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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  ≠ 0  and  ≠ 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    ≠ 0  and  ≠ 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.

 

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

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12th Mathematics : UNIT 6 : Applications of Vector Algebra


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