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Math Help - Vector valued functions : Determine a formula for ...

  1. #1
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    Vector valued functions : Determine a formula for ...

    Let r(t) be a v.v.f -with the first and second derivatives r' and r''. Determine formula for
    d/dt [r.(r' x r'')] -in terms of r:

    How do we approach this one?

    maybe:
    d/dt [r.(r' x r'')] = d/dt [r.r' x r.r''] = ((r'.r' + r.r'') x r'.r'') + (r.r' x (r'.r'' + r.r''')) ??
    ...and then what? I probably looking at this completely wrong -- there must be some simple vector calculus identities that make this easy or something.

    Is anyone able to give me a starting point -- or starting direction? -- thanks heaps
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  2. #2
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    We must assume that r has a third derivative.
    \frac{d}{{dt}}\left[ {r \cdot \left( {r' \times r''} \right)} \right] = r' \cdot \left( {r' \times r''} \right) + r \cdot \left[ {\left( {r'' \times r''} \right) + \left( {r' \times r'''} \right)} \right].
    But both r' \cdot \left( {r' \times r''} \right) = 0\quad \& \quad \left( {r'' \times r''} \right) = 0. Do you see why?
    So the final answer would be what?
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  3. #3
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    Thumbs up Ahh, ...thanks heaps

    r'' x r'' = 0 since they are parallel (the same!), and,

    r'.(r' x r '') = 0, since r' x r'' is orthogonal to r' (and r'') ...

    so the final answer is r.(r' x r''')
    Last edited by WalkingInMud; April 22nd 2008 at 08:24 AM.
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