Given the length of vector v ||v||, vector u ||u||, and ||v + u||, how would I find ||2v + u||?

[gracie614]

I am working on a homework problem for multivariable calculus that gives ||v|| = 1, ||w|| = 1, and ||v + w|| = 1. I was wondering how I could then use this to find ||2v + u||. A step by step would be appreciated so I could generalize it onto other problems. Thank you in advance

 

[Harry_the_cat]

Draw a triangle diagram showing addition of vectors. The only way that ||v|| = 1, ||w|| = 1, and ||v + w|| = 1 can hold is if the triangle is ???

 

[pka]

I am working on a homework problem for multivariable calculus that gives ||v|| = 1, ||w|| = 1, and ||v + w|| = 1. I was wondering how I could then use this to find ||2v + u||. A step by step would be appreciated so I could generalize it onto other problems.

Recall that $\|\vec{v}\|^2=\vec{v}\cdot\vec{v}$ therefore if $\|\vec{u}\|=1$ then $\vec{u}\cdot\vec{u}=1$
Now $(2\vec{v}+\vec{u})\cdot(2\vec{v}+\vec{u})=4\vec{v}\cdot\vec{v}+4\vec{v}\cdot\vec{u}+\vec{u}\cdot\vec{u}$
OR that is $5+4\vec{v}\cdot\vec{u}$ explain why that is true!
From that can you get $\|2\vec{v}+\vec{u}\|~?$