Line Of Intersection Of Two Planes Calculator - Direction, Point, and Parametric Form

The line of intersection of two planes calculator is the fastest way to find the straight line that lies in both planes at the same time. Enter each plane in general form a x + b y + c z + d = 0, and the calculator returns the direction vector of the shared line, a specific point on that line, and the parametric form (x, y, z) = (x₀, y₀, z₀) + t · d. Use this line of intersection of two planes calculator when a homework problem, a CAD model, or an engineering sketch hands you two flat surfaces and asks for the line where they meet.

• • Multivariable calculus homework: Verify line-of-intersection answers without re-deriving the cross product or solving the 2x2 system by hand.
• • Computer graphics and CAD: Find where two clipping planes meet so you can stitch a boundary edge or solve a 3D constraint in a mesh or model.
• • Physics and engineering constraints: Solve for the line where two flat surfaces, hinges, or supports intersect in space, like the ridge of two roof planes.

The result is always a real line in 3-space when the two plane normals are not parallel. When the normals are parallel the line of intersection of two planes calculator reports 'parallel' or 'coincident' so you do not have to guess whether a line exists.

When the follow-up question is how far a known point sits from one of the two planes, Distance From Point to Plane Calculator gives the perpendicular distance in the same general-form input style.

The calculator takes the cross product of the two normal vectors to find the line's direction, then solves a 2x2 system to recover a point on the line.

• a₁, b₁, c₁, d₁: Coefficients of plane 1 in a₁ x + b₁ y + c₁ z + d₁ = 0; the first three form the normal vector (a₁, b₁, c₁).
• a₂, b₂, c₂, d₂: Coefficients of plane 2 in a₂ x + b₂ y + c₂ z + d₂ = 0; the first three form the normal vector (a₂, b₂, c₂).
• d = n₁ × n₂: Direction vector of the intersection line, equal to the cross product of the two normal vectors.
• x₀, y₀, z₀: Coordinates of a specific point on the line, found by fixing one coordinate and solving the resulting 2x2 system with Cramer's rule.

The calculator picks the largest direction component as the pivot, sets that coordinate to 0, and uses Cramer's rule on the remaining 2x2 system. If the cross product is the zero vector, the calculator reports 'parallel' or 'coincident' instead of fabricating a direction.

According to Wolfram MathWorld, the line of intersection of two planes in 3D is parallel to the cross product of the two plane normal vectors.

According to Wikipedia, every plane in 3D can be written as ax + by + cz + d = 0, where (a, b, c) is the plane's normal vector.

The direction vector of the line of intersection is the cross product of the two plane normals, which is exactly the calculation Cross Product Calculator performs on two 3D vectors.

These four ideas turn the line of intersection into a real line you can use.

Reversing the two planes flips the sign of the direction vector but leaves the geometric line unchanged.

Once you have the two plane normals and the intersection direction, the dihedral angle between the planes is the same as the angle between the normals, which Angle Between Two Vectors Calculator computes directly.

Work through these steps any time you need a quick line of intersection.

1. 1 Rewrite both planes in general form: If a plane is given as 2x − y + 3z = 4, move the constant to the left so it reads 2x − y + 3z − 4 = 0.
2. 2 Read off the four coefficients of plane 1: Enter a₁, b₁, c₁, d₁ exactly as they appear in a₁ x + b₁ y + c₁ z + d₁ = 0. The constant d carries the sign of the moved term, so d = −4 for 2x − y + 3z = 4.
3. 3 Read off the four coefficients of plane 2: Do the same for plane 2, entering a₂, b₂, c₂, d₂ into the second row of inputs.
4. 4 Read the direction vector and a point on the line: The first three results are the components of d = n₁ × n₂; the next three are a specific point (x₀, y₀, z₀) on the line.
5. 5 Use the relationship label to handle special cases: If the calculator reports 'parallel' or 'coincident', the two planes do not meet in a single line; if it reports 'invalid', one plane has a zero normal.

A CAD model has a tilted floor plane 0.4x + y + 0.2z − 1.2 = 0 meeting a wall plane x + 0y + 0z − 3 = 0. Enter a₁ = 0.4, b₁ = 1, c₁ = 0.2, d₁ = −1.2 and a₂ = 1, b₂ = 0, c₂ = 0, d₂ = −3 to get a direction vector of (0, 0.2, −1) and a point on the line.

For the 2D version of the same problem on the plane, Intersection Of Two Lines Calculator solves a 2x2 system for the meeting point of two standard-form lines using Cramer's rule.

Using this line of intersection of two planes calculator gives you a faster and more reliable workflow than re-running the formula by hand.

• • Skip the algebra: Computing the cross product, solving the 2x2 system, and stitching the parametric form is three steps of error-prone arithmetic; the line of intersection of two planes calculator does them in milliseconds.
• • Catch sign and sign-convention errors: Showing the relationship label makes it obvious when the planes share a line and when they do not.
• • Works for any plane orientation: Horizontal, tilted, vertical, or planes with only one or two non-zero coefficients all use the same general form.
• • Pairs with related geometry tools: Once you have the direction and the point, the same 3D vectors feed naturally into distance, dot-product, and cross-product tools for the next step.

If your homework also asks for the angle between the two planes, the angle-between-two-vectors calculator handles that calculation.

If your next step is to project a known point onto the intersection line, the same direction vector feeds into a dot-product step, and Dot Product Calculator handles that arithmetic for you.

The line of intersection changes in predictable ways when the two plane coefficients change.

• • The calculator assumes standard Cartesian 3-space. It does not apply directly to curved surfaces, latitude-longitude coordinates, or non-orthogonal coordinate systems without a transform.
• • Numerical precision is limited by IEEE 754 double-precision arithmetic. For inputs larger than about 1e15 in magnitude, rounding error becomes visible in the last decimals of the direction and point.

Plane 1 × plane 2 is the consistent convention used here. Swap the two planes and the direction vector changes sign, but the line of intersection is the same set of points.

According to Paul's Online Math Notes, a line in 3D can be written as the parametric equations x = x₀ + a t, y = y₀ + b t, z = z₀ + c t using one point on the line and its direction vector (a, b, c).

When you have two points on the line of intersection and want the parametric form from those, Line Equation From Two Points Calculator accepts two points and returns the same direction-and-point structure this calculator produces.