When does equality hold in this case? Give example of two vectors $x$ and $y$ such that $$||x+y||_2^2 = ||x||_2^2+||y||_2^2$$
and
$$<x,y>\neq0$$
I can't seem to find any two vectors $x$ and $y$ that satisfied both conditions at the same time.

In $\mathbb{C}$ as $\mathbb{C}$-space, with $(z,w)=z\overline{w}$

 $$|1+i|^2=2=1+1=|1|^2+|i|^2$$
 and $$(1,i)=1\overline{i}=-i\neq0$$

