Line of Intersection of Two Planes Calculator (direction vector & point)

Understanding the line of intersection of two planes calculator — concept and principle

This calculator finds the line where two planes meet in 3D space. It gives the direction vector, a point on the line (the one closest to the origin), the parametric equations, the angle between the planes and the distance from the origin to the line. Use it for multivariable calculus, vector geometry, computer graphics and CAD problems.

The line lies in both planes, so it is perpendicular to both normals: their cross product.

How it is calculated

For planes a₁x + b₁y + c₁z = d₁ and a₂x + b₂y + c₂z = d₂ with normals n₁ = ⟨a₁, b₁, c₁⟩ and n₂ = ⟨a₂, b₂, c₂⟩:

Direction: v = n₁ × n₂
Point P₀: solve both plane equations together with v · P₀ = 0, which gives the point of the line nearest the origin
Line: (x, y, z) = P₀ + t·v
Angle: cos θ = |n₁ · n₂| ÷ (|n₁||n₂|)
Distance from origin to line = |P₀|

If v = 0, the planes are parallel or identical and there is no single line.

The principle

The normal vector of a plane is perpendicular to every direction lying in that plane. The intersection line lies in both planes, so its direction is perpendicular to both normals, and the cross product of two vectors is exactly a vector perpendicular to both. Any point that satisfies both equations lies on the line; requiring the point to be perpendicular to v selects the one closest to the origin, which makes the answer unique. The angle between two planes equals the angle between their normals, taken as the acute one.

Worked example

Default planes: x + y + z = 6 and x − y + 2z = 5.

n₁ × n₂ = ⟨1·2 − 1·(−1), 1·1 − 1·2, 1·(−1) − 1·1⟩ = ⟨3, −1, −2⟩.
Solve x + y + z = 6, x − y + 2z = 5, 3x − y − 2z = 0: P₀ = (29/14, 23/14, 16/7) ≈ (2.07143, 1.64286, 2.28571).
Line: x = 2.07143 + 3t, y = 1.64286 − t, z = 2.28571 − 2t.
n₁ · n₂ = 1 − 1 + 2 = 2; cos θ = 2 ÷ (√3·√6) ≈ 0.4714, θ ≈ 61.874°.
Distance from the origin = |P₀| ≈ 3.49489.

How to read the result

The direction vector can be scaled by any non-zero number, and P₀ can be replaced by any point on the line, so different books may show different-looking but equivalent answers. Check by substituting a point (any t) into both plane equations. An angle of 90° means the planes are perpendicular; an angle near 0° means they are almost parallel, and the line may be very sensitive to small changes.

Common mistakes

Swapping the order in the cross product; it reverses the direction but still describes the same line.
Setting z = 0 to find a point when the line never reaches z = 0 (direction with no z-component).
Writing the plane with d on the wrong side; the form here is ax + by + cz = d.
Forgetting to check for parallel planes.

Keep in mind: Very small cross products indicate nearly parallel planes, where rounding errors grow. Treat such results with caution.

Frequently asked questions

How do I know if two planes intersect?

They intersect in a line whenever their normals are not parallel, that is, when n₁ × n₂ ≠ 0.

Is the parametric form unique?

No. Any point on the line and any non-zero multiple of the direction vector describe the same line.

How do I get the symmetric form?

Solve each equation for t: (x − 2.07143)/3 = (y − 1.64286)/(−1) = (z − 2.28571)/(−2).

Sources: Northern Kentucky University MAT 229, "Section Summary: Equations of Lines and Planes" §12.5

Checked 9 Nov 2026. Standard vector method (direction of the common line = n₁ × n₂; the line of intersection is found by solving both plane equations simultaneously, as summarised in Northern Kentucky University MAT 229, “Section Summary: Equations of Lines and Planes” §12.5; planes are parallel when their normals are parallel).

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