# Plane Equation Calculator Given 3 Points in 3D Space

 

## Formulas Used in the Calculator

A free online calculator, showing all steps, to calculate the equation of a plane in 3 D given 3 points $A = (A_x,A_y,A_z)$, $B = (B_x,B_y,B_z)$ and $C = (C_x,C_y,C_z)$ is presented.

Let vectors $\vec {AB}$ and $\vec {AC}$ be defined by their endpoints as follows:
$\vec {AB} \; = \; \lt B_x - A_x , B_y - A_y , B_z - A_z \gt$
$\vec {AC} \; = \; \lt C_x - A_x , C_y - A_y , C_z - A_z \gt$
The cross product of vectors $\vec {AB}$ and $\vec {AC}$ is orthogonal to the plane defined by the three points $A$, $B$ and $C$
The cross product of vectors $\vec {AB}$ and $\vec {AC}$ is given by
$\vec{AB} \times \vec{AC} \; = \; \lt AB_y \cdot AC_z - AB_z \cdot AC_y , AB_z \cdot AC_x - AB_x \cdot AC_z , AB_x \cdot AC_y-AB_y \cdot AC_x \gt$
Let $M$ be any point on the plane defined by its coordinates as follows
$M \; = \; ( x , y , z )$
Let vector $\vec {AM}$ be defined by
$\vec {AM} \; = \; \lt x - A_x, y -A_y , z - A_z \gt$
For a point $M = (x,y,z)$ to be on the plane defined by the three points $A$, $B$ and $C$, we need to have the dot product of the vectors $\vec {AM}$ and $\vec{AB} \times \vec{AC}$ equal to zero. Hence the equation of the plane:
$\vec {AM} \cdot (\vec{AB} \times \vec{AC}) = 0$
Substitute the components and rewrite the above equation as
$(x - A_x) \cdot (AB_y \cdot AC_z - AB_z \cdot AC_y) + (y -A_y) \cdot (AB_z \cdot AC_x - AB_x \cdot AC_z) + (z - A_z) \cdot (AB_x \cdot AC_y-AB_y \cdot AC_x) = 0$
Let
$a \; = \; (AB_y \cdot AC_z - AB_z \cdot AC_y)$
$b \; = \; (AB_z \cdot AC_x - AB_x \cdot AC_z)$
$c \; = \; (AB_x \cdot AC_y-AB_y \cdot AC_x)$
$d \; = \; - A_x \cdot a - A_y \cdot b - A_z \cdot c$
Rewrite the equation of the plane as:
$a x + by + cz + d = 0$

## Use of the Calculator

Enter the coordinates $x, y$ and $z$ of each point and press "Calculate". The outputs are vectors vectors $\vec {AB}$ and $\vec {AC}$ and their cross product $\vec{AB} \times \vec{AC}$ , the coefficients $a, b , c, d$ and the equation of the plane.

 $A =$ (1 , 2 , 3 ) $B =$ (4 , 5 , 5 ) $C =$ (-2 , 2 , -1 )

## Outputs

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