13. Planes in space
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Suppose that P0 has coordinates x0,y0,z0 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 P0 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 ax0 + by0 + cz0. 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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