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Introduction to Vector calculations (Calculus 3)

  1. Jan 21, 2012 #1
    1. The problem statement, all variables and given/known data
    Find nonzero scalars a, b, c, such that au+b(u-v)+c(u+v)=0 for every pair of vectors u and v

    This isn't a homework question, more of a conceptual exercise, but I want to understand it thoroughly.





    3. The attempt at a solution
    I've gone to u(a+b+c) + v(c-b)=0
    then I couldn't quite figure where to go next. There is so many unknowns at once it's a little disorienting where to start first.

    Then I figured to try splitting it into the vector pairs,
    a(u1,u2)+b(u1-v1, u2-v2)+c(u1+v1, u2+v2)=0

    but I am still stumped as to where to go from here. It seems like it's painfully simple but I'm not quite seeing it.

    EDIT1: I've attempted putting the vectors in a system of equations;

    u1a+u1b+u1c-v1b+v1c=0
    u2a+u2b+u2c-v2b+v2c=0

    but once again I hit a dead end and only get the scalars equaling zero.
     
    Last edited: Jan 21, 2012
  2. jcsd
  3. Jan 21, 2012 #2

    SammyS

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    au+b(u-v)+c(u+v)=0 has to be true for any pair of vectors u & v.

    u(a+b+c) + v(c-b)=0 is equivalent to au+b(u-v)+c(u+v)=0. So you can't count on anything from u(a+b+c) cancelling anything from v(c-b). Therefore, each of those has to be zero:
    u(a+b+c) = 0

    v(c-b) = 0​
    The solution for a, b, and c, isn't unique, but you can find a solution that does the job.
     
  4. Jan 21, 2012 #3
    So what I get is

    a=-2c
    and
    b=c

    the actual numbers inside will be arbitrary? just as long as a=-2c and b=c above are true?
     
  5. Jan 21, 2012 #4

    SammyS

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    That looks good.
     
  6. Jan 21, 2012 #5
    Thanks, I think I just looked deeper than what the question actually pertained to. I need to learn to drop expectations I suppose. :P

    Thanks again!
     
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