Free Line of Intersection of Two Planes Calculator

Planes and Their Line of Intersection

In three-dimensional space, a plane is a flat, two-dimensional surface with zero curvature. When two such planes are not parallel, their intersection automatically forms a straight line of intersection — a fundamental result in spatial geometry. The Intersection of Two Planes calculator offers a fast way to derive this line, presenting it as a parametric line equation that can be used for further analysis in engineering, physics, or computer graphics.

Representing a Plane

A plane is fully described by a point and a normal vector that is perpendicular to the surface. If the point is $P_0 = (x_0, y_0, z_0)$ and the normal is $\mathbf{n} = \langle a, b, c \rangle$, the Cartesian equation is:

$$a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 \quad \text{or} \quad a x + b y + c z = d,$$

where $d = a x_0 + b y_0 + c z_0$. The normal vector governs the plane's orientation, while the point determines its distance from the origin.

Intersection Geometry

Two planes can be:

• Parallel (same normal direction) — no intersection line exists; they are either coincident or separated.
• Intersecting (normal vectors are not parallel) — the intersection is a straight line.

The line belongs to both planes simultaneously and is essentially the set of all points that satisfy both plane equations.

Computing the Intersection Line – Parametric Method

The line of intersection can be expressed in parametric form using three steps:

1. Direction vector – Because the line lies in both planes, it is perpendicular to each plane's normal. The cross product of the normal vectors gives the line's direction:

$$\mathbf{r} = \mathbf{n}_1 \times \mathbf{n}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix}.$$

2. Common point – Solve the two plane equations together (usually by setting one coordinate to zero) to find a point $P_0 = (x_0, y_0, z_0)$ that lies on both planes.

3. Parametric equation – Combine the point and direction:

$$\mathbf{l}(\lambda) = \langle x_0, y_0, z_0 \rangle + \lambda \langle r_x, r_y, r_z \rangle,$$

which is equivalent to:

\begin{cases} x(\lambda) = x_0 + \lambda r_x,\$$2pt] y(\lambda) = y_0 + \lambda r_y,\$$2pt] z(\lambda) = z_0 + \lambda r_z. \end{cases}

The parameter $\lambda$ can take any real value; each value yields a corresponding point on the intersection line.

Example

Take these two planes:

$$\begin{aligned} A &: -2x + 3y + 4z = -1,\\ B &: 2x - y - 3z = 2. \end{aligned}$$

Normals: $\mathbf{n}_1 = \langle -2, 3, 4 \rangle$, $\mathbf{n}_2 = \langle 2, -1, -3 \rangle$.

Cross product:

$$\mathbf{r} = \langle (-3)(-3) - (4)(-1),\; (4)(2) - (-2)(-3),\; (-2)(-1) - (3)(2) \rangle = \langle -5, 2, -4 \rangle.$$

To get a common point, set $x = 0$ and solve the system:

$$\begin{cases} 3y + 4z = -1,\\ -y - 3z = 2. \end{cases}$$

This yields $y = 1$, $z = -1$, so $P_0 = (0, 1, -1)$. The parametric line is:

$$\mathbf{l}(\lambda) = \langle 0, 1, -1 \rangle + \lambda \langle -5, 2, -4 \rangle.$$

Intersection Line in Symmetric Form

The parametric equation can be converted into a symmetric form by eliminating $\lambda$, provided that no component of $\mathbf{r}$ is zero:

$$\frac{x - x_0}{r_x} = \frac{y - y_0}{r_y} = \frac{z - z_0}{r_z}.$$

Alternatively, the intersection can be found directly by elimination without using a parameter.

Example Using Elimination

Consider:

$$\begin{aligned} &2x - 4z = -1,\\ &x - 2y + z = 2. \end{aligned}$$

From the first equation, isolate $z$:

$$z = \frac{1}{2}x + \frac{1}{4}.$$

Substitute into the second:

$$x - 2y + \frac{1}{2}x + \frac{1}{4} = 2 \;\Longrightarrow\; \frac{3}{2}x - 2y = \frac{7}{4}.$$

Solve for $x$:

$$x = \frac{4}{3}y + \frac{7}{6}.$$

Also from the first equation, express $x$ in terms of $z$:

$$x = 2z + \frac{1}{2}.$$

Thus the line is described symmetrically by the two equations:

$$x = \frac{4}{3}y + \frac{7}{6} = 2z + \frac{1}{2}.$$

Using the Calculator

The Line of Intersection of Two Planes Calculator simplifies every step. You enter the coefficients $(a,b,c,d)$ for each plane in the standard form $ax+by+cz = d$. The tool automatically computes the cross product of the normals to obtain the direction vector, finds a convenient common point, and displays the result as a parametric equation. It also alerts you when the planes are parallel—meaning no intersection exists.