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Transformation matrix problem

  1. May 24, 2009 #1
    1. The problem statement, all variables and given/known data

    7. (a) A transformation, T1 of three dimensional space is given by r'=Mr, where

    [itex]r=\left(
    \begin{array}{c}
    x \\
    y \\
    z
    \end{array}
    \right)[/itex]

    [itex]r'=\left(
    \begin{array}{c}
    x' \\
    y' \\
    z'
    \end{array}
    \right)[/itex]

    and

    [itex]
    M=\left(
    \begin{array}{ccc}
    1 & 0 & 0 \\
    0 & 0 & -1 \\
    0 & 1 & 0
    \end{array}
    \right)[/itex]

    Describe the transformation geometrically.

    (b)

    Two other transformations are defined as follows: T2 is a reflection in the x-y plane, and 3 is a rotation through 180 degrees about the line x=0, y+z=0. By considering the image under each transformation of the points with position vectors, i,j,k or otherwise find a matrix for each T2/

    (c) Determine the matrixes for the combined transformations of T3T1 amd T1T3 amd describe each of these tranformations geometrically.


    2. Relevant information

    [itex]\left(
    \begin{array}{ccc}
    1 & 0 & 0 \\
    0 & \text{cos$\theta $} & -\text{sin$\theta $} \\
    0 & \text{sin$\theta $} & \text{cos$\theta $}
    \end{array}
    \right),\left(
    \begin{array}{ccc}
    \text{cos$\theta $} & 0 & \text{sin$\theta $} \\
    0 & 1 & 0 \\
    -\text{sin$\theta $} & 0 & \text{cos$\theta $}
    \end{array}
    \right),\left(
    \begin{array}{ccc}
    \text{cos$\theta $} & -\text{sin$\theta $} & 0 \\
    \text{sin$\theta $} & \text{cos$\theta $} & 0 \\
    0 & 0 & 1
    \end{array}
    \right).[/itex] represent rotations of theta degrees about the x-,y- and z-axes.

    3. Attempt
    [itex]T=\left(
    \begin{array}{ccc}
    1 & 0 & 0 \\
    0 & \text{cos$\theta $} & -\text{sin$\theta $} \\
    0 & \text{sin$\theta $} & \text{cos$\theta $}
    \end{array}
    \right)[/itex]

    Rotation about the x-axis 90 degrees.

    (b)

    [itex]T_2:{x,y,z} \to {x,-y,z} [/itex]

    [itex] T_2 =\left(
    \begin{array}{ccc}
    1 & 0 & 0 \\
    0 & -1 & 0 \\
    0 & 0 & 1
    \end{array}
    \right)[/itex]


    (b)
    I am stuck here on how to do a rotation about the line x=0, y+z=0. Does this imply it is about the 3D line x=y+z.

    (c) This will be simple once I have done the other part.
     
  2. jcsd
  3. May 24, 2009 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi Gregg! :smile:

    (very nice LaTeX, btw! :wink:)

    your (a) is right.

    your (b) is a reflection, but about the wrong axis

    for (b) part2, the line x=0, y+z=0 is in the y-z plane (so not x = y + x) :wink:
     
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