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Math Help - Equation of plane.

  1. #1
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    Equation of plane.

    Find the equation of the plane which passes through the points A(-1,-3,1), B(1,-2,0) and C(2,1,-1).
    Calculate the shortest distance between the plane and the origin of the coordinate system.
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  2. #2
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    Since A, B, and C lie in the plane, the vectors

    \overline{AB}=[2,1,-1]

    and \overline{AC}=[3,4,-1] are parallel to the plane.

    Therefore, \overline{AB}\times \overline{AC}=\begin{vmatrix}i&j&k\\2&1&-1\\3&4&-1\end{vmatrix}

    =2i+j+5k

    is normal to the plane, since it is perpendicular to \overline{AB}, \;\ and \;\ \overline{AC}.

    By using the normal and the point (-1,-3,1) in the plane, we get the

    point-normal form:

    2(x+1)+(y+3)+5(z-1)

    2x+y+5z=0

    To find the distance between a point and a plane, use the formula:

    D=\frac{|ax_{0}+by_{0}+cz_{0}+d|}{\sqrt{a^{2}+b^{2  }+c^{2}}}
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