete intersection, show that, for $i>0$, we have $Tor_i^{O_Y}(O_Z,O_{Z'}) = 0$. Here $O_Z$ is the structure sheaf. I thought possibly to try to show that the morphism from $Z \to Y$ is flat, but that seems wrong.