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How to find angle after two rotations

  1. Jan 23, 2013 #1
    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.

  2. jcsd
  3. Jan 24, 2013 #2


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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 y-axis, 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.
  4. Jan 24, 2013 #3


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