Equation of a plane

The equation of a plane can be expressed in three different forms: vector form, cartesian form, and scalar (dot) product form. Set forms depend on different conditions and often used in different circumstances.
The general equation of a plane can be written as ax + by + cz + d = 0 where a, b and c are the coefficients of x, y and z accordingly, and provide a vector normal (perpendicular) to the plane. d is the constant on the right-hand side of the equation.

Vector Form of a Plane

In the vector form, a plane is represented as r = a + λb + μc where r represents any point (x, y, z) on the plane, a is the position vector to a known point on the plane and b and c are two non-parallel direction vectors lying on the plane.
These direction vector b and c must not be parallel or equivalent to each other, otherwise they would represent the same line and not a plane.

The cartesian form of a plane is when the equation is expressed in terms of x, y and z. It has the form ax + by + cz = d.
The numbers a, b and c here define a normal vector to the plane.
d is a constant that determines where the plane cuts the z-axis. If d equals to zero, it means the plane passes through the origin.

In the scalar (dot) product form, the plane’s equation is given as n.r = p, where n is the normal vector to the plane, r is any position vector from the origin to the plane and p is the perpendicular distance from the origin to the plane.
The dot product signifies the projection of one vector onto another. So n.r gives the projection of the position vector r onto the normal vector n, which is equal to the distance p.

To find the point of intersection between a line and a plane, substitute the equations of the line into the plane equation. Solve for the values of x, y, and z.
If these values satisfy the line equations for a specific value of the scalar parameter, then that point is an intersection between the line and the plane.