## Equations of Planes

A line, which is a two dimensional object, is fixed by two points on it – two dimensions, two points. The equation of a line can be written given by – this is the cartesian form of the line. The cartesian form of a plane is where and are constants To find the equation of a plane we need three points. Each point determines an equation in We solve these simultaneous equations to find the constants in terms of and write down the equation of the plane. Finally we cancel the constant d which appears throughout.

Example: A plane passes through the three points and Find the equation of the plane.

Substituting the first point into the equation of the plane gives Similarly the second and third give and We solve the simultaneous equations, (1) (2) (3)

(1)+(3) gives Sub into (2) to give Sub and into (1) to give The equation of the plane is then Cancel the factor to give and clear all the fractions to give the final answer There is an alternative form for the equation of a plane to terms of vectors: where and are parameters and is a point in the plane. For the plane give above we can find and by subtracting points in the plane from each other: and  .

The vector form is not unique since any points in the plane can be used.