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How to find angle after two rotations 
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#1
Jan2313, 06:29 PM

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I have coordinate system A with bases a, b, c.
Say I rotate the whole system 30 degrees, so that the angle between a and a' is 30 degrees. Then I make another rotation so that this plane of rotation is perpendicular to that of the old one. What is the angle between a and a' now? I am trying to find the angles to use in a tensor transformation law, but I am having problems understanding what the angles will be between the old and new axes when a transformation isn't just a single rotation in one plane of the system. Tia 


#2
Jan2413, 06:55 AM

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You can write any rotation as a matrix multiplication. Then two rotations is given by the product of the two matrices.
For example, if you wrote 30 degrees around the z axis, the rotation is given by [tex]\begin{bmatrix} cos(30) & sin(30) & 0 \\ sin(30) & cos(30) & 0 \\ 0 & 0 & 1\end{bmatrix}= \begin{bmatrix}\frac{\sqrt{3}}{2} & \frac{1}{2} & 0\\ \frac{1}{2} & \frac{\sqrt{3}}{2}& 0 \\ 0 & 0 & 1\end{bmatrix}[/tex] A rotation around the yaxis, through 30 degrees is given by [tex]\begin{bmatrix} cos(30) & 0 &sin(30)\\ 0 & 1 & 0 \\ sin(30) & 0 & cos(30) \end{bmatrix}= \begin{bmatrix}\frac{\sqrt{3}}{2} & 0 & \frac{1}{2} \\ 0 & 1 & 0 \\ \frac{1}{2} & 0 & \frac{\sqrt{3}}{2}\end{bmatrix}[/tex] The two rotations together would be given by the product of the two matrices. 


#3
Jan2413, 06:58 AM

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