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

(\(\)\(\vec{r}\) \(\) – \(\)\(\vec{a}\) \(\)). \(\)\(\vec{N}\) \(\) = 0

Here, \(\)\(\vec{r}\) \(\) and \(\)\(\vec{a}\) \(\) represent the position vectors.

The Cartesian equation of such a plane is represented by:

\(\)\(A(x-{x}_{1}) + B (y- {y}_{1}) + C (z -{z}_{1}) = 0\) \(\)

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 \(\)\(\vec{a}\) \(\) , \(\)\(\vec{b}\) \(\) and \(\)\(\vec{c}\) \(\) as shown in the figure given below.

The vectors \(\)\(\vec{PQ}\) \(\) and \(\)\(\vec{PR}\) \(\) lie in the same plane. The vector lying perpendicular to plane containing the points P, Q and R is given by \(\)\(\vec{PQ}\) \(\) × \(\)\(\vec{PR}\) . If is the position vector of any point A lying in the plane containing P, Q, R then using the vector equation of a plane as mentioned above, the equation of the plane passing through P and perpendicular to the vector \(\)\(\vec{PQ}\) \(\) × \(\)\(\vec{PR}\) is given by

(\(\)\(\vec{r}\) \(\) – \(\)\(\vec{a}\) \(\)). (\(\)\(\vec{PQ}\) \(\) × \(\)\(\vec{PR}\)) = 0

Also, from the above figure, and . Substituting these values in the above equation, we have

\(\)(\(\vec{r}\) – \(\vec{a}\)).[(\(\vec{b}\)- \(\vec{a}\)) × (\(\vec{c}\)- \(\vec{a}\))] = 0 \(\)

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 (x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}) and (x_{3}, y_{3}, z_{3}) 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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