I'm attempting to distribute vectors with the same origin with a maximum angle of separation. Then if given a set of vectors, I want to determine how far from maximum separation they are. For instance in 2D, two vectors would be 180 degrees apart, 3 would be 120, 4 90, etc. So my two problems are determining maximum separation in 3D and giving some sort of feedback on how close they are to maximum separation.
Thanks
You are opening very deep problems about sphere packings, kissing numbers and so on here. A very rough estimate would be the following: The curved surface of a sperical cap on the unit ball with angle $\alpha$ is $2\pi(1-\cos\alpha)$. Hence if you have $n$ vectors and all angles between them are $\ge2\alpha$, then we can compare areas and find $2n\pi(1-\cos\alpha)\le 4\pi$, i.e. the number $$ \frac{n(1-\cos\alpha)}2$$ can be taken as a simple estimation for the "quality": The closer the value is to $1$, the better. Note however, that $1$ cannot be reached (except for $n=2$) and in fact for $n\to \infty$, we cannot beat the hexagonal density $\frac\pi{\sqrt{12}}\approx0.9$.
