I've come across the following formula and I've been told that it's expressing a dot product in terms of length, but I can't find any sources or derivations for it online.
$$⟨u,v⟩ = \frac{|u+v|^2 - |u|^2 - |v|^2 }{2}$$
I found some similar-looking formulae, but nothing exactly of this form. Can you tell me how to derive this? I know about finding the Pythagorean length, but stumped to find the exact formula above.
There is also this question which seems to be almost the same, but again, it's not quite there.
Calculate dot product without the use of angles
And in a comment to that question, the hint
$$(a+b)^2−(a−b)^2=4ab$$
This is a polarization identity, it can be shown by rewriting the norms in terms of the inner product. In this case,
\begin{align*} |u+v|^2 - |u|^2 - |v|^2 &= \langle u+v,u+v \rangle - |u|^2 - |v|^2 \\ &= \langle u,u\rangle + 2 \langle u,v \rangle + \langle v,v \rangle - |u|^2 - |v|^2 \\ &= |u|^2 + 2 \langle u,v \rangle + |v|^2 - |u|^2 - |v|^2 \\ &= 2 \langle u,v \rangle. \end{align*}