I was told in class that a rotation matrix is defined by a rotation angle and rotation axis, if we call the rotation axis $v$ and take a basis of $\mathbb{R}^{3}=\{v\}\bigoplus\{v\}^{\perp}$ then the matrix is similar by an orthogonal matrix to a matrix of the form $$\begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\\ & & 1 \end{pmatrix}$$
I asked my self the following question: If I rotate in the $xy$ plain (i.e. rotation axis is $z$) in angle $\theta$, and then rotate in the $yz$ plain (i.e. rotation axis is $x$) in angle $\varphi$ , what rotation matrix I get ?
I tried multiplying the corresponding matrices but that did not produce anything useful, I can't also thing of a vector $v\in\mathbb{R}^{3}$that is invariant under the composition...
What is the rotation axis, and the rotation angle of these two compositions ? Help is appreciated!
When composing two rotations, it is useful to know that a rotation about $\alpha$ about an axis $\ell$ can be written as the composition of two reflections in planes containing $\ell$, the first being chosen arbitrarily and the second being at an (oriented) angle $\frac\alpha2$ with respect to the first. Now in the composition of $4$ reflections you get, you can make your choices so that the second and third planes of reflection (the second reflection for the first rotation and the first reflection for the second rotation) are both equal to the unique plane passing through the two axes. Then poof!, those second and third reflections annihilate each other, and you are left with the composition of the first and the fourth reflection, which is a rotation with axis the intersection of those planes, and angle twice the angle between those planes.
If you want to calculate the axis and angle in terms of the original angles, formulas get a bit complicated (even for very easy choices of initial axes as in the question), but such is life, the concrete answer isn't really very easy to write down or remember.