Point of Intersection of Three Planes Calculator

\( \) \( \)\( \)\( \)\( \)\( \) An online calculator to find the point of intersection of three planes in 3D is presented.

Solve System of Equations to Find the Intersection Point

Each plane is given by the equation: \[ a x + b y + c z = d \] Given three planes: \[ a_1 x + b_1 y + c_1 z = d_1,\quad a_2 x + b_2 y + c_2 z = d_2,\quad a_3 x + b_3 y + c_3 z = d_3 \] The intersection point \((x, y, z)\) is the solution to the 3×3 linear system: \[ \begin{cases} a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3 \end{cases} \] Geometrically, it is the point where all three planes meet (unique intersection).

Graphical illustration of three planes intersecting at a single point in 3D

Calculator

Enter the twelve coefficients (including decimals) and click "Calculate Intersection".

Three Planes Intersection

Input coefficients (a, b, c, d) for each plane

Plane 1

a₁

b₁

c₁

d₁

Plane 2

a₂

b₂

c₂

d₂

Plane 3

a₃

b₃

c₃

d₃

Intersection Result

x —

y —

z —

Notes

The calculator solves the linear system using Gaussian elimination (via numeric.solve).
If the three planes do not intersect at a single point (e.g., parallel or infinite solutions), the system may be singular. In such cases, an error message will appear.
You can enter decimal numbers (e.g., 1.5, -0.75) with the step="any" input.
The example coefficients (default values) correspond to three planes intersecting at a unique point.