If ${\bf v},{\bf w}\in\mathbb R^n$, $\lVert{\bf v}+{\bf w}\rVert=\lVert{\bf v}\rVert+\lVert{\bf w}\rVert$, what can you say about $\bf v$ and $\bf w$?
Hint: Squaring the equation results in $$\langle v+w,v+w\rangle=\|v+w\|^2=(\|v\|+\|w\|)^2$$ or $$ \langle v,w\rangle=\|v\|\|w\|. $$