• Intuitively, it
makes sense that 3D rotations can be expressed as 3
separate rotations about fixed axes. Rotations have 3 degrees of freedom; two describe an axis of rotation, and one the amount.
• Rotations
preserve the length of a vector, and the angle between two vectors.
Therefore, (1,0,0), (0,1,0), (0,0,1) must be orthonormal after rotation. After rotation, they are the three columns of R. So these columns must be orthonormal
vectors for R to be a rotation.
Similarly, if they are orthonormal
vectors (with determinant 1) R will have the
effect of rotating (1,0,0), (0,1,0), (0,0,1).
Same reasoning as 2D tells us
all other points rotate too.
• Note if R has
determinant -1, then R is a rotation plus a
reflection.