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Cross product derivative

  1. Feb 5, 2007 #1
    hi all.

    my homework question is what is the derivative of:

    [(a + t * b) x (a + t * b + t^(2) * c)]

    a, b, and c are vectors, and t is a constant. * is multipy, ^(2) is squared, and x is cross product.

    i've been working on it for hours and i have no idea what to do.

    there's another similar problem which i have the answer to. it asks to find the derivative of:

    [(a + t * b) x (c + t * d)]

    it has the same specs, with d being another vector.

    the answer to this one is:

    (a x d) + (b x c) + 2t(b x d)

    i just don't see how these connect and can actually be equivalent.

    help would be ridiculously appreciated.
  2. jcsd
  3. Feb 5, 2007 #2


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    You have a (cross) product there. So you should use the (cross) product rule.
  4. Feb 6, 2007 #3


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    First, the product rule, for cross product: (uxv)'= uxv'+ u'xv.

    Second, cross product is anti-commutative: uxv= -vxu and, in particular, uxu= 0.
    Finally, cross product is distributive: ux(v+ w)= uxv+ uxw (though is it NOT associative).

    Look at your second, simpler, problem: [(a + t * b) x (c + t * d)] '
    = (a+ tb)' x(c+ td)+ (a+tb)x(c+ td)'= bx(c+td)+ (a+tb)x(d)= bxc+ t(bxd)+ axd+ t(bxd)= axd+ bxc+ 2t(bxd).
    Last edited: Feb 6, 2007
  5. Feb 6, 2007 #4
    ok thank you so much you two.
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