Analytic Geometry

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Given two planes, discuss the methods used to determine if the planes are parallel, perpendicular, coincident, or none of these.

  • Analytic Geometry -

    Given two planes:
    Π1 : Ax+By+Cz+D=0 and
    Π2 : ax+by+cz+d=0

    The following uses the notation that
    <a,b,c> represents a vector with three components.

    The normal vectors are:
    N1 : <A,B,C>
    N2 : <a,b,c>

    If Π1 is perpendicular to Π2, then
    N1.N2=0 (dot-product).

    If the planes are parallel, we can find a numeric factor α such that
    <A,B,C>=α<a,b,c>
    If in addition, D=αd, then the planes are coincident.

    If no such α can be found, the planes are not parallel.

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