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Math Help - Plane in 3 dimensions (Please check my work)

  1. #1
    Bar0n janvdl's Avatar
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    Plane in 3 dimensions (Please check my work)

    Find scalar equations of the form ax + by + cz = 0 for the following plane in R^3:

    The plane through the points:

    • P(0;1;1)
    • Q(1;0;1)
    • R(1;1;0)

    {PQ}^{\rightarrow} = (1;-1;0)

    {QR}^{\rightarrow} = (0;1;-1)

    The perpendicular vector = {PQ}^{\rightarrow} \times {QR}^{\rightarrow}

    Then I just picked point Q to substitute into the equation:

    1(x -1) + 1(y - 0) + 1(z - 1) = 0

    Multiplying it out:

    x + y + z = 2

    And that should be the equation of the plane?
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  2. #2
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    Quote Originally Posted by janvdl View Post
    Find scalar equations of the form ax + by + cz = 0 for the following plane in R^3:

    The plane through the points:

    • P(0;1;1)
    • Q(1;0;1)
    • R(1;1;0)

    {PQ}^{\rightarrow} = (1;-1;0)

    {QR}^{\rightarrow} = (0;1;-1)

    The perpendicular vector = {PQ}^{\rightarrow} \times {QR}^{\rightarrow}

    Then I just picked point Q to substitute into the equation:

    1(x -1) + 1(y - 0) + 1(z - 1) = 0

    Multiplying it out:

    x + y + z = 2

    And that should be the equation of the plane?

    Correct.
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  3. #3
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    Opalg's Avatar
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    That's fine. Notice that (1) x+y+z = 2 is a linear equation, so it represents a plane; (2) the equation is obviously satisfied when (x,y,z) is P, Q or R. So that has to be the right answer!
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  4. #4
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    As Opalq said, it is easy to check it yourself.
    Do P: (0, 1, 1), Q: (1, 0, 1), and R: (1, 1, 0) all satisfy x+ y+ z= 2? If yes, that is the equation of the plane containing those three points and if not, it is not.

    (In fact, you could have found the equation just by noting that the three components of each of those points sums to 2.)
    Last edited by mr fantastic; January 3rd 2009 at 07:49 PM. Reason: Disabled the smilies
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  5. #5
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    Quote Originally Posted by HallsofIvy View Post
    As Opalq said, it is easy to check it yourself.
    Do P0, 1, 1), Q1, 0, 1), and R1, 1, 0) all satisfy x+ y+ z= 2? If yes, that is the equation of the plane containing those three points and if not, it is not.

    (In fact, you could have found the equation just by noting that the three components of each of those points sums to 2.)
    or : ( ...... The space makes a difference if you don't disable the smilies .... *ahem* or did I mean : )
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