# Two subspaces of different dimensions and the orthogonal complement

I need help getting started with this problem. Please just give me a nudge in the right direction.

Suppose $$U_1,U_2$$ are subspaces of the euclidean space $$V$$ such that $$dimU_1.
Show that there is a nonzero vector $$u\in U_2$$ such that $$u$$ is in the orthogonal complement of $$U_1$$.

Abridged solution. If $$U_1^\perp \cap U_2 = \{0\}$$ then $$U_1 \subset U_2.$$ Hence, the dimension hypothesis completes the result. Q.E.D.