Statement: “Consider vectors $u, v \in \Bbb R_n$. If $\Vert u−v \Vert = \Vert u+v \Vert$, then $u$ is orthogonal to $v$. Prove true or false."

Question

I attached my proof below and I am curious if there is any truth to it.my solution

Answer 1

We have $\|u-v\|=\|u+v\|$ iff $$\|u\|^2-2\langle u,v\rangle+\|v\|^2 =\|u\|^2+2\langle u,v\rangle+\|v\|^2$$ iff $\langle u,v\rangle=0$. Avoid coordinate whenever possible.