text: Equation of a Plane - 3 Points - Maple Help
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  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
  
   More MathApps
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