Let R, Q and M be three non-collinear points on the plane with position vectors \(\vec a,\vec b \ and \ \vec c\) respectively. The vectors \(\overrightarrow {RQ} \ and\ \overrightarrow {RM}\) are in the given plane. Therefore, the vectors \(\overrightarrow {RQ} \ and\ \overrightarrow {RM}\) is perpendicular to the plane containing points R, Q and M.
Let the position vector of any point P(x, y, z) is \(\vec r\)
The equation of the plane passing through R and perpendicular to the vector \(\overrightarrow {RQ} \ and\ \overrightarrow {RM}\) is :
\((\vec r -\vec a).\overrightarrow {RQ} \ \times\ \overrightarrow {RM} = 0\)
This is the equation of the plane in vector form.
Cartesian Form of plane:
Let coordinate of point R, Q and M are (x1 y1 z1), (x2, y2, z2) (x3 y3 z3) respectively.
Substituting these values in equation (1) of the vector form and expressing it in the form of a determinant, we have
which is the equation of the plane in cartesian form passing through three non-collinear points (x1 y1 z1), (x2, y2, z2) (x3 y3 z3).
