Plane-Plane Intersection — Definition, Formula & Examples

A plane-plane intersection is the line where two non-parallel planes in three-dimensional space cross each other. If two planes are not parallel, they always intersect along exactly one straight line.

Given two distinct planes in $\mathbb{R}^3$ defined by $a_1x + b_1y + c_1z = d_1$ and $a_2x + b_2y + c_2z = d_2$ , their intersection is the set of all points $(x, y, z)$ satisfying both equations simultaneously. This set forms a line if and only if the normal vectors $\mathbf{n}_1 = \langle a_1, b_1, c_1 \rangle$ and $\mathbf{n}_2 = \langle a_2, b_2, c_2 \rangle$ are not parallel (i.e., $\mathbf{n}_1 \times \mathbf{n}_2 \neq \mathbf{0}$ ).

Key Formula

\mathbf{d} = \mathbf{n}_1 \times \mathbf{n}_2

Where:

$\mathbf{d}$ = Direction vector of the line of intersection
$\mathbf{n}_1$ = Normal vector of the first plane
$\mathbf{n}_2$ = Normal vector of the second plane

How It Works

To find the line of intersection, you solve the two plane equations as a system. The direction vector of the line is given by the cross product

\mathbf{n}_1 \times \mathbf{n}_2

of the two normal vectors. To find a specific point on the line, set one variable (such as

z = 0

) and solve the remaining two equations for the other variables. Combine the point and direction vector to write the line in parametric form.

Worked Example

Problem: Find the line of intersection of the planes x + 2y + z = 6 and 2x - y + z = 3.

Find the direction vector: Compute the cross product of the normals. Here n₁ = ⟨1, 2, 1⟩ and n₂ = ⟨2, −1, 1⟩.

\mathbf{d} = \mathbf{n}_1 \times \mathbf{n}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 2 & 1 \\ 2 & -1 & 1 \end{vmatrix} = \langle 3, 1, -5 \rangle

Find a point on the line: Set z = 0 and solve the system x + 2y = 6 and 2x − y = 3.

x + 2y = 6 \quad\text{and}\quad 2x - y = 3 \implies x = \frac{12}{5},\; y = \frac{9}{5}

Write the parametric equations: Combine the point and direction vector.

x = \tfrac{12}{5} + 3t,\quad y = \tfrac{9}{5} + t,\quad z = -5t

Answer: The line of intersection is

(x, y, z) = \left(\tfrac{12}{5} + 3t,\; \tfrac{9}{5} + t,\; -5t\right)

Why It Matters

Finding where planes intersect is essential in multivariable calculus and linear algebra when solving systems of three equations. In computer graphics and CAD software, plane-plane intersections are used constantly to calculate edges of 3D objects and determine how surfaces meet.

Common Mistakes

Mistake: Assuming two distinct planes can intersect at a single point.

Correction: In 3D, two distinct planes either don't intersect (parallel), intersect along a line, or are the same plane. A single intersection point is not possible — that requires three planes.