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Math Help - Intersection of two planes

  1. #1
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    Intersection of two planes

    I'm given the planes x + y + z = 18 and 4x + 3y - z = -3. With these planes I am asked to find a parametric equation of the line in which these two planes intersect.

    Then, I am asked to find the distance between this line and the point (-5, 10, 13).

    How should I go about this problem?
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  2. #2
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    Quote Originally Posted by Ares_D1 View Post
    I'm given the planes x + y + z = 18 and 4x + 3y - z = -3. With these planes I am asked to find a parametric equation of the line in which these two planes intersect.

    Then, I am asked to find the distance between this line and the point (-5, 10, 13).

    How should I go about this problem?
    Hi

    You have several ways to find a parametric equation of the line of intersection

    You can find 2 points on the line
    Take for instance x=3, solve the system
    y + z = 15
    3y - z = -15

    Then take x=-1, solve the system
    y + z = 19
    3y - z = 1

    These 2 points define one direction vector of the line
    Since you already have 1 point of the line, you can get a parametric equation

    Second method, solve the system
    x + y + z = 18
    4x + 3y - z = -3
    using z (for instance) as parameter

    You will find x, y (and z) function of z (parameter)
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  3. #3
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    Quote Originally Posted by running-gag View Post
    Hi

    You have several ways to find a parametric equation of the line of intersection

    You can find 2 points on the line
    Take for instance x=3, solve the system
    y + z = 15
    3y - z = -15

    Then take x=-1, solve the system
    y + z = 19
    3y - z = 1

    These 2 points define one direction vector of the line
    Since you already have 1 point of the line, you can get a parametric equation

    Second method, solve the system
    x + y + z = 18
    4x + 3y - z = -3
    using z (for instance) as parameter

    You will find x, y (and z) function of z (parameter)
    Would solving for each variable by Gaussian elimination work in this case?
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  4. #4
    MHF Contributor
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    The system
    x + y + z = 18
    4x + 3y - z = -3

    is a linear system with 2 equations and 3 unknowns
    You can calculate each of the unknowns with respect to only one
    You can use elimination if you want
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