# Point of Intersection of Three Planes Calculator

An online calculator to calculate the coordinates of the point of intersection of three planes in 3D is presented.

## Solve System of Equations to Find the Point of Intersection

The equation of each plane is of the form

\[ a x + b y + c z = d \]

Given the equations of three planes,

\( \quad a_1 x + b_1 y + c_1 z = d_1 \) , \( a_2 x + b_2 y + c_2 z = d_2 \) and \( a_3 x + b_3 y + c_3 z = d_3 \),

the coordinates \( (x,y,z) \) of point of intersection of the three planes given by the three equations above, is found by solving the 3 by 3 system of equation given by

\[
\begin{array}{lcl} a_1 x + b_1 y + c_1 z & = & d_1 \\ a_2 x + b_2 y + c_2 z & = & d_2 \\ a_3 x + b_3 y + c_3 z & = & d_3 \end{array}
\]
An example is shown in the graph below.

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## Use of Calculator

Enter the twelve coefficients describing the three planes and press "Calculate Intersection".

\(a_1\) | \(b_1\) | \(c_1\) | \(d_1\) | ||||

\(a_2\) | \(b_2\) | \(c_2\) | \(d_2\) | ||||

\(a_3\) | \(b_3\) | \(c_3\) | \(d_3\) |

Solution