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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

    [tex]
    \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)



    [/tex]

    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:

    Picture1-43.png

    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

    lanedance

    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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