Proof with orthogonal vectors in real analysis If $x,y \in \mathbb{R}^n$, then $x$ and $y$ are called perpendicular (or orthogonal) if $\langle x,y\rangle =0$. If $x$ and $y$ are perpendicular, prove that $|x+y|^2=|x|^2+|y|^2$.
Seems pretty basic, but I'm missing something.

$$\big|x+y\big|^2= \langle x+y,x+y \rangle=\langle x,x+y\rangle+y\langle x+y\rangle$$ $$= \langle x,x \rangle + \langle x,y \rangle + \langle y,x\rangle+ \langle y,y\rangle$$ $$=\big|x\big|^2+\big|y\big|^2$$
as $\langle x,y \rangle = \langle y,x \rangle =0$
