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Math Help - Line of intersection of 3 planes.

  1. #1
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    Line of intersection of 3 planes.

    Please help.

    Find the vector equation of the line of intersection of the 3 planes represented by this system of equations.

    2x - 7y + 5z = 1
    6x + 3y - z = -1
    -14x - 23y + 13z = 5

    Thank you very much!
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  2. #2
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    Quote Originally Posted by mathstudent43 View Post
    Please help.

    Find the vector equation of the line of intersection of the 3 planes represented by this system of equations.

    2x - 7y + 5z = 1
    6x + 3y - z = -1
    -14x - 23y + 13z = 5

    Thank you very much!
    Solve the system for x and y dependent from z. I've got:

    x=-\frac1{12} - \frac16 z and y = -\frac16 + \frac23 z

    Set z = t. Then you have the parametric equation of the intersection line:

    \left|\begin{array}{l}x = -\frac1{12} - \frac16 t \\ y= -\frac16 + \frac23 t \\z = t \end{array} \right.

    That means the vector equation of the line is:

    (x, y, z) = \left(-\frac1{12}~,~-\frac16~,~0\right) + t \cdot \left(-\frac16~,~ \frac23~,~1\right)

    The first vector is the stationary vector of a point which is located in all three planes and the second vector is the direction vector of the line.
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