$|u+w| = |u| + |w|$ iff $\langle u,w \rangle =0$.

Question

I was asked what needs to hold such that $|u+w|=|u|+|w|$. Where $u,w \in \mathbb R^n$.

Well, first notice that if $|u+w|=|u|+|w|$ then $|u+w|^2 = |u|^2 + 2|u||w|+|w|^2$

if we go by the definition $|v|=\sqrt{\langle v,v \rangle}$ which I think is the most standard definition (where $\langle .\rangle$ is some inner product), then we have:

$|u+w|^2 = \langle u+w , u+w \rangle=\langle u ,u \rangle+\langle u , w \rangle +\langle w , u \rangle+\langle w,w \rangle=|u|^2+2\langle u,w \rangle +|w|^2$

And so for $|u+w|^2 = |u|^2 + 2|u||w|+|w|^2$ to hold, we must have $2\langle u,w \rangle = 2|u||w|$ or in other words,

$\langle u,w \rangle = \sqrt{ \langle u,u \rangle \langle w,w \rangle }$

Why is this true iff $\langle u,w \rangle = 0$? Doesn't seem correct, and that was the answer the teacher gave.

Answer 1

It seems that the question that the teach meant to ask was find a condition such that $$ |u+v|^2 = |u|^2 + |v|^2 $$ Note that the desired if and only if statement holds only for real vector spaces.