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Homework Help: Vector Transformations

  1. Feb 20, 2006 #1
    Problem 1.10(a) of DJGriffiths asks: "How do the components of a vector transform under a translation of coordinates?"

    This is confusing me (not hard to do) since the translation is given, then isn't it just:

    x' = x + A

    where A = [tex]\left(\begin{array}{c}
    0 \\ -a \\ 0 \end{array}\right)[/tex]

    Problem 1.10(b): The same "... inversion ..." so that x' = -I x?

    Problem 1.10(c): "How does the cross-product of two vectors transform under inversion?"

    Once again, if A is a vector, then it transforms as always A' = RA.

    So how is it any different if the vector is generated by a cross-product or is made up by me? It's a vector! Unless the question is not asking about A', but rather about how does BxC transform??? ...how would I apply the ransformation to the actual cross-product? I mean, do I take RBxRC or R(BxC)??

    sorry if my questions are annoying or too frequent. but it's always the same thing - I read the chapter and have no problem following the theory. Then I get to the Problems section and suddenly it's like the questions have nothing to do with the chapter I just read! :frown:

    Last edited: Feb 20, 2006
  2. jcsd
  3. Feb 20, 2006 #2

    Tom Mattson

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    Staff Emeritus
    Science Advisor
    Gold Member

    No, vectors are invariant under translation. For instance if you and I are at rest relative to each other but standing at different locations, and one of us observes a car zipping by at 50 mph due east, then the other of us will agree with that velocity measurement.

    Just carry out the matrix multiplication and you'll have your answer.

    Strictly speaking the cross product of two vectors is not a vector: It's an axial vector or pseudovector.

    Yes, that's what they're asking. You would transform the vectors B and C first, then take their cross product.
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