When discussing a rotation, there are two possible conventions:
- rotation of the object relative to fixed axes
- rotation of the axes
1. Rotation of the object relative to fixed axes
In R2 space, consider the matrix that rotates a given vector v0 by a counterclockwise angle θ in a fixed coordinate system as shown below.
Then the rotation matrix is:
v' = Rθ v0
2. Rotation of the Axes
On the other hand, consider the matrix that rotates the coordinate system through a counterclockwise angle θ.
The coordinates of the fixed vector v in the rotated coordinate system are now given by a rotation matrix which is the transpose of the fixed-axis matrix.
As can be seen in the above diagram, this is equivalent to rotating the vector by a counterclockwise angle of -θ relative to a fixed set of axes.
So the rotation matrix is:
This is the convention commonly used in textbooks such as Arfken (1985, p. 195).