TITLE: Prove that $u\cdot v = 1/4||u+v||^2 - 1/4||u-v||^2$ for all vectors $u$ and $v$ in $\mathbb{R}^n$
QUESTION [5 upvotes]: I need some help figuring out how to work through this problem.

Prove that $ u \cdot v  = 1/4 ||u + v||^2 - 1/4||u - v||^2$ for all vectors $u$ and $v$ in $\mathbb{R}^n$. 

Sorry, forgot to include my work so far:
I decided to ignore the 1/4 and deal with it later once I had a better understanding of the question.
$= ||u+v||^2 - ||u-v||^2$
$= (u+v)(u+v) - (u-v)(u-v)$
$= u(u+v) + v(u+v) - u(u-v) + v(u-v)$
$= uu + uv + uv + vv - uu + uv + uv - vv$
$u \cdot v= 3(uv)$
This is as far as I've gotten, not sure if I'm on the right track or where to go next.

REPLY [2 votes]: Here is a start

$$||u+v||^2= \langle u+v, u+v \rangle= \langle u, u \rangle+\dots \,. $$

Do the same with the other and multiply both eqs and by $\frac{1}{4}$ and subtract. See my answer. 
Note:

$$\langle u, v\rangle = u.v  \,.  $$