Equation Of A Plane: 3 Non-Collinear Points

The physical quantities which have magnitude, as well as directly attached to them, are known as vectors. Position vectors simply denote the position or location of a point in the three dimensional Cartesian systems with respect to a reference origin. In the upcoming discussion, we shall study about vectors and the Cartesian equation of a plane in three dimensions.

In the previous section we have already discussed that to a given vector there lie infinite planes which are perpendicular to it. But for a particular point there exist one and only one unique plane which is perpendicular to it passing through the given point. The vector equation of a plane in this case is given as

Here,  

The Cartesian equation of such a plane is represented by:

Here, A, B, and C are the direction ratios.

Let us now discuss the equation of a plane passing through three points which are non-collinear.

Equation of a Plane Passing through Three Non Collinear Points

Let us consider three non collinear points P, Q, R lying on a plane such that their position vectors are given by

The vectors  

(

Also, from the above figure,  and . Substituting these values in the above equation, we have

This represents the equation of a plane in vector form passing through three points which are non- collinear.

To convert this equation in Cartesian system, let us assume that the coordinates of the point P, Q and R are given as (x₁, y₁, z₁), (x₂, y₂, z₂) and (x₃, y₃, z₃) respectively. Also let the coordinates of point A be  x, y and z.

Substituting these values in the equation of a plane in Cartesian form passing through three non-collinear points, we have

This is the equation of a plane in Cartesian form passing through three points which are non- collinear.

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