Equation of a plane containing three points
Find the equation of the plane containing the three points A, B, and C: ax + by + cz + d = 0.
| Equation of the plane: ax + by + cz + d = 0 | |
| x+ y+ z+ =0 | |
The cross product of vectors AB and AC yields the normal vector of the plane, and its components become the coefficients of x, y, and z.
Substituting the coordinates of point A into the equation of the plane yields the constant terms.
Substituting the coordinates of point A into the equation of the plane yields the constant terms.
| \(\normalsize Plane\ equation\hspace{20px}{\large ax+by+cz+d=0}\\ (1)\ \vec{AB}=(B_x-A_x,B_y-A_y,B_z-A_z)\\ \hspace{25px} \vec{AC}=(C_x-A_x,C_y-A_y,C_z-A_z)\\ (2)\ \vec{AB}\times \vec{AC}=(a,b,c)\\ \hspace{10px} a=(B_y-A_y)(C_z-A_z)-(C_y-A_y)(B_z-A_z)\\ \hspace{10px} b=(B_z-A_z)(C_x-A_x)-(C_z-A_z)(B_x-A_x)\\ \hspace{10px} c=(B_x-A_x)(C_y-A_y)-(C_x-A_x)(B_y-A_y)\\ \hspace{10px} d=-(aA_x+bA_y+cA_z)\\\) |
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