I am haunted by a question. Consider a vector $v=\begin{bmatrix} a\\ b \\ c \end{bmatrix}$ is firstly multiplied by $R_1= \begin{bmatrix} \cos(\theta_1) &-\sin(\theta_1)&0\\ \sin(\theta_1) &\cos(\theta_1)&0 \\ 0&0&1 \end{bmatrix}$ and secondly is rotated along rotated axe $x$ by $R_2= \begin{bmatrix} 1&0&0\\0&\cos(\theta_2) &-\sin(\theta_2)\\0& \sin(\theta_2) &\cos(\theta_2)\end{bmatrix}$ and finally is rotated along rotated axe $y$ by $R_3= \begin{bmatrix} \cos(\theta_3)&0 &-\sin(\theta_3)\\0&1&0\\ \sin(\theta_3)&0 &\cos(\theta_3)\end{bmatrix}$. At last, $v$ is transformed into new position $v'=\begin{bmatrix} a'\\ b' \\ c' \end{bmatrix}$. We could express the angles of rotation relative to the 3 axis of inertial reference by projecting the original vector and transformed vector to $XOY$,$ZOY$ and $XOZ$. But if we do not use this vector, imagining that at the origin of the inertial reference there is a small ball or just a point which experiences these three rotations one by one, how could we give or simplify an expression of angles of rotation relative to the three inertial reference axis which the ball or point experiences. I find out a solution too complex and looks like wrong.
Could you give me a hint or guide to treat this haunting (just for me) question. Thank you very much for taking a look.