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.

Vectorial Equation of the Plane

The point P belongs to the plane π if the vector Vector PX is coplanar with the vectors Vector U and Vector V.

Vector Equation of the Plane

Vector Equation of the Plane

Vector Equation of the Plane


Parametric Equations of the Plane

Parametric Equations of the Plane

Parametric Equations of the Plane


Cartesian Equation of the Plane

A point is in the plane π if the system has the solution:

Cartesian Equation of the Plane

Cartesian Equation of the Plane

Cartesian Equation of the Plane

The values are given as:

Cartesian Equation of the Plane

Plane Equation

Plane Equation

Equation


Intercept Form

Equation of the Plane

A(a, 0, 0), B(0, b, 0) and C(0, 0, c).

Intercept Form Equation

Coordinate Coefficients


Examples

1.Find the equations of the plane that pass through point A = (1, 1, 1) and their direction vectors are: Plane Equation Example and Plane Equation Example.

Plane Equation Example

Cartesian Equation of the Plane

Plane Equation Solution


2.Find the equations of the plane that pass through points A = (−1, 2, 3) and B = (3, 1, 4) and contains the vector Plane Equation Example.

Plane Equation Example

Parametric Equations

Cartesian Equation of the Plane


3.Find the equations of the plane that pass through points A = (−1, 1, −1), B = (0, 1, 1) and C = (4, −3, 2).

Plane Equation Example

Plane Equation Example

Parametric Equations

Cartesian Equation of the Plane


4. π is the plane of parametric equations:

Parametric Equations

Confirm whether the points A = (2, 1, 9/2) and B = (0, 9, −1) belong to this plane.

Plane Equation Example

Plane Equation Example

Plane Equation Solution


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).

Vectorial Equation of the Plane Example

Vector Plane Example

Plane Equation

Divide by −2, and the equation is obtained:

Plane Solution


6.Find the equation of the plane that passes through the point A = (2, 0, 1) and contains the line with the equation:

Vectorial Equation of the Plane

From the equation of the line, a second point and the vector vector u is obtained.

Vector AB

Solutions

Cartesian Equation of the Plane


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:

Vectorial Equation of the Plane

Linear Determination

Equation


8.Given the lines:

Equations of Lines

Determine the equation of the plane that contains the line r and is parallel to the line s.

Linear Determination

Vectorial Equation of the Plane Solution