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Finding the solution of three planes

  1. Mar 9, 2012 #1
    1. The problem statement, all variables and given/known data
    given the planes with equations:
    x + y + 7z = -7
    2x + 3y + 17z = -16
    x + 2y + (a^2 + 1) z = 3a

    find values for the constant a for which:
    -there are no solutions
    -the planes meet in a line. in this case find the parametric equation of the line
    -meet at a point. then find the parametric equation of the point.

    2. Relevant equations

    gaussian elimination? I dont think there are any. =S

    3. The attempt at a solution

    I put the planes into an augmented matrix, and then into echelon form, and got (I think its correct) the general solution, which is attached. From the I can read off the equation of the line, right? The parts I'm really stuck on are parts one and three. I realise that for the planes to meet in a point, there has to be 3 pivots but beyond that I'm a bit stumped.

    Well actually, I have a bit of a clue for part one, that a =/= 3 because then 0z = 18 which is inconsistent. is that right? But for part 3, I'm not sure how to do it, really.
     

    Attached Files:

    Last edited: Mar 9, 2012
  2. jcsd
  3. Mar 9, 2012 #2

    Mark44

    Staff: Mentor

    You have a mistake. In your first augmented matrix, the bottom row is 0 0 a2 - 9 | 3a + 9

    Right after that, you say let a = -3i. There are two real values that make a2 - 9 equal to 0.

    For your other questions, there will be no solutions if the last equation is 0z = k, with k not equal to 0.
    There will be multiple solutions (points along a line) if the last equation is 0z = 0.
    Finally, there will be a unique solution (a single point) if the last equation is kz = <whatever>, with k not equal to 0.
     
  4. Mar 9, 2012 #3
    Ooops. I guess this is what happens when I try do maths at obscene times of day. Thankyou, i worked it out now!
     
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