Uniform distribution on sphere

Question

Let $U = (u_1, u_2, u_3)$ is random vector uniformly distributed on unit sphere $S^{2} \subset \mathbb{R}^3$. Are $u_1, u_2, u_3$ mutually independent ? I guess not, but I have no idea to prove it.

Answer 1

A simple counter-example can show dependence. For example:

$P(0\lt u_1\lt \frac{1}{2} \;\cap\; 0\lt u_2\lt \frac{1}{2} \;\cap\; 0\lt u_3\lt \frac{1}{2}) = 0$

but

$P(0\lt u_1\lt \frac{1}{2})P(0\lt u_2\lt \frac{1}{2})P(0\lt u_3\lt \frac{1}{2}) \gt 0$.