Let $vec uinmathbb{R}^d$ be a random unit vector, with uniform distribution on the surface of the unit sphere. For a **fixed** unit vector $vec v$, what is the following probability?

$$ Prleft( left| langle vec u, vec v rangle right| geq xi right) $$

My guess is that it would be something like $1-xi$, since (presumably, I’m not sure about that either) $langle vec u, vec v rangle$ also has a uniform distribution for fixed unit vector $vec v$ and uniformly-distributed-on-surface-of-unit-sphere $vec u$, but I’m not aware of distributions on the surface of spheres at all, so if someone could enlighten me, please do.