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Planar Intersections (Answer Check)

  1. May 13, 2010 #1
    1. The problem statement, all variables and given/known data


    Solve the following systems and interpret the result geometrically
    x - y - 2z - 3 = 0
    2x - 3y - 3z + 15 = 0
    x - 2y - z + 10 = 0




    2. Relevant equations



    3. The attempt at a solution

    x - y - 2z - 3 = 0…………….(1)
    2x - 3y - 3z + 15 = 0……….(2)
    x - 2y - z + 10 = 0…………..(3)

    first multiply equation (1) by -2
    getting:

    2x-2y-4z-6=0


    Use elimination:

    2x-2y-4z-6=0
    -(2x - 3y - 3z + 15 = 0)
    y-z-21=0

    y-z=21

    Elimination:

    x-y-2z-3=0
    -(x - 2y - z + 10 = 0)

    y-z-13=0

    ==> y-z=13

    Use elimination:

    y-z=+21
    y-z=13

    Use elimination

    (y-z=21)
    -(y-z=13)
    0=8


    The answer is 0=number..This means that the system is inconsistent, and the planes never intersect.

    I would really appreciate it if someone could take a look over my working, and point out any mistakes.

    Thanks! :smile:
     
  2. jcsd
  3. May 13, 2010 #2
    Your working looks right to me, so the system is inconsistent like you said. However, be careful. In this case every pair of planes does intersect in a line. You can see this because parallel planes have proportionate coefficients for each independent variable and different constant terms. What is the actual (more precise) geometric interpretation?
     
  4. May 13, 2010 #3
    Thanks Tedjn

    So, there is an intersecting line? because I'm really confused; isn't 0=8 a false statement, meaning that the planes are neither parallel, nor they intersect. Is this an example of planes intersecting in pairs? could you please elaborate a little.

    Thanks! :smile:
     
  5. May 13, 2010 #4
    Yes, the planes do intersect in pairs. Most books have a picture of this occurring but where the three planes do not intersect together at any point or line, so that the system has no (simultaneous) solution.
     
  6. May 13, 2010 #5
    Thank you.

    Just one more question: when it says to interpret the result geometrically, do I have to graph it? or is it just stating the facts that we discussed above?
     
  7. May 13, 2010 #6
    I believe just explaining the facts would be enough. If you are artistic, you might draw a simple picture illustrating how such pairwise intersections might look, but nothing accurate is probably required.
     
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