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
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
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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Frequently Asked Questions on Equation of a Plane Passing Through 3 Non-collinear Points
What is meant by the equation of a plane?
The equation of a plane defines the plane surface in the three-dimensional space.
Mention two different forms of the equation of a plane?
The equation of a plane can be expressed in two different forms such as vector form and cartesian form.
What are the different methods to derive the equation of a plane?
The equation of a plane can be derived using different methods, such as
Equation of a plane in normal form
Equation of a plane perpendicular to a given vector and through a point
Equation of a plane passing through three non-collinear points
Equation of a plane passing through the intersection of two given planes.
Is one normal and one point sufficient to form the equation of a plane?
Yes, one normal and one point is sufficient to form the equation of a plane.
What is the intercept form of the equation of a plane?
The intercept form of the equation of a plane is (x/a) +(y/b) +(z/c) =1. Here, a, b and c are the x-intercept, y-intercept and z-intercept respectively.