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Commutative Matrices Associated with Vector Rotations

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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.

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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.

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Contributed by: D. Meliga, A. Ratti and S. Z. Lavagnino (June 2019)
Additional contribution by: L. Lavagnino
Open content licensed under CC BY-NC-SA

Details

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.

Reference

[1] D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed., Upper Saddle River, NJ: Pearson Prentice Hall, 2005.

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