$u+v$ is orthogonal to $u-v$ if and only if $\|u\| = \|v\|$

Question

I'm doing an introductory linear algebra course and I'm stuck on this question.

Show that with respect to any inner product, $u+v$ is orthogonal to $u-v$ if and only if $\|u\| = \|v\|$.

I'm trying to prove the forward implication and I don't know where to go from $\langle u+v,u-v \rangle=0$

I tried working with the cosine formula or with the fact that $\langle u+v,u-v \rangle = \langle u-v,u+v \rangle$ but I don't really know where I'm going...

Could someone show me how to prove both forward and backward implications?

Thanks in advance!

Answer 1

Expand the inner product using the linearity rules (twice): $\langle a + b, c \rangle = \langle a,c\rangle + \langle b,c\rangle$. Can you take it from here?

Answer 2

You can start like this:
$\langle u+v,u-v\rangle =0$ $\Leftrightarrow$
$\langle u+v,u\rangle - \langle u+v,v\rangle=0$ $\Leftrightarrow$
Can you take it from there? (At some place you will have to use that $\langle u,v \rangle = \langle v,u\rangle$ and that $\langle u,u\rangle = \|u\|^2$.)