Grade 12 Vectors Problem

[vectors123]

I need some help with this question:

Prove in three space, that non-zero vectors u and v are orthogonal if and only if |u+v| = |u-v|. You must prove both directions.

Thanks

 

[pka]

I need some help with this question:
Prove in three space, that non-zero vectors u and v are orthogonal if and only if |u+v| = |u-v|. You must prove both directions.

HINTS:
\(\displaystyle \|u\|^2=u\cdot u \)

\(\displaystyle (u+v)\cdot(u+v)=u\cdot u+2u\cdot v+v\cdot v \)

\(\displaystyle (u-v)\cdot(u-v)=u\cdot u-2u\cdot v+v\cdot v \)

If \(\displaystyle \|u+v\|=\|u-v\| \) then \(\displaystyle \|u+v\|^2=\|u-v\|^2 \)

If \(\displaystyle u \bot v \) then \(\displaystyle u\cdot v=0 \)