13. Planes in space





Suppose that P_{0} has coordinates x_{0},y_{0},z_{0} and n has components a,b,c.
Let P, with coordinates x,y,z, be an arbitrary point on the plane. The position vectors of P_{0} and P are and Substituting into the vector equation, we obtain which, when multiplied out, gives
This is called a Cartesian equation of the plane. It simplifies to
where d is the constant ax_{0} + by_{0} + cz_{0}. An equation of the form where a,b,c and d are constants and not all a,b,c are zero, can be taken to be an equation of a plane in space. The coefficients a, b and c are the components of a normal vector for the plane described by the equation. 

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