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Cross product proof

  1. Dec 16, 2007 #1
    1. The problem statement, all variables and given/known data
    We know that T(t) = r'(t)/||r'(t)||, or equivalently, r'(t) = ||r'(t)||·T(t). Differentiate this equation to find r''(t), then show that

    r'(t) x r''(t) = ||r'(t)||^2 · (T(t) x T'(t)).


    2. Relevant equations
    r''(t) = ||r'(t)||'·T(t) + T'(t)·||r'(t)||


    3. The attempt at a solution
    I was able to get as far the differentiation above but from there I got stuck. I attempted to cross both sides, which i'm pretty sure needs to be done, but I'm confused how to get the right side of the equation to simplify to the desired form. Any help would be greatly appreciated...Thanks a lot
     
  2. jcsd
  3. Dec 17, 2007 #2

    Defennder

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    Actually I don't see any puzzle here. Are you sure this is what the question is asking for? All you have to do is to differentiate r'(t)=|r'(t)|T(t) to get r''(t)=|r'(t)|T'(t). Then simply do the cross product for r'(t) and r''(t) from the above 2 expressions. You should get the answer as shown.
     
  4. Dec 17, 2007 #3
    Use the fact that the cross product of a vector with itself is zero.

    Assaf
    Physically Incorrect
     
  5. Dec 17, 2007 #4

    HallsofIvy

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    No. r''(t) is not |r'(t)|T'(t). Since |r'(t)| is a function of t itself you have to use the product rule.
     
  6. Dec 17, 2007 #5

    Defennder

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    Oh, yeah. I kept thinking |r'(t)| could be treated as a constant.
     
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