Finding the Equation of a Plane from Three Points

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  • 0:04 Noncollinear Points
  • 0:52 General Procedure
  • 2:25 3-Point Plane Equation Example
  • 3:09 Lesson Summary
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Lesson Transcript
Instructor: Russell Frith
In this lesson we will learn the technique for computing the equation of a plane when the coordinates of three noncollinear points in three dimensional space are given.

Noncollinear Points

In this lesson we'll learn the technique for computing the equation of a plane when the coordinates of three noncollinear points - or points that aren't on the same line - in three dimensional space are given. Let the points be designated as P(x1,y1,z1),Q(x2,y2,z2), and R(x3,y3,z3).

We'll then show that the equation of the plane through those points is:

a(x - x0) + b(y - y0) + c(z - z0) = 0, where ( x0, y0, z0) are the coordinates of any one of the points P, Q, or R, and <a,b,c> is a vector perpendicular to the plane.

General Procedure

The following method outlines the general procedure for computing the equation of a plane passing through three given points in space:

(1) Let the three points be designated as P(x1,y1,z1), Q(x2,y2,z2), and R(x3,y3,z3).

(2) Let (x0,y0,z0) be a point on the plane. You can use the coordinates from either P,Q, or R.

(3) Let the equation of the plane be a(x - x0) + b(y - y0) + c(z - z0) = 0 . <a,b,c> is a vector perpendicular to the plane. We need to find values for a,b, and c.

(4) Define the following vectors that you can see below:


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(5) Find the vector perpendicular to those two vectors by taking their cross product, which is the product of two values, represented by perpendicular lines.


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Evaluating the cross product gives coefficients for a vector perpendicular to the plane.


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The coefficients of the vector computed from the cross product are determinants which must be simplified into numbers.


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The numbers from the vector <a b c> are the coefficients of the vector computed from the cross product.


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(6) Using any of the three given points, write the equation of the plane.

a(x - x0) + b(y - y0) + c(z - z0) = 0

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