Prove that if $\Vert v+w \Vert^2 = \Vert v\Vert^2 + \Vert w\Vert^2$, then $v\cdot w = 0$

Question

Let $v,w \in \Bbb{R}^n$.

Prove that if $\Vert v+w \Vert^2 = \Vert v\Vert^2 + \Vert w\Vert^2$, then $v\cdot w = 0$.

Do I prove this using axioms of vector spaces?

Any help is appreciated, thanks!

Answer 1

Note that $\|v+w\|^{2}=\langle (v+w), (v+w)\rangle=\langle v, v)\rangle+\langle v, w\rangle+\langle w, v\rangle+\langle w, w)\rangle=\|v\|^{2}+\|w\|^{2}+2 \langle v, w\rangle$.