Commutativity of Spatial Rotations

Question

I know that in general spatial rotations (rotations in $\Bbb R^3$) are not commutative. But what if we restricted our possible rotations to only those around orthogonal axes? For instance, what if we only allowed rotations around the $x$-axis, the $y$-axis, and the $z$-axis. Would those rotations still not commute?

Answer 1

These rotations definitely don't commute with each other. (This is one of the 'small' problems with Euler angles - one needs to define an ordering to apply them in.) Try a thought experiment: take a book and lay it down in front of you, with the cover up and the spine facing left. I'm going to use a coordinate system where positive Z is towards the ceiling, positive Y is away from you, and positive X is towards your right. Start by rotating it $90^\circ$ counterclockwise about the positive X axis; the cover will be facing you. Now, rotate it $90^\circ$ counterclockwise about the positive Y axis; the cover will still be facing you and the spine will be facing up.

Now, start again with your book on the table, cover up and spine facing left, but perform the two rotations in the other order: after rotating $90^\circ$ CCW about positive Y the spine of the book will be facing up and the cover will be facing right. Then rotating $90^\circ$ CCW about positive X will have the spine facing you and the cover facing right, clearly a different position than you ended up in when doing the rotations in the other order.

Answer 2

A rotation of an angle $2\theta$ in space, around an axis passing through the origin, is represented by a quaternion $e^{\mathbf{u}\theta}$, where $\mathbf{u}$ is the imaginary quaternion that correspond to the unit vector oriented along the axis of rotation. So we have the correspondence: $$ \vec{w}=R_{\mathbf{u},\theta} \; \vec{v} \quad \longleftrightarrow \quad \mathbf{w}= e^{\mathbf{u}\theta/2}\mathbf{v}e^{-\mathbf{u}\theta/2} $$

In this notation the question in OP can be reduced to the problem:

There exists $\alpha,\beta,\gamma$ such that $e^{\alpha \mathbf{i}}$, $e^{\beta \mathbf{j}}$ and $e^{\gamma \mathbf{k}}$ commute? (where $\mathbf{i},\mathbf{j},\mathbf{k}$ are three immaginary units).

We can prove that these exponentials commute iff at least two exponent coefficients are integer multiples of $\pi$, i.e. two rotations are in fact identites. You can see Quaternion exponential for the proof and for a more general discussion about commuting exponentials of quaternions.