Rotation Matrix

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:

So

v' = R_θ 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).