• 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.