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# 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