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Math Help - Distance Between a Line and a Plane

  1. #1
    xiz
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    Distance Between a Line and a Plane

    Hey everyone, this is my first post here, so here goes:

    I'm stuck on a question right now for an assignment, the question is:

    Show that the line:
    x = 2 + t, y = -3 + 2t, z = 1 + 4t
    is parallel to the plane
    2y - z = 1.
    What is the distance between the line and the plane?



    I determined that the vector of the line is (1, 2, 4), from the given equation. With a provided point of (2, -3, 1) from the equation.

    I determined the normal of the plane to be (0, 2, -1) from the provided plane equation.

    To be parallel, the dot product of the normal and the vector of the line must be zero. This proves the line and the normal of the plane are orthogonal. So: (0, 2, -1) dot (1, 2, 4) must = 0.

    4 - 4 = 0, so yes, the line and the plane are parallel.

    Here's where I am having difficulty: How do I determine the distance between the line and the plane? Do I substitute the point (2, -3, 1) into the plane equation?

    I'm lost here, thanks for your help!
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  2. #2
    Forum Admin topsquark's Avatar
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    Re: Distance Between a Line and a Plane

    Here's a procedure:

    The distance between the line and the plane will be another line...this one at right angles to both. So find the equation of a line perpendicular to the line. It must, then, intersect the plane at a right angle since the plane are parallel. Then find the length of the perpendicular.

    -Dan
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