Nonnegativity condition for equality within Triangle Inequality

Question

Suppose $u, v \in V$, where $V$ is an inner product space over $\mathbb{C}$. Then $\begin{equation*} ||u + v|| \leq ||u|| + ||v||.\end{equation*}$

This relation is an equality if and only if one of $u, v$ is a nonnegative multiple of the other.

In the case of equality, we can use Cauchy-Schwarz to conclude that one of $u, v$ is a multiple of the other. However, I am not understanding the additional nonnegativity condition.

Answer 1

If $u$ is nonzero, $v=-u$, then $\|u+v\|=0$ but $\|u\|+\|v\|>0$.

In general, if $v=\lambda u$, then $\|u+v\|=|1+\lambda|\|u\|$ while $\|u\|+\|v\|=(1+|\lambda|)\|u\|$.

Answer 2

An hint with a picture:

let $\overrightarrow{AB}=\vec u$

$\overrightarrow{CD}=\vec v$ is a positive multiple of $\vec u$ and $\overrightarrow{EF}=\vec u +\vec v$

$\overrightarrow{CG}=\vec v'$ is a negative multiple of $\vec u$ and $\overrightarrow{EH}=\vec u +\vec v'$

can you formalize the general result from this intuition?