Coordinate Geometry (Analytic Geometry) in Three-dimensional Space
Plane in a three-dimensional (3D) coordinate system The equation of a plane in a 3D coordinate system The Hessian normal form of the equation of a plane
The intercept form of the equation of a plane

Equations of a plane in a coordinate space
The equation of a plane in a 3D coordinate system
A plane in space is defined by three points (which don’t all lie on the same line) or by a point and a normal vector to the plane.
Then, the scalar product of the vector P1P = r - r1, drawn from the given point P1(x1, y1, z1) of the plane to any point P(x, y, z) of the plane, and the normal vector N = Ai + Bj + Ck, is zero, that is which is called the vector equation of a plane.
 Or, Therefore,
(x - x1) · A + (y - y1) · B + (z - z1) · C = 0
 giving A · x + B · y + C · z + D = 0
where,   D - (Ax1+ By1+ Cz1)
the general equation of a plane in 3D space.
If plane passes through the origin O of a coordinate system then its coordinates, x = 0, y = 0, and z = 0 plugged into the equation of the plane, give
A · 0 + B · 0 + C · 0 + D = 0   =>   D = 0.
Thus, the condition that plane passes through the origin is  D = 0. The Hessian normal form of the equation of a plane
Position of a plane in space can also be defined by the length of the normal, drawn from the origin to the plane and by angles, a, b and g, that the normal forms with the coordinate axes.
If  N°  is the unit vector of the normal then its components are,  cosa, cosb and cosg.
Then the projection of the position vector r, of any point P(x, y, z) of the plane, onto the normal has the length p.
Since the projection is determined by expression that is,  r · N° = p or written in the coordinates
 x · cosa + y · cosb + z · cosg - p = 0
represents the equation of a plane in the Hessian normal form. The intercept form of the equation of a plane
If, l, m and n are the intercepts of x, y and z axes and a plane respectively, then projections of these segments in direction of the normal drawn from the origin to the plane are all equal to the length of the normal, that is By plugging these values of cosines into Hessian normal form of the equation of plane, obtained is the intercept form of the equation of plane.    Coordinate geometry contents 