# [Linear Algebra]Proving the magnitude of sums is equal to the sum of the magnitudes

## Homework Statement

*v and u are vectors where ||u|| is the magnitude of u and ||v|| is the magnitude of v

Prove that ||u + v|| = ||u|| + ||v|| if and only if u and v have the same direction.

## The Attempt at a Solution

At first, I tried using what it means for two vectors to have the same direction: u = v/||v||

u + v = v/||v|| + v (added v to both sides)
||u + v|| = ||(v/||v||) + v|| (took the magnitude of both sides)

From here, if I substitute (v/||v||) with u, I would just have ||u + v|| equal to itself.

I also tried looking up Properties of Dot Product but couldn't find a place to apply them. I'm kinda stuck on what else I can do so if anyone can provide tips or pointers in the right direction, I'd be grateful.

You should definitely use the dot product for this one. You will need two things. Firstly, remember that $<a,b>=|a||b|\cos \theta$. What is the angle between two vectors that point in the same direction? Secondly, $|a|^2=<a,a>$, write the left hand side like this.