Given three points \[ A=(A_x,A_y,A_z),\quad B=(B_x,B_y,B_z),\quad C=(C_x,C_y,C_z) \] define the vectors
\[ \vec{AB}=\langle B_x-A_x,B_y-A_y,B_z-A_z\rangle \] \[ \vec{AC}=\langle C_x-A_x,C_y-A_y,C_z-A_z\rangle \]
The cross product of the vetcors \( \vec{AB} \) and \( \vec{AC} \) is perpendicular to the plane and is given by:
\[ \vec{AB}\times\vec{AC}= \langle AB_yAC_z-AB_zAC_y,\; AB_zAC_x-AB_xAC_z,\; AB_xAC_y-AB_yAC_x \rangle \]Let \(M=(x,y,z)\) be any point on the plane and
\[ \vec{AM}=\langle x-A_x,y-A_y,z-A_z\rangle \]The plane equation comes from
\[ \vec{AM}\cdot(\vec{AB}\times\vec{AC})=0 \]which expands to
\[ (x-A_x)a+(y-A_y)b+(z-A_z)c=0 \] where \[ a=AB_yAC_z-AB_zAC_y \] \[ b=AB_zAC_x-AB_xAC_z \] \[ c=AB_xAC_y-AB_yAC_x \] \[ d=-A_xa-A_yb-A_zc \]Final plane equation:
\[ ax+by+cz+d=0 \]