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Image of a Linear Transformation

  1. Mar 24, 2014 #1
    T2 projects orthogonally onto the xz-plane

    T3 rotates clockwise through an angle of 3π/4 radians about the x axis

    The point (-3, -4, -3) is first mapped by T2 and then T3. what are the coordinates of the resulting point?

    this question is on a program call Calmaeth. My answer for this question is (-3,0,-√2/2). The program says its wrong but i have checked thoroughly many times and cannot find my mistake.

    My transformation matrix for T2 is ##
    \begin{pmatrix}
    1 & 0 & 0 \\
    0 & 0 & 0 \\
    0 & 0 & 1
    \end{pmatrix}
    ## and for T3 is ##
    \begin{pmatrix}
    1 & 0 & 0 \\
    0 & \frac{-1}{√2} & \frac{1}{√2} \\
    0 & \frac{-1}{√2} & \frac{-1}{√2}
    \end{pmatrix}
    ##
    To get the resulting standard matrix, i did T3*T2 and then multiplied this matrix by the point (-3, -4, -3) to get the resulting point.

    Can anyone see where i went wrong if i did? (also to let you know, the program said my matrices for T2&T3 were correct)
     
  2. jcsd
  3. Mar 24, 2014 #2

    jbunniii

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    Your method is fine. Assuming your matrices are right, it looks like you made an error with the matrix multiplication. What did you get for ##T3 * T2##?
     
  4. Mar 24, 2014 #3

    HallsofIvy

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    I agree with jbuniii. The "-3" and "0" are correct but I do not get [itex]-\sqrt{2}/2[/itex] as the third component when I multiply your matrices.

    (As a check, rather than multiplying T3*T2 first and then multiplying that by the vector, you can multiply the vector by T2 and then multiply the result by T3.)
     
    Last edited: Mar 24, 2014
  5. Mar 24, 2014 #4
    How is the "0" correct?
     
  6. Mar 24, 2014 #5

    micromass

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    Yes, only the ##-3## is correct.

    Anyway, please post this in the homework forum next time! Thanks :smile:
     
  7. Mar 25, 2014 #6
    Hi guys It was a rookie mistake on my part. I was doing [-3,-4,-3]*[standard matrix] . when i should have been doing [standard matrix]*##\begin{pmatrix}
    -3 \\
    -4 \\
    -3
    \end{pmatrix}##
    And sorry, in the future ill post in the homework section.
     
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