Commutative Matrices Associated with Vector Rotations
Rotations of vectors in three dimensions are, in general, not commutative. The one exception occurs when the two rotations occur in the same plane. In such cases, the commutative law for multiplication, , is valid.
Select the "3D graphics" button to see a graphic representation of the rotations; the "matrices" button shows the corresponding mathematical representation. The red dashed arrow marks the initial position and the brown arrow marks the position after the first rotation. The black arrow marks the position after the second rotation.
Snapshot 1: The box on the right shows a rotation of following a rotation of , while the box on the left shows an initial rotation of and then a rotation. Since the rotations are in the same plane, the final result is the same.
Snapshot 2: The box on the right shows a rotation of following a rotation of , while the box on the left shows an initial rotation of and then a rotation. Again, the final result is the same.
Snapshot 3: The rotation is written in matrix form; in this case, the matrix multiplication is commutative.
 D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed., Upper Saddle River, NJ: Pearson Prentice Hall, 2005.