Prove that if the dot product is the product of the norms, then the vectors are linearly dependent. [duplicate]

Let u, v be n-tuples in R^n. Recall the Cauchy-Schwarz Inequality: |<u,v>| <= |u||v|.

Prove that |<u,v>| = |u||v| if and only if the u,v are linearly dependent, that is u = 0 or v = au for some a in R.

The only solution I could think of involved using <u,v> = |u||v|cos(theta), which I cannot use. I read another solution to this on this forum, and I couldn't parse it out either. I managed to prove the reverse case.