When does equality hold in this case?

Question

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.

Answer 1

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

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

Answer 2

If $x\perp y$, then $\|x+y\|^2=\|x\|^2+\|y\|^2$.

E.g., for $x,y \in \mathbb{R}^n$,

$\|x+y\|^2=(x+y)^T(x+y)=x^Tx+2x^Ty+y^y=\|x\|^2+\|y\|^2+2*x^Ty$

if the dot product between the two is zero (i.e. $<x,y>=x^Ty=0$), then you have: $\|x+y\|^2=\|x\|^2+\|y\|^2$