The equation of a plane in the three-dimensional space is defined with the normal vector and the known point on the plane. A Vector is a physical quantity that with its magnitude also has a direction attached to it. A normal vector means the line which is perpendicular to the plane. With reference to an origin, the position vector basically denotes the location or position (in a 3D Cartesian system) of a point. The Cartesian equation of a plane in 3 Dimensional space and vectors are explained in this article.

Equation of a Plane in Three Dimensional Space

Generally, the plane can be specified using four different methods. They are:

• Two intersecting lines
• A line and point (not on a line)
• Three non-collinear points (Three points are not on the line)
• Two parallel and the non-coincident line
• The normal vector and the point

There are infinite planes that lie perpendicular to a specific vector. But only one unique plane exists to a specific point which remains perpendicular to the point while going through it

Let us consider a plane passing through a given point A having position vector  

Position vector simply denotes the position or location of a point in the three-dimensional Cartesian system with respect to a reference origin.

For point P to lie on the given plane it must satisfy the following condition:

From the figure given above it can be seen that,

Substituting this value in  

This equation represents the vector equation of a plane.

We will assume that P, Q and R points are regarded as x₁, y₁, z₁ and x₂, y₂, z₂ in respectively to change the equation into the Cartesian system. A, B and C will be the assumed direction ratios. Thus,

Substituting these values in the vector equation of a plane, we have

This gives us the Cartesian equation of a plane. To learn more about the equation of a plane in three dimensions and three-dimensional geometry download BYJU’S – The Learning App.