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Linear Algebra problem (Least Squares?)

  1. Jun 28, 2012 #1
    Linear Algebra problem (Least Squares? - Distance between lines)

    1. The problem statement, all variables and given/known data

    We have two points R = (x,x,x) and S = (y,3y,-1). All we know is that they are on lines somewhere in 3-space and that they don't cross. Need to find an x and y that minimize || R - S ||2

    2. Relevant equations

    ATAx = ATb

    3. The attempt at a solution

    I tried using the equation above, i.e. inverting (ATA) and multiplying both sides with that, but the resulting matrix that I got was a 2x1 matrix of zeros. This is definitely not the right answer. I also tried using (C+D(t)-b)2... for each coord and doing a partial derivative for C and D, but I ended up getting the same equation for both derivatives, which I am sure is not right.

    I am very confused and not sure where to go from here.
     
    Last edited: Jun 28, 2012
  2. jcsd
  3. Jun 29, 2012 #2
    Why don't you minimize it in the usual way, [tex] \nabla ||R-S||^2 = 0 [/tex] ?
     
  4. Jun 29, 2012 #3
    That is what I tried. At least that's what I think I tried. That was where the (C+D(t)-b)2... etc, was about in my previous post. (C+Dx - y)2 + (C+Dx - 3y)2 + (C+Dx + 1)2.

    But, since the t (x) values are all x's, they cancel with the two's after I do the partial derivative w/respect to D, and both derivatives end up the same. Is there something I'm missing, or am I doing something wrong?
     
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