• 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.