JEE Main & Advanced Mathematics Three Dimensional Geometry Equation of Plane Passing Through the Given Point

(1) Equation of plane passing through a given point : Equation of plane passing through the point \[({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})\] is \[A(x-{{x}_{1}})+B(y-{{y}_{1}})+C(z-{{z}_{1}})=0\], where \[A,\,\,B\] and \[C\] are d.r.’s of normal to the plane.

(2) Equation of plane through three points : The equation of plane passing through three non-collinear points \[({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})\], \[({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})\] and \[({{x}_{3}},\,{{y}_{3}},\,{{z}_{3}})\] is \[\left| \,\begin{matrix} x & y & z & 1  \\ {{x}_{1}} & {{y}_{1}} & {{z}_{1}} & 1  \\ {{x}_{2}} & {{y}_{2}} & {{z}_{2}} & 1  \\ {{x}_{3}} & {{y}_{3}} & {{z}_{3}} & 1  \\ \end{matrix}\, \right|=0\] or \[\left| \,\begin{matrix} x-{{x}_{1}} & y-{{y}_{1}} & z-{{z}_{1}}  \\ {{x}_{2}}-{{x}_{1}} & {{y}_{2}}-{{y}_{1}} & {{z}_{2}}-{{z}_{1}}  \\ {{x}_{3}}-{{x}_{1}} & {{y}_{3}}-{{y}_{1}} & {{z}_{3}}-{{z}_{1}}  \\ \end{matrix}\, \right|=0\].