Home | | Maths 12th Std | Equation of a plane passing through three given non-collinear points

Definition, Theorem, Proof - Equation of a plane passing through three given non-collinear points | 12th Mathematics : UNIT 6 : Applications of Vector Algebra

Chapter: 12th Mathematics : UNIT 6 : Applications of Vector Algebra

Equation of a plane passing through three given non-collinear points

(a) Parametric form of vector equation (b) Non-parametric form of vector equation (c) Cartesian form of equation

Equation of a plane passing through three given non-collinear points


(a) Parametric form of vector equation

Theorem 6.17

If three non-collinear points with position vectors  are given, then the vector equation of the plane passing through the given points in parametric form is


Proof

Consider a plane passing through three non-collinear points A, B, C with position vectors  respectively. Then atleast two of them are non-zero vectors. Let us take  â‰  0  and  â‰  0 . Let  be the position vector of an arbitrary point P on the plane. Take a point D on AB (produced) such that  is parallel to  and  is parallel to . Therefore,


This is the parametric form of vector equation of the plane passing through the given three non-collinear points.


(b) Non-parametric form of vector equation

Let A, B, and C be the three non collinear points on the plane with  position vectors  respectively. Then atleast two of them are  non-zero vectors. Let us take    â‰  0  and  â‰  0 .

Now  = -  and  = - . The vectors (  -  and  -  lie on the plane. Since  are non-collinear,  is not parallel to . Therefore,   is perpendicular to the plane.

If  is the position vector of an arbitrary point P(x, y, z) on the plane, then the equation of the plane passing through the point A with position vector  and perpendicular to the vector  is given by


This is the non-parametric form of vector equation of the plane passing through three non-collinear points.


(c) Cartesian form of equation

If (x1 , y1 , z1 ), (x2 , y2 , z2 ) and (x3 , y3 , z3 ) are the coordinates of three non-collinear points A, B, C with position vectors  respectively and (x, y, z) is the coordinates of the point P with position vector  , then we have


Using these vectors, the non-parametric form of vector equation of the plane passing through the given three non-collinear points can be equivalently written as


which is the Cartesian equation of the plane passing through three non-collinear points.

 

Tags : Definition, Theorem, Proof , 12th Mathematics : UNIT 6 : Applications of Vector Algebra
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
12th Mathematics : UNIT 6 : Applications of Vector Algebra : Equation of a plane passing through three given non-collinear points | Definition, Theorem, Proof

Related Topics

12th Mathematics : UNIT 6 : Applications of Vector Algebra


Privacy Policy, Terms and Conditions, DMCA Policy and Compliant

Copyright © 2018-2023 BrainKart.com; All Rights Reserved. Developed by Therithal info, Chennai.