I have 2 unit-length vectors in $\mathbb{R}^3$ : $d_1$ and $d_2$ which are distributed uniformly over the unit-sphere. What will be the distribution of the direction of cross product of these vectors ?
By intuition, it seems that the cross product should be uniformly distributed over a unit sphere too. I tried to prove this starting from the way $d_1$ and $d_2$ were produced (by sampling 3 values from a standard normal distribution and getting a unit-length vector from the vector of those 3 values), but could not reach anywhere.
I am looking for a formal proof, so if anyone can help directing with that, it would be great.