Prove $ |\langle u,v\rangle| = \lVert u \rVert \cdot \lVert v \rVert$

Question

If V is the finite dimensional inner product space, then prove the following:

If $u, v \in V$ are linearly dependent, then $ |\langle u,v\rangle| = \lVert u \rVert \cdot \lVert v \rVert$

Thanks.

Answer 1

Hint: Try to prove that if $u,v$ are linearly dependent, then there exists $\alpha$ such that $v = \alpha u$.

Answer 2

If they are linearly dependent then $u = cv$ for some constant $c$. Then you have that $|\langle u, v \rangle| = |c|\cdot ||v||^2$. Can you see where to go from here?