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

Please don't use pictures for text or formulas. See here, why. Use MathJax. Here is a tutorial. — Dietrich Burde, May 02 '23 at 15:08
I didn't check your solution's validity, but it looks overly complicated. Hint: all you need is $\vec a \cdot \vec a = |\vec a|^2$. — Deepak, May 02 '23 at 15:12
I know the correct solution at this point, I am asking as it was an exam question, and I am curious if that answer has any validity. — vitaliymsh, May 02 '23 at 15:16
Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. — Community, May 02 '23 at 15:17

score 1 · Answer 1 · answered May 02 '23 at 16:04

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.

answered May 02 '23 at 16:04

Michael Hoppe

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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."

1 Answers1