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

  • Thread starter butterfli
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  • #1
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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.
 

Answers and Replies

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

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