Rotation Matrix
When discussing a Rotation, there are two possible conventions: rotation of the axes and rotation of the objectrelative to fixed axes.
In 
, let a curve be rotated by a clockwise Angle 
, so that the original axes of the curve are
and 
, and the new axes of the curve are 
and 
. The Matrix transforming the original curve to the rotated curve, referred to the original and axes, is
 | (1) |
 | (2) |
On the other hand, let the axes with respect to which a curve is measured be rotated by a clockwise Angle, so that the original axes are 
and 
, and the new axes are and . Then the Matrix transforming the coordinates of the curve with respect to and isgiven by the Matrix Transpose of the above matrix:
 | (3) |
 | (4) |
In 
, rotations of the 
-, 
-, and 
-axes give the matrices
 |  |  | (5) |
 | |  | (6) |
 | |  | (7) |
See also Euler Angles, Euler's Rotation Theorem, Rotation