I have the following problem that I can only partially solve: We are in $\mathbb{R}^3$. Let $v_1,v_2$ be two normed (of length 1) and linearly independent vectors. Let $R_i,\ i\in \{1,2\}$ be the two reflections on $\langle v_i\rangle ^\perp$ i.e. $S|_{\langle v_i\rangle}=-\text{id},\ S|_{\langle v_i\rangle^\perp}=\text{id}$.
Show the following:
$S_1S_2$ is a rotation
The rotation axis is $\langle v_1,v_2\rangle^\perp$
The rotation angle is $\cos\frac{\alpha}{2}=\pm (v_1,v_2)$ where $(v_1,v_2)$ denotes the inner product of these vectors.
My attempt: I am fairly certain that I have gotten the first two correct: As each reflection is orthogonal and $O(3)$ is a group, their product $S_1S_2\in O(3)$. Furthermore, the determinant of a reflection is -1 and, thus $S_1S_2\in SO(3)$ is a rotation.
I have also been able to show the rotation axis: As, for something to be the rotation axis, it has to be the eigenvector of the eigenvalue 1. That is $(S_1S_2-id)(w)= S_1S_2(w)-id(w)=w-w=0$ for any $w\in \langle v_1,v_2\rangle^\perp$. Implying that $\langle w\rangle=\langle v_1,v_2\rangle^\perp$ is indeed the rotation axis.
It is the third part I am struggling on. I should perhaps mention that I know that the rotation angle $\alpha$ of any rotational matrix is equal to $\frac{1}{2}\cdot(tr(A)-1)=\cos\alpha$. However, I am not able to progress from here.
Any help is much appreciated!