
The components of a free vector change as the perspective (reference frame) changes. Figure 1 shows two different reference frames: the XY system and the X'Y' system. Vector v in Figure 1 can be expressed as (x, y) in the XY system, or (x', y') in the X'Y' system. The relationships between x & y and x' & y' can be obtained from the geometric relationships:
Expanding [1] to 3 dimensions:
[2] is the axis rotation matrix for a rotation about the Z axis. Applying the same method to the rotations about the X and the Y axis, respectively:
These matrices for the axis rotations about particular coordinate axes are essential in developing the concept of the Eulerian/Cardanian angles. See Eulerian Angles for the details. The rotation matrices fulfill the requirements of the transformation matrix. See Transformation Matrix for the details of the requirements. Axis Rotation vs. Vector Rotation Figure 2 shows a situation slightly different from that in Figure 1. This time, the vector rather than the axes was rotated about the Z axis by f. This is called the vector rotation. In other words, vector v_{1} was rotated to v_{2} by angle f.
Again, one can obtain the following relationships:
since
where r = length of the vector, and a = the angle v_{1} makes with the X axis. Expanding [5] to 3dimension:
Similarly,
and
From [2]  [4] and [7]  [9]:
Vector rotation is equivalent to the axis rotation in the opposite direction. One should not be confused by the axis rotation and the vector rotation. In vector transformation, the axis rotation matrices should be used instead of the vector rotation matrices because vector transformation means change in the perspective. 
© YoungHoo Kwon, 1998 