Question

Prove that `(veca + vecb).(veca + vecb)` = `|veca|^2 + |vecb|^2` if and only if `veca . vecb` are perpendicular, given `veca != vec0, vecb != vec0.`

Answer 1

`(veca + vecb) xx (veca + vecb) = |veca|^2 + |vecb|^2`

`veca xx veca + veca xx vecb + vecb xx veca + vecb xx vecb = |veca|^2 + |vecb|^2`

`|veca|^2 + 2veca xx vecb + |b|^2 = |veca|^2 + |vecb|^2`

`2veca xx vecb = 0`

`veca xx vecb = 0`

`veca, vecb` are perpendicular.