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Matrix intersection of planes

  1. Jan 4, 2007 #1
    1. The problem statement, all variables and given/known data


    Find a necessary condition for the three planes given below to have a line of intersection.

    -x +ay+bz=0
    ax-y+cz=0
    bx+cy-z=0


    2. Relevant equations

    in order to get a line of intersection between the planes..i know i need one line of the matrix to be [0 0 0|0]


    3. The attempt at a solution

    well heres the attempt..and its wrong

    [ -1 a b | 0
    a -1 c | 0
    b c -1| 0 ]

    =>

    [-1 a b | 0
    0 (a^2-1) ba+c | 0 (aRow1 + Row2)
    0 (ab+c) b^2+1 | 0 ] (brow1 + Row 2)


    =>

    [ -1 a b | 0
    0 a^2 -1 ba+c |0
    0 0 2abc +c^2 - a^2 + b^2 +1) |0 ] (ab+c row2- a^2-1 Row1)


    then what i did ..by inspection i made 2abc+c^2 -a^2 +b^2 +1 = 0 by letting a=b=1, and c=-1......

    but that doesnt work becasue that owuld make plane 1 and 2 the same plane.

    i need help

    thanks
     
    Last edited: Jan 4, 2007
  2. jcsd
  3. Jan 4, 2007 #2

    AKG

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    The third row in your original matrix should be "b -c -1 | 0" not "b c -1 | 0".
     
  4. Jan 4, 2007 #3
    my bad..edited...i mistyped the question

    but still need help
     
  5. Jan 4, 2007 #4

    AKG

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    In your last matrix, the 3rd element of the third row is "2abc +c2 - a2 + b2 +1" but then you start looking at the equation "2ab + c2 - a2 + b2 +1 = 0".
     
  6. Jan 4, 2007 #5
    another typo on my part i have that c there
     
  7. Jan 4, 2007 #6

    Hurkyl

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    So? You weren't asked to find a sufficient condition, you were asked to find a necessary condition.

    Incidentally, you have either the polynomial wrong, or the matrix wrong: I think determinants are a simpler approach to the problem than Gaussian elimination.
     
    Last edited: Jan 4, 2007
  8. Jan 5, 2007 #7
    Im not sure how to do it the dertiminant way. I do not think my math is wrong so far.

    Help
     
  9. Jan 5, 2007 #8

    AKG

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    "a=b=1, c=-1" is a sufficient condition, not a necessary condition. In fact, "2abc + c2 - a2 + b2 +1 = 0" is also just a sufficient condition, not a necessary condition, since it isn't necessary for the third line to be all zeroes (the second line could be all zeroes).
     
  10. Jan 5, 2007 #9

    ok thanks

    what would be an example as a necessary conditon and how would i go about finding it
     
  11. Jan 6, 2007 #10

    HallsofIvy

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    If the matrix of coefficients were invertible then the only simultaneous solution to the three equations would be (0, 0, 0), the POINT of intersection of the three planes. In order that the three planes intersect in a line it is necessary that the matrix not be invertible: in other words that the determinant be 0. Find the determinant and set it equal to 0.
     
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