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A Transformation Matrix question

  1. May 4, 2010 #1
    1. The problem statement, all variables and given/known data

    Find the transformation matrix 'R' that describes a rotation by 120 degrees about an axis from the origin through the point (1,1,1). The rotation is clockwise as you look down the axis toward the origin.

    2. Relevant equations

    \left( \begin{array}{c} A'_x \\ A'_y \\ A'_z \end{array} \right) =

    \left( \begin{array}{ccc}
    R_{xx} & R_{xy} & R_{xz} \\
    R_{yx} & R_{yy} & R_{yz} \\
    R_{zx} & R_{zy} & R_{zz}
    \end{array} \right)

    \left( \begin{array}{c} A_x \\ A_y \\ A_z \end{array} \right)


    3. The attempt at a solution

    I am just very confused by the wording of the question. I am used to talking about the transformation of a vector... I am not sure what is being transformed here... the coordinate system?

    Here is the solution:


    I am having a hard time deciphering the problem statement even looking at the solution. It is clear that he wanted us to "swap" axes. But I am not sure exactly what is happening here.
  2. jcsd
  3. May 4, 2010 #2


    User Avatar
    Homework Helper

    you can imagine it a few different ways, i would think of it as follows:
    - The global coordinate frame does not change.
    - The operation is a transformation of a vector within that frame (it maps each vector to a vector).

    so imagine the line from the origin to (1,1,1)

    Now start with the vector (1,0,0), this will be transformed to (0,0,1).

    The global co-ordinate frame doesn't change, but a vector on the x axis is mapped to a vector on the z axis.

    By considering the action on each of the basis vectors it shoudl be pretty startightforward to write down the matrix.
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