How to Find the Equation of a Plane Through Three Points

If you know the coordinates of three distinct points in three-dimensional space, you can determine the equation of the plane that contains the point. In the xyz-coordinate system, equations of planes have the form

ax + by + cz = d,

where a, b, c, and d are real numbers. The method below will show you how to find the values for a, b, c, and d if you know the coordinates of three points. You can also use the calculators on the left. The first calculator will directly compute the equation of a plane if you input the three points. The second calculator finds the normal vector perpendicular to two vectors, i.e. the cross product. (Finding the normal vector is a step in the process detailed below.)

Step 1

First convert the three points into two vectors by subtracting one point from the other two. For example, if your three points are (1,2,3), (4,6,9), and (12,11,9), then you can compute these two vectors:

(12,11,9) - (1,2,3) = ‹11, 9, 6›
(4,6,9) - (1,2,3) = ‹3, 4, 6›

Step 2

Find the cross product of the vectors found in Step 1. The cross product of ‹11, 9, 6› and ‹3, 4, 6› is a vector that is perpendicular to both, i.e., the normal vector to the plane. You can use the short-cut calculator above to find the cross product of two vectors, or do the arithmetic by hand. Respectively, the x, y, and z components of the normal vector are

(9)(6)-(6)(4) = 30
(6)(3)-(11)(6) = -48
(11)(4)-(9)(3) = 17,

so the normal vector is ‹30, -48, 17›

Step 3

The coefficients a, b, and c of the planar equation are 30, -48, and 17. So we have 30x - 48y + 17z = d. To find d, we simply plug one of the three points into the equation. For example, if we select the point (1,2,3), we get

(30)(1) - (48)(2) + (17)(3) = -15

So the final answer is 30x - 48y + 17z = -15.

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