3D Plane Through Three Points – Step-by-Step + Simplified Equation

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:

\[ \vec{AB}\times\vec{AC}= \langle AB_y AC_z-AB_zAC_y,\; AB_z AC_x-AB_xAC_z,\; AB_x AC_y-AB_yAC_x \rangle \]

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 \[ a=AB_yAC_z-AB_zAC_y \] \[ b=AB_zAC_x-AB_xAC_z \] \[ c=AB_x AC_y-AB_yAC_x \] \[ d=-A_xa-A_yb-A_zc \]

Final plane equation:

\[ ax+by+cz+d=0 \]