# 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:

(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.

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

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

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

(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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