Working with Planes

Understanding Planes

A plane is a flat, two-dimensional surface that extends indefinitely in all directions. In mathematics, we generally consider planes in three-dimensional space.
The equation of a plane is in the form Ax + By + Cz = D, where A, B and C are the coefficients of x, y, z respectively, and D is the constant term.
The coefficients A, B, and C in the equation of a plane are the components of a normal vector to the plane. This vector is perpendicular to the plane.
A plane is uniquely defined by a point and a normal vector, or by three non-collinear (not in a single line) points.

Intersection of two planes can result in a line, if the planes are not parallel. If the planes are coincident (the same plane) their intersection is the plane itself. If they are parallel and distinct, they do not intersect.
The angle between two planes can be found using the dot product of their normal vectors.
The distance from a point to a plane can be found using the formula Ax1 + By1 + Cz1 - D / sqrt(A^2 + B^2 + C^2), where (x1, y1, z1) are the coordinates of the point.
Two planes are parallel if and only if their normal vectors are proportional (they are the exact same vector up to scaling). They are perpendicular if the dot product of their normal vectors is zero.

A plane can be represented by a system of linear equations. Each linear equation represents a plane, and the system represents the intersection of the planes.
When solving a system of linear equations you are finding the set of points that meet all plane conditions concurrently.
Row operations on the system of linear equations correspond to geometric operations on the corresponding planes.
The solution of the system can be a unique point (all planes intersect in one point), a line (all planes intersect along a line), a plane (all planes coincide), or there may be no solution (the planes do not all intersect at the same place).