Equation of a Plane - 3 Points - Maple Help

Equation of a Plane - 3 Points

Main Concept

A plane can be defined by four different methods:

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

The Cartesian equation of a plane is , where is the vector normal to the plane.

How to find the equation of a plane using three non-collinear points

Three points (A,B,C) can define two distinct vectors AB and AC. Since the two vectors lie on the plane, their cross product can be used as a normal to the plane.

 1 Determine the vectors
 2 Find the cross product of the two vectors
 3 Substitute one point into the Cartesian equation to solve for d.

Example

Find the equation of the plane that passes through the points .

 1 Determine the vectors



  



  
 2 Determine the normal vector



 3 The equation of the plane is



 4 Plug in any point to find the value of d

 $-d=$  $d=$ $-4$
 5 The equation of the plane is

Change the three points on the plane and see how it affects the plane.

 Point A Point B ${x}_{B}=$ ${y}_{A}=$ ${y}_{B}=$ ${z}_{A}=$ ${z}_{B}=$

 Point C

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