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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 $\theta_1,\theta_2,\theta_3$.
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  • $\begingroup$ What do you mean by "the resultant of $\theta_1,\theta_2,\theta_3$" ? $\endgroup$ – Emilio Novati Nov 11 '14 at 19:01
  • $\begingroup$ Means vector sum $\endgroup$ – Nirvana Nov 11 '14 at 21:57
  • $\begingroup$ But $\theta_1,\theta_2,\theta_3$ are not vectors. $\endgroup$ – Emilio Novati Nov 12 '14 at 15:57
  • $\begingroup$ They are angles represented by vectors..As explained in the question magnitude of this is the angle in radian $\endgroup$ – Nirvana Nov 13 '14 at 1:22
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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

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