Prove that if $V$ is an inner product space, then $|\langle x,y\rangle| = \|x\| \cdot \|y\|$ if and only if one of the vectors $x$ or $y$ is a multiple of the other.
Given the hint: identity holds and $y \neq 0$, let $a=\frac{\langle x,y\rangle}{\|y\|^2}$, and let $z=x-ay$. Prove $y$ and $z$ are orthogonal and $|a|=\frac{\|x\|}{\|y\|}$. Then use Pythagorean theorem
I am not able to appreciate this hint, can someone explain the general idea behind this? especially why do we need a $z$ a have to prove $z$ and $y$ are orthogonal? Want the intuition to understand why to do this.
