(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**

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

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.

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

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.

If (*x*_{1} , *y*_{1} , *z*_{1}
), (*x*_{2} , *y*_{2} , *z*_{2} ) and (*x*_{3}
, *y*_{3} , *z*_{3} ) 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

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

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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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