Q: Prove that $\displaystyle ||\vec{v}+\vec{u}||=||\vec{u}||+||\vec{v}||$ if and only if $\displaystyle \vec{u}$ and $\displaystyle \vec{v}$ have the same direction.

A: Well, if $\displaystyle \vec{u}$ the same direction of $\displaystyle \vec{v}$, then $\displaystyle \vec{u}$ must be a multiple of $\displaystyle \vec{v}$. Thus, $\displaystyle \vec{u}=k\vec{v}$ where $\displaystyle k\in{\mathbb{R}}$ (I am not sure if I lost any generality by choosing $\displaystyle \vec{u}$). From here I expanded $\displaystyle ||\vec{v}+\vec{u}||$ and ended up with some messy algebra and not the result I was looking for.

In the other direction I started with $\displaystyle ||\vec{v}+\vec{u}||=...$ and worked my way down to $\displaystyle =||\vec{u}||^{2}+|2<\vec{u},\vec{v}>|+||\vec{v}||^ {2}$ and figured I have to introduce the Cauchy-Schawrz Ineguality here, but I am not sure how.

I am having a hard time formalizing this proof. All I have is a bunch of dead und scratch work. It seems that this proof should flow with ease, because, geometrically, it is pretty obviouse.

I would appreciate some guidence.

Thank you.