Composition of Two Rotations About the Same Point

The composition of two rotations about the same point is another rotation about that same point.

First, let us work in the plane. Suppose we have the following two rotations: $$R_1\text{: center P, directed angle $\theta_1$}$$ $$R_2\text{: center P, directed angle $\theta_2$}$$ Then we know the following about their composition: $$R_1\circ R_2=R_2\circ R_1\text{: center P, directed angle $\theta_1+\theta_2$}$$ The proof of this fact is a trivial application of the Angle Addition Postulate, which says the following: $$m\angle AOB+m\angle BOC=m\angle AOC$$ where all of the angles are directed.

Now, let us suppose that we are in a space of more than two dimensions. Now, things are not quite as nice.

The composition of two rotations about the same point is still a rotation about that point.

To prove this, suppose that the point $P$ is moved to the point $Q$ by the two rotations. Suppose that the center of the rotation is $O$. Consider the plane $PQO$. Clearly, $P$ and $Q$ lie on a circle centered at $O$, and the circle is in this plane. Thus, a single rotation in this plane, centered at $O$, can map $P$ to $Q$.

However, rotations are no longer necessarily commutative. Also, since the rotations may be in different planes, the angle measures do not necessarily add up.