vectors -- is this a recognizable quantity?

Question

I'm working on a problem with two vectors, $u=\left<a,b\right>$ and $v=\left<c,d\right>$. One quantity that crops up in solving my problem is $a^2+b^2-c^2-d^2$. I know that's $\left|u\right|^2-\left|v\right|^2$, but my question is whether there's a geometric visualization of how to simplify this any further or if there's a useful identity I might be missing.

Answer 1

You can simplify it as the scalar product $(u+v, u-v)$. Geometrically you can say that it measures the orthogonality of these two vectors (i.e. this is equal to zero iff $u+v$ and $u-v$ are orthogonal).