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Constraints the elements of the 3D-rotation matrix must satisfy

  1. Aug 4, 2014 #1
    1. The problem statement, all variables and given/known data
    Taken from "Introduction to Electrodynamics" by David J. Griffiths p.12 problem 1.8 (b):
    What constraints must the elements [itex]R_{ij}[/itex] of the three-dimensional rotation matrix satisfy, in order to preserve the length of vector A (for all vectors A)?


    2. Relevant equations
    The 3D rotation matrix around an arbitrary axis:
    [tex]
    \begin{pmatrix}
    \bar{A_x}\\
    \bar{A_y}\\
    \bar{A_z}\\
    \end{pmatrix} = \begin{pmatrix}
    R_{xx} & R_{xy} & R_{xz}\\
    R_{yx} & R_{yy} & R_{yz}\\
    R_{zx} & R_{zy} & R_{zz}\\
    \end{pmatrix} \begin{pmatrix}
    A_x\\
    A_y\\
    A_z\\
    \end{pmatrix}
    [/tex]
    Thus,
    [itex]\bar{A_i} = \sum_{j=1}^3 R_{ij}A_j[/itex]
    where the index 1 stands for x, 2 for y and 3 for z.

    3. The attempt at a solution
    Since the length of the vector before and after transformation must be equal:
    [itex]\sum_{i=1}^3 (\bar{A_i})^2 = \sum_{i=1}^3 (A_i)^2[/itex]
    [itex]\sum_{i=1}^3 (\bar{A_i})^2 = \sum_{i=1}^3 (\bar{A_i})(\bar{A_i}) = \sum_{i=1}^3 \left( \sum_{j=1}^3 R_{ij}A_j \right) \left( \sum_{k=1}^3 R_{ik}A_k \right) = \sum_{i=1}^3 (A_i)^2 [/itex]
    The one before the last equality I obtain from substituting the above equation for [itex]\bar{A_i}[/itex] .
    I'm not sure how to proceed from here. I guess that I lack understanding of the summation notation. I tried looking it up but I still can't understand how to continue.

    Any help will be greatly appreciated!
     
  2. jcsd
  3. Aug 4, 2014 #2

    Ray Vickson

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    Google "orthogonal matrix"
     
  4. Aug 4, 2014 #3

    vela

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    First thing you should do is learn Einstein's summation convention (Einstein's greatest contribution to physics): repeated indices imply summation. It'll save you a lot of tedious writing. Using this convention, you have
    $$\bar{A}_i \bar{A}_i = (R_{ij}A_j)(R_{ik}A_k).$$ No annoying sigmas to keep writing.

    The relationship has to hold for all vectors, so try using ##A = (1, 0, 0)##, for example, to show that the first column of ##R## has to be a unit vector. For other choices of ##A##, you can show that different columns are orthogonal to each other.
     
  5. Aug 5, 2014 #4
    Thanks, I didn't know about these matrices!

    Thank you! So if I understand correctly, I just pick 2 convenient vectors like ##\vec{A} = <1,0,0>## and ##\vec{B} = <1,1,0>## and generalize the results I get for all choices? Am I allowed to do it without proving these relations in general?

    EDIT: After some more readings I think I finally get it. Thanks again.
     
    Last edited: Aug 5, 2014
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