# Equation of a plane passing through three points

This online calculator calculates the general form of the equation of a plane passing through three points

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.1

The general form of the equation of a plane is
$ax+by+cz+d=0$

A plane can be uniquely determined by three non-collinear points (points not on a single line). And this is what the calculator below does. You enter coordinates of three points, and the calculator calculates the equation of a plane passing through three points. As usual, explanations with theory can be found below the calculator.

#### Third point

Digits after the decimal point: 2
Equation of the plane

Coefficient vector

### A plane passing through three points

If we know three points on a plane, we know that they should satisfy the equation of a plane. We can express this mathematically:
$ax_1+by_1+cz_1+d=0 \\ ax_2+by_2+cz_2+d=0 \\ ax_3+by_3+cz_3+d=0$

Points are known, and a, b, c, d coefficients are what we need to find. It means that we have system of three linear equations with four variables a, b, c, d:

$x_1a+y_1b+z_1c+d=0 \\ x_2a+y_2b+z_2c+d=0 \\ x_3a+y_3b+z_3c+d=0$

Or, in matrix form:
$\begin{array}{|cccc|} x_1 & y_1 & z_1 & 1 \\ x_2 & y_2 & z_2 & 1 \\ x_3 & y_3 & z_3 & 1\\ \end{array} * \begin{array}{|c|} a \\ b \\ c \\ d \\ \end{array} = \begin{array}{|c|} 0 \\ 0 \\ 0 \\ \end{array}$

Though we have only three equations for four unknowns, which means that the system has infinitely many solutions, we still can use Gaussian elimination to get a solution in general form with independent variables (meaning that they are allowed to take any value).

In our case, we have only one independent variable. If all coordinates are integers, the calculator chooses the independent variable's value to be the lowest common multiple (LCM) of all denominators in other coefficients to get rid of fractions in the answer. If any coordinate is not an integer, the value of the independent variable is set to one.

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PLANETCALC, Equation of a plane passing through three points