The vector equation of a plane is derived from a point on the plane and a normal vector, using the condition that the vector connecting the given point to any point in the plane must be perpendicular (dot product zero) to the normal, yielding the linear equation A(x−x0)+B(y−y0)+C(z−z0)=0. The vector equation of a line is derived from a point on the line and a direction vector, using the condition that the vector connecting the given point to any point on the line must be a scalar multiple of the direction vector, yielding the standard (symmetric) form (x−x0)/A=(y−y0)/B=(z−z0)/C. This belongs to three-dimensional analytic/vector geometry within the broader study of multivariable calculus, where planes generalize the role tangent lines play for curves in single-variable calculus, serving as the linear approximation building blocks for surfaces.
The vector equation of a plane is derived from a point on the plane and a normal vector, using the condition that the vector connecting the given point to any point in the plane must be perpendicular…