If $\langle x,y \rangle=\|x\|\|y\|$ then $y=0$ or $x=\alpha y$

Question

Let X be a (complex) inner product space. Proof that if for some $x,y \in X$ we have $\langle x,y \rangle=\|x\|\|y\|$ then $y=0$ or there exists complex $\alpha$ such that $x=\alpha y$

I have tried to proof it using different properties of inner product but that led me nowhere. I appreciate any hints how to solve it.

Answer 1

Case 1: $y=0$, then we are done.

Case 2: let $y \ne 0$ and put $ \alpha = \frac{||x||}{||y||}$ and compute $||x-\alpha y||^2$ with the inner product ( $||z||^2=\langle z,z \rangle$). See what happens !