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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.

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It's not correct since a rotation with one subspace preserved followed by another rotation with another subspace preserved does not result in the intersection being preserved. I'm not sure what is meant by "geometrically describe". We know that the only norm-preserving transformations are translations, rotations and reflections, and since norm-preserving transformations are closed under composition, we know that combining two rotations must be another norm-preserving transformation. Also we know that rotations preserve an orthogonal basis and its orientation (determinant), so combining them also does. In particular, we know that rotations are essentially orthogonal matrices with determinant one, so a combination is also. To find the axis of rotation is equivalent to finding an eigenvector.

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    $\begingroup$ I think the accepted answer here is a good answer to this question. $\endgroup$
    – Jose27
    Feb 8 '15 at 6:50
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    $\begingroup$ @Jose27: That's a very nice answer, but the same problem of how to actually find the axis in practice arises.. $\endgroup$
    – user21820
    Feb 8 '15 at 6:57
  • $\begingroup$ I agree, but as far as geometric intuition goes, I think the answer paints a fairly clear picture of what's going on, which is what I think Artin's asking for in that question. $\endgroup$
    – Jose27
    Feb 8 '15 at 7:54
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    $\begingroup$ @Jose27: Ah okay. By the way, what's the easiest way to justify the fact that a rotation can be written as two reflections given one of the reflections? $\endgroup$
    – user21820
    Feb 8 '15 at 8:18
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    $\begingroup$ 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). $\endgroup$
    – Jose27
    Feb 8 '15 at 8:31

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