Rotation about an axis by matrix multiplication

Question

Suppose I have three axis of rotation vectors $\vec{v_1},\vec{v_2},\vec{v_3}$ and angle of rotation as vectors $\theta_1,\theta_2,\theta_3$.

1. Take a vector $P$ then apply rotation around $\vec{v_1}$ with rotation angle $|\theta_1|$
2. From new orientation of $P$ apply rotation around $\vec{v_2}$ with rotation angle $|\theta_2|$
3. From new orientation of $P$ apply rotation around $\vec{v_3}$ with rotation angle $|\theta_3|$
4. Now you get final orientation of $P$

Question

1. Shall we get the same orientation of P if rotate around resultant of $\vec{v_1},\vec{v_2},\vec{v_3}$ let us call V with an angle which is equal to the magnitude of the resultant of $\theta_1,\theta_2,\theta_3$.

Answer 1

No (it is called “non-commutativity”).

First hint: rotate 90° around z, then 90° around x, then −90° around z.

When the first exercise was complete, try to rotate −90° around z, then 90° around x, then 90° around z ☺