Euclidean norm of complex vectors

Question

I am working on a proof: One has two vectors, $u,v \in \mathbb C^n$, such that $u \cdot v=0$ . I am trying to prove that

$$|u + v|^2 = |u|^2 + |v|^2.$$

I am a little stuck on how to do $u + v$ dotted with the conjugate of $u + v$. Is there anything special I can do with this?

Answer 1

$|u+v|^2 = \langle u+v,u+v\rangle = \langle u,u\rangle + 0 + 0 + \langle v,v \rangle = |u|^2 + |v|^2$

Answer 2

Here is how.

$$ ||u+v||^2 = \langle u+v,u+v \rangle = \langle u,u \rangle + \langle v,v \rangle + \langle u,v \rangle + \langle v,u \rangle = ||u||^2+||v||^2 +0+0. $$

Note:

$$ \langle u,v \rangle = u \cdot v $$