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If we have three points like $A(x_1,y_1,z_1)$, $B(x_2,y_2,z_2)$ and $C(a,b,c)$. Then, $A$ and $B$ determines a line like $l$. After that, we rotate $C$ around $l$ by $\omega$ degree (anti-clockwise). How can be calculated new position of $C$ "$C'(a',b',c')$"?

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    $\begingroup$ How do you define "anti-clockwise" in 3 dimensions? $\endgroup$ – JimmyK4542 Jul 31 '14 at 11:03
  • $\begingroup$ yes I forgot to write it, it is right hand rule thumb point B to A $\endgroup$ – user161943 Jul 31 '14 at 11:15
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    $\begingroup$ Do you want to use only elementary analytic geometry? If not, see en.wikipedia.org/wiki/Quaternions_and_spatial_rotation $\endgroup$ – user35603 Jul 31 '14 at 11:15
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There are several ways to do it, it depends on what you want to do in practice. With linear algebra, this can be achieved using a simple matrix multiplication. You can define a unit vector from $A$ to $B$ to define a rotation axis, then apply the axis-angle rotation matrix on the vector of the point $C$.

This can also be done using quaternions.

See axis-angle representation for more information.

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It would be clockwise if the perspective was from a point on the line on the correct side of the perpendicular from C that intersects the line. Anticlockwise would be the perspective from the other side. As for how to do it - this is what quaternions are all about.

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