RandomRotationMatri​x

Generate a random 3 x 3 rotation matrix in SO(3).

Note that each column of a 3 x 3 rotation matrix can be thought of as the x, y and z-axis of some reference frame placed with some orientation.

Using this fact, we do the following:

1. We set the x-axis to be a random 3D unit vector.
2. Then, we generate the y-axis by finding a unit vector that is perpendicular to the x-axis. This is done by generating another random 3D vector and then computing the cross product with the x-axis.
3. Finally, the z-axis must be the cross product of the x-axis and the y-axis.

Puting these three axis in one matrix side by side, we get a 3 x 3 rotation matrix.