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Math Help - Finding the equation of a Plane

  1. #1
    Super Member craig's Avatar
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    Finding the equation of a Plane

    Hi, another vector question I'm afraid

    I have the 4 lines:

    BC = \begin{pmatrix} -2 \\ -4 \\ 1 \end{pmatrix} + \lambda\begin{pmatrix} 3 \\ 6 \\ -3 \end{pmatrix}, AC = \begin{pmatrix} 1 \\ 2 \\ -2 \end{pmatrix} + \sigma\begin{pmatrix} -2 \\ 2 \\ -4 \end{pmatrix}.

    l2 = \begin{pmatrix} 3 \\ 0 \\ 2 \end{pmatrix} + \beta\begin{pmatrix} -2 \\ 2 \\ -4 \end{pmatrix}, AD = \begin{pmatrix} 0 \\ 0 \\ -1 \end{pmatrix} + \alpha\begin{pmatrix} -3 \\ 0 \\ -3 \end{pmatrix}.

    I need to show that all 4 lines lie in a single plane and fine the Cartesian Equation of the plane.

    The Cartesian equation of a plane is in the form Ax+By+Cz=d, so from this I can set up the following equations:

    -2A-4B+C=d

    3A+2C=d

    A+2B-2C=d

    From these I can get all A,B,C in terms of d.

    A=d, B=C=-d

    Putting these values back into the original Cartesian Equation I got:

    dx-dy-dz = d, which can be simplified to x-y-z=1.

    Is this correct so far or have I gone horribly wrong?

    Thanks
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  2. #2
    Super Member craig's Avatar
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    Over 20 views and none of you have any ideas

    Anyone got any comments at all?
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  3. #3
    MHF Contributor

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    You are correct so far.
    With each line there is a point and a direction vector
    Test each line with with respect to the plane x-y-z=1.
    Does the point satisify that equation?
    Is the direction vector perpendicular to <1,-1,-1>?

    If the answer is yes in all cases you are done.
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  4. #4
    Super Member craig's Avatar
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    Quote Originally Posted by Plato View Post
    You are correct so far.
    With each line there is a point and a direction vector
    Test each line with with respect to the plane x-y-z=1.
    Does the point satisify that equation?
    Is the direction vector perpendicular to <1,-1,-1>?

    If the answer is yes in all cases you are done.
    Yes and yes.

    I was pretty sure that I'd gone about it in the right way but just wanted to make sure.

    Thanks again for the reply.
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