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    Relations between Non-Parallel Planes

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    Let P1 and P2 be two dimensional planes that are not parallel. If these planes are contained in R3 then they must intersect in a line. Prove that if they are contained in R4 instead then they can intersect either along a line, or at a single point. (HINT)- A two dimensional plane in R4 is determined by two equations in four unknowns. Two planes in R4 are not parellel if the left hand side of the equations which define one of them are not a constant mutiple of the left hand side of the equation which define the other).

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    Solution Preview

    Please see the attached file for the fully formatted problems.

    Please see the attached file for the full solution.

    Suppose is determined by the equations , is determined by the equations .
    Let , , . Then the intersection of and is determined by the equation
    (*)
    Let are vectors ...

    Solution Summary

    It is proven planes in R4 intersect at a line or a single point. The dimensional planes that are not parallel are given.

    $2.19

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