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Geometry proof

  1. Oct 4, 2005 #1
    Ok I know that isometries preserve distance and in order for a fn to be an isometry || f(u) - f(v) || = || u - v ||
    and in this question it asks to prove

    prove that if an isometry satisfies f(0) = 0 then we have
    f(u) x f(v) = +- f(u x v)
    and what property of f determines the choice of sign

    "x" is the cross product

    Now i know that this space must be R^3 because its the cross product
    and i know that f(0) = 0 because
    || f(v) - f(u) || = || f(0) - f(0) || = 0

    I just dont know how to connect this knowledge to the cross product.
    A push in the right direction would be awsome!! I just need a start.
    thank you very much
  2. jcsd
  3. Oct 4, 2005 #2


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    Homework Helper

    What do you mean that you "know f(0) = 0 because..."? You know f(0) = 0 because they tell you so, you didn't (can't) deduce that fact, since translation is an isometry for which that equation doesn't hold. Also, why in the world would you have:

    ||f(v) - f(u)|| = ||f(0) - f(0)||?

    That appears to come out of nowhere. Moreover, it appears to have nothing to do with f(0) = 0, although you claim to use it as your reason for justifying it. Here's what you know:

    ||f(u) - f(v)|| = ||u - v|| because f is an isometry
    f(0) = 0 because they tell you so
    "|| f(v) - f(u) || = || f(0) - f(0) ||" is false in general
  4. Oct 4, 2005 #3
    oh ok, i was just trying anything really.
    my biggest problem is using ||f(u) - f(v)|| = ||u - v||
    to show that f(u) x f(v) = +- f(u x v)
    i just dont know how to relate the two,
    I dont want you to give me the answer AKG, i would much rather understand what i am suppose to do then get something free.
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