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 iswhereandare constants To find the equation of a plane we need three points. Each point determines an equation inWe solve these simultaneous equations to find the constantsin terms ofand write down the equation of the plane. Finally we cancel the constant d which appears throughout.
Example: A plane passes through the three pointsandFind the equation of the plane.
Substituting the first pointinto the equation of the planegives
Similarly the second and third giveandWe solve the simultaneous equations,
Subinto (2) to give
Subandinto (1) to give
The equation of the plane is thenCancel the factorto giveand clear all the fractions to give the final answer
There is an alternative form for the equation of a plane to terms of vectors:whereandare parameters andis a point in the plane. For the plane give above we can findandby subtracting points in the plane from each other:
The vector form is not unique since any points in the plane can be used.