triangle inequality and dot product

Question

So I have this question here which says:

If $u,v$ are vectors in $\mathbb{R^n}$, such that $||u+v||=2$ and $||u-v||=\sqrt{8}$ then $u\cdot v=$

$a) -1$

$b)$ $4$

$c)-4$

$d)\sqrt{2}$

$e)$ $0$

It seems like a really obvious question but there are a a few issues. First I can't do a system of equation on this because I end up with a negative square root if I try that. I then tried to incorporate the triangle inequality into this question but I can't really solve for anything.

It is an inequality and not an equality so I am not sure if it would be appropriate use use the triangle inequality here. I feel like I'm missing something super obvious here but I can't figure out what it is.

Answer 1

If you calculate, $$ \|u+v\|^2-\|u-v\|^2=4u\cdot v. $$