Angle between non parallel unit vectors

Question

Suppose that $ u $ and $ v $ are non-parallel unit vectors, $ a = u+\sqrt{3}v $ and $ b = u- \sqrt{3}v $. If the angle $ \theta $ between $ u $ and $ v $ is the same as that between $ a $ and $ b $, find $ cos(\theta) $.

I'm stuck. Please help.

Thanks in advance!

Answer 1

Hint: Note that $\cos(\theta)\lVert a\rVert\lVert b\rVert=\langle a,b\rangle=\left\langle u+\sqrt3v,u-\sqrt3v\right\rangle=-2$. Now, what can you say about $\lVert a\rVert$ and $\lVert b\rVert$?