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Formula for determining if one vector is a multiple for two others

  1. May 13, 2012 #1
    1. The problem statement, all variables and given/known data

    Screenshot2012-05-13at111042AM.png
    Screenshot2012-05-13at111113AM.png


    3. The attempt at a solution

    Is there a formula for determining

    x = 3v1 - 2v2

    It looks like you just have to do trial and error. I tried Gaussian Elimination on the two vectors and got -1.5 and .5 so that didn't work.
     
  2. jcsd
  3. May 13, 2012 #2

    sharks

    User Avatar
    Gold Member

    Do you mean, how to find a1 and a2? Just substitute the given vectors and solve.
     
  4. May 13, 2012 #3
    Quite simply, by comparing the three components of the vector on each side of the equation, the following things must be true:
    [tex]x^1 = a_1 v^1_1 + a_2 v^1_2[/tex]
    [tex]x^2 = a_1 v^2_1 + a_2 v^2_2[/tex]
    [tex]x^3 = a_1 v^3_1 + a_2 v^3_2[/tex]
    where the superscript denotes the first, second or third element of the vector. This leads to the set of equations
    [tex]-1 = a_1 + 2 a_2[/tex]
    [tex]-3 = -a_1[/tex]
    [tex]4 = 2a_1 + a_2[/tex]
    From there it should be obvious how that solution was obtained.
     
  5. May 13, 2012 #4
    I mean how they found out that 3 times vector 1 - 2 times vector 2 = x = (-1,-3,4)
     
  6. May 13, 2012 #5
    Steely Dan, thanks, I got it.

    when i did gaussian elimination i wrote the vector horizontally rather than vertically, that's why I was wrong.
     
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