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Thread: The intersection of two planes

  1. #1 The intersection of two planes 
    Forum Junior anticorncob28's Avatar
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    There are two main postulates describing the nature of planes:
    1) Through any three non-collinear points, there is exactly one plane.
    2) If two points lie on a plane, the entire line containing those two points also lies on that plane.
    Using these, I can show that if two planes intersect, their intersection is either a point, or a line. Suppose that the intersection of two planes contained a set of three non-collinear points. Then, since these three points are common to both planes, through these three points there are two planes, contradicting the first postulate. Thus, any pair of three points on the intersection of two planes are collinear and thus all the points are collinear.
    If all the points in the intersection of two planes are collinear, the points are either a line, a ray, line segment, or a set line segments and points, etc. If the intersection is at least two points, the intersection must be a line. This is because if two points lie on the plane, the entire line containing those points also lies on that plane, so the entire line must be common to both planes, and the intersection must be a line.
    Now, the intersection of two planes must then be either a line, or a single point, as was demonstrated. In three dimensions, it is impossible for the intersection of two planes to be exactly one point. In higher dimensions, it is actually possible. The question is, how can you prove that two planes that lie in the same three-dimensional space cannot intersect at just one point?
    I've been working on this for a while and could not figure it out. Any help would be greatly appreciated. If it is possible to show this using the following axioms about solids (three-dimensional "planes"), I would also like that.
    1) Through any four non-coplanar points there is exactly one solid.
    2) If three non-collinear points lie in a solid, the entire plane containing those three points also lies in that solid.


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  3. #2  
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    The cross product of the normals to the two planes is a vector X (free floating) parallel to the two planes. If you have one point of intersection, then anchor X at that point. It is parallel to both planes, and has a point in both planes, so it must be in both planes.


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  4. #3  
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    Quote Originally Posted by anticorncob28 View Post
    The question is, how can you prove that two planes that lie in the same three-dimensional space cannot intersect at just one point?
    Well I don't think I can answer that using the principles you've given, but I can explain why this isn't possible using some basic facts from linear algebra. Construct a 2x3 matrix using the coefficients of 2 non-parallel planes. This matrix will have rank = 2 (because the planes are non-parallel), leaving 1 free variable, so the subspace created by the intersection of these two planes will be 1-dimensional. Interestingly enough a similar argument shows that three planes CAN intersect at a point.
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    Malignant Pimple shlunka's Avatar
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    This is a very uncommon occurrence, as airports closely monitor their incoming/outgoing flights.
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