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Homework Help: Orthogonal transformation problem

  1. Jun 21, 2010 #1
    1. The problem statement, all variables and given/known data

    Lets say I fix 3 mutually orthogonal unit vectors i, j and k. Consider an orthogonal transformation F of vectors defined by [tex] F(a_1i+ a_2j + a_3k)=a_1'i+a_2'k+a_3'k [/tex] where

    [tex] \left( \begin{array}{ccc} a_1 \\ a_2 \\ a_3\end{array}\right) = A\left( \begin{array}{ccc} a_1' \\ a_2' \\ a_3'\end{array}\right) [/tex] for a fixed orthogonal matrix A.

    How can I show that F(v).F(w)=v.w?

    I tried to do this by letting

    [tex] v = \left( \begin{array}{ccc} a_1 \\ a_2 \\ a_3\end{array}\right), w= \left( \begin{array}{ccc} a_1' \\ a_2' \\ a_3'\end{array}\right) [/tex]

    but the fact that I don't know A is holding me back from doing this, so I think there must be some other approach.

    Any help would be much appreciated
     
  2. jcsd
  3. Jun 21, 2010 #2

    gabbagabbahey

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    Re: Transformations

    I wouldn't use [tex]w=\begin{pmatrix} a_1' \\ a_2' \\ a_3'\end{pmatrix}[/tex] if I were you, since it is not a general vector, it is related to [tex]v=\begin{pmatrix} a_1 \\ a_2 \\ a_3\end{pmatrix}[/tex] by [itex]v=Aw[/itex]. Instead, you want to use a general vector like [tex]w=\begin{pmatrix} b_1 \\ b_2 \\ b_3\end{pmatrix}[/tex].

    What is [itex]v\cdot w=v^Tw[/itex] if [itex]v=Av'[/itex] and [tex]w=Aw'[/itex]? What is [itex]A^TA[/itex] for an orthogonal matrix?
     
  4. Jun 21, 2010 #3
    Re: Transformations

    Ok, so I'll name my vectors differently. However how did you get [itex]v\cdot w=v^Tw[/itex] ?

    Also, in reply to your question [itex]A^TA=I[/itex]
     
  5. Jun 21, 2010 #4

    lanedance

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    Re: Transformations

    [tex]v\cdot w=v^Tw[/tex]
    this is just a dot product written in matrix notation, the transpose comes about as to satisy matrix multiplication, you must multiply a 1xn matrix (row vector) with a nx1 matrix (column vetcor)
     
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