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Homework Help: Transformation Matrix Problem

  1. May 7, 2007 #1
    1. The problem statement, all variables and given/known data

    Consider the quadrilateral (namely Q) in R^3 formed by the points
    (1, 0, 0), (2, 0, 0), (1, 1, 3), and (2, 1, 3).

    a) What should the coordinates be for the figure R we get by rotating Q counterclockwise in the x-y plane by 45 degrees, then dilating it by a factor of 3/2, then translating it along the vector (-2, 1, -1)?

    b) Find the matrix that transforms Q into R.

    3. The attempt at a solution

    Okay, so what I did for (a) was I used the matrix

    cosθ -sinθ 0
    sinθ cosθ 0
    0 0 1

    Then substituted 45 for θ.

    After this, multiplied the identity matrix for R^3 by 3/2 and then multiplied it by the matrix with 45 substituted for θ.

    Then T(x,y,z) = (-2+3sqrt(2)x/4-3sqrt(2)y/4,1+3sqrt(2)x/4+3sqrt(2)y/4,-1+3z/2).

    I substituted each of the quadrilateral points in for T(x, y, z) to come up with the four points and got:


    (3/(2√2) - 2, 3/(2√2) + 1, -1),
    (3/√2 - 2, 3/√2 + 1, -1),
    (-2, 3/√2 + 1, 3.5),
    (3/(2√2) - 2, 9/(2√2) + 1, 3.5)

    I was wondering if someone could show me how to find the matrix that transforms Q into R. It would be greatly appreciated!
     
  2. jcsd
  3. May 8, 2007 #2

    HallsofIvy

    User Avatar
    Science Advisor

    Translation in 3 dimensions cannot be written as a 3 by 3 matrix- it is a vector addition. However, using "projective coordinates", you can write any such transformation as a 4 by 4 matrix. Have you done anything with that?
     
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