In the chapter on Vectors, we have already learnt how to write the equations for a plane, in different forms. In this section, we will extend that discussion and learn how to write the equation of a plane in three dimensional coordinates form.
The general vector equation of a plane is of the form
: is a constant
where is the variable vector representing any point on the plane, while is a fixed vector, say which is perpendicular to the plane. Thus, the equation of the plane can be written as
This is the most general equation of a plane in coordinate form. Note that this equation of the plane contains only three arbitrary constants, for, it can be written as
Thus, three independent constraints are sufficient to uniquely determine a plane. For example, three non collinear points are sufficient to uniquely determine the plane passing through them.
Write the equation of an arbitrary plane passing through the point
Let us denote the position vector of by is therefore Now, assume that the normal to the plane is where , , are variable :
Thus, for any variable point on the plane, since is perpendicular to we have
This is the required equation of an arbitrary plane through the point
We could have arrived at this equation alternatively as follows: we assume the general equation of a plane which is