Suppose we have a product $$\vec v=\left(\vec x^T \cdot R(\vec\varphi_1)\cdot R(\vec\varphi_2)\cdot ...\right)^T,$$ where $R(\vec\varphi_i)$ is a matrix of rotation by $3D$ angle $|\vec\varphi_i|$ around $\vec \varphi_i$.
Assuming that this product of rotation matrices converges, how can one, given $\vec\varphi_i\forall i$, find $\vec v$ or the resulting matrix of transformation?