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.