Plane Equation
Vector Equation of the Plane
To determine the equation of a plane in 3D space, a point P and a pair of vectors which form a basis (linearly independent vectors) must be known.
The point P belongs to the plane π if the vector 
is coplanar with the vectors 
and 
.
Parametric Equations of the Plane
Cartesian Equation of the Plane
A point is in the plane π if the system has the solution:
The values are given as:
Intercept Form
A(a, 0, 0), B(0, b, 0) and C(0, 0, c).
Examples
1.Find the equations of the plane that pass through point A = (1, 1, 1) and their direction vectors are: 
and 
.
2.Find the equations of the plane that pass through points A = (−1, 2, 3) and B = (3, 1, 4) and contains the vector 
.
3.Find the equations of the plane that pass through points A = (−1, 1, −1), B = (0, 1, 1) and C = (4, −3, 2).
4. π is the plane of parametric equations:
Confirm whether the points A = (2, 1, 9/2) and B = (0, 9, −1) belong to this plane.
5.Find the equation of the plane in intercept form that passes through the points A = (1, 1, 0), B = (1, 0, 1) and C = (0, 1, 1).
Divide by −2, and the equation is obtained:
6.Find the equation of the plane that passes through the point A = (2, 0, 1) and contains the line with the equation:
From the equation of the line, a second point and the vector is obtained.
7.Find the equation of the plane that passes through the points A = (1, −2, 4) and B = (0, 3, 2) and is parallel to the line:
8.Given the lines:
Determine the equation of the plane that contains the line r and is parallel to the line s.
