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Inverse of inner product

  1. Jan 19, 2009 #1
    my question is:

    if a.b=c

    a=any vector
    b=any vector
    .=inner product
    c=resulting scalar

    is there a way to describe a=f(b,c)?

  2. jcsd
  3. Jan 19, 2009 #2


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    Staff Emeritus
    Science Advisor
    Gold Member

    No. a isn't completely determined by b and c, since you can replace a with a+d in your first equation, if d is orthogonal to b.
  4. Jan 19, 2009 #3

    D H

    Staff: Mentor

    No. Think of it this way: Suppose you find vectors a and d such that a.b = c and d.b=0 (i.e., d is orthogonal to b). Then for any scalar k, (a+k*d).b=c also. In other words, the solution to a.b=c is not unique.
  5. Jan 19, 2009 #4
    Ok thanks!

    but it did not solve my problem regarding vortex dynamics....

    I would like to use the the law of Biot and Savart to determine the velocity field induced by a vortex filament. I am trying to do this by rewriting the equation of the circulation to a function for the vorticity.

    To do this, I am using the Dirac Delta function for another integral for the circulation. This way i can equal the two integrands. But from that expression I would like to write the vorticity as a function of the rest (to substitute it in Biot ans Savart). But from your explanations I see that this is not possible??

    Attached Files:

  6. Jan 19, 2009 #5
    already solved it....

    w.n = the magnitude of the vector=length(w)
    dx/ds / length(dx/ds) = unity direction of the vector

    so the vector is w is determined to be (w.n)*(dx/ds / length(dx/ds))
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