Suppose a position vector(adsbygoogle = window.adsbygoogle || []).push({}); vis rotated anticlockwise at an angle ##\theta## about an arbitrary axis pointing in the direction of a position vectorp, what is the rotation matrixRsuch thatRvgives the position vector after the rotation?

Supposep= ##\begin{pmatrix}1\\1\\1\end{pmatrix}## and ##\theta## = -120##^\circ##.

My approach is as follows.

First, perform transformation ##T_1##: rotate the xy plane 45##^\circ## anticlockwise about the z axis. (I treat this as keepingvfixed but expressingvin terms of the new coordinate system CS1.)

Next, perform transformation ##T_2##: rotate the xz plane ##\tan^{-1}\frac{1}{\sqrt2}## clockwise about the y axis (clockwise when the y axis points into your eye) so that the x axis is now pointing in the direction ofp. (Again, I treat this as keepingvfixed but expressingvin terms of the new coordinate system CS2.)

Then, perform transformation ##T_3##: rotate the yz plane 120##^\circ## anticlockwise about the x axis, since rotatingvby -120##^\circ## has the same effect as rotating the yz plane 120##^\circ## (turning the coordinate system about the rotation axis by 120##^\circ## while keepingvfixed).

We have ##T_1=\begin{pmatrix}\frac{1}{\sqrt2}&\frac{1}{\sqrt2}&0\\-\frac{1}{\sqrt2}&\frac{1}{\sqrt2}&0\\0&0&1\end{pmatrix}## , ##T_2=\begin{pmatrix}\frac{\sqrt2}{\sqrt3}&0&-\frac{1}{\sqrt3}\\0&1&0\\\frac{1}{\sqrt3}&0&\frac{\sqrt2}{\sqrt3}\end{pmatrix}## , ##T_3=\begin{pmatrix}1&0&0\\0&-\frac{1}{2}&\frac{\sqrt3}{2}\\0&-\frac{\sqrt3}{2}&-\frac{1}{2}\end{pmatrix}##

Next, perform the inverse of transformation ##T_2##. (Changing the expression ofvfrom coordinate system CS2 back to CS1.)

Lastly, perform the inverse of transformation ##T_1##. (Changing the expression ofvfrom coordinate system CS1 back to CS0, the original coordinate system.)

Thus,R= ##T_1^{-1}T_2^{-1}T_3T_2T_1 = \begin{pmatrix}0&0&-1\\1&0&0\\0&-1&0\end{pmatrix}##

ButRshould be ##\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}## since rotatingvaboutpby -120##^\circ## has the effect of turning the x axis to the y axis, and the y axis to the z axis, and the z axis to the x axis (turning the coordinate system about the rotation axispby 120##^\circ## while keepingvfixed).

Why do I get different answers? What's wrong with my approach?

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# Rotation matrix about an arbitrary axis

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