If two vectors are orthogonal then $\|a + b\|^2=\|a\|^2 + \|b\|^2$

Question

Can we say that if $a$ and $b$ are orthogonal (meaning $\langle a,b \rangle = 0$) then

$\|a + b\|^2=\|a\|^2 + \|b\|^2$

must be correct?

Reasoning:

$\|a+b\|^2 = \langle{a+b,a+b}\rangle \underset{Orthogonality}{=} \langle{a,a}\rangle + \langle{b,b}\rangle = \|a\|^2 + \|b\|^2$

Answer 1

Take @GEdgar's advice. In particular, $\langle a+b,\,a+b\rangle=\langle a,\,a\rangle+\langle a,\,b\rangle+\langle b,\,a\rangle+\langle b,\,b\rangle$ by bilinearity, but two of the terms are $0$.