Composition of two rotations in $\mathbb{R}^3$

It is an exercise in Artin's Algebra that "describe geometrically composition of two rotations (along axis through $0$) in $\mathbb{R}^3$".

I thought in the following linear algebra way: if $l,m$ are axis of rotations (passing through $0$) then we can think of $l$, $m$ as one dimensional subspaces. Since rotations stabilize $l$ and $m$ respectively, it follows that they stabilize $l^{\perp}$ and $m^{\perp}$, which are two-dimensional subspaces, so their intersection is $1$-dimensional, say a line $n$ passing through $0$.

Then composition of rotations along $l$ and $m$ is rotation along line $n$.

Is this the correct way? Otherwise, give me a hint.

• An orthogonal transformation in $\mathbb{R}^3$ is completely determined by its action on two orthogonal vectors. If we know both transformations fix the axis of rotation, then they'll be equal as soon as they agree on one orthogonal vector. So take a point in the plane given by the first reflection, then the second plane is determined by asking that the reflection of this point with respect to this second plane is the image under the rotation (i.e. the plane containing the axis and bisecting the angle formed by the point and its image under the rotation). Feb 8 '15 at 8:31