Proving ||u + v|| = ||u|| + ||v|| if Vectors U & V Have Same Direction

[butterfli]

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.

Homework Equations

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.

 

[Cyosis]

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.