Suppose C and D are vector spaces over a field. Is the intersection of C and D a vector space? Give a proof or counterexample.

Okay, from the simple examples I've tried, it appears to be true, but I cannot figure out exactly how to prove it. My professor says I need to show it contains a zero element, is closed under scalar multiplication, and is closed under addition. I know that since both are vector spaces, they both must contain the zero element, so their intersection does as well. However, I don't know where to start with the other two requirements.