Find the line where two planes intersect in 3D space. Computes direction vector, parametric equations, a point on the line, and the angle between the planes.
The **Line of Intersection of Two Planes Calculator** determines every detail about where two planes meet in three-dimensional space. Given the general-form equations a₁x + b₁y + c₁z = d₁ and a₂x + b₂y + c₂z = d₂, it computes the direction vector of the line of intersection, finds a specific point on that line, and writes out the full parametric and symmetric equations.
Two planes in 3D either intersect along a line, are parallel with no intersection, or are coincident (the same plane). This calculator detects all three cases automatically and displays a clear status indicator. When the planes intersect, the direction of the line is the cross product of the two normal vectors — a fundamental result from linear algebra and analytic geometry that this tool computes instantly.
Beyond the core result, the calculator shows the dihedral angle between the planes (both the acute angle and its supplement), the magnitudes of both normal vectors, and a sample table of seven points along the intersection line for different parameter values. A direction-vector component chart visualizes positive and negative components with color-coded bars, making it easy to see how the line is oriented in space.
Eight presets cover common scenarios: standard planes, parallel planes, coincident planes, perpendicular planes, and coordinate-aligned examples. The tool is invaluable for multivariable calculus, linear algebra, and 3D geometry courses where plane-intersection problems are routine exercises.
Line of Intersection of Two Planes Calculator helps you avoid repetitive setup mistakes when solving trigonometric and coordinate-geometry problems. Instead of recalculating conversions, signs, and edge cases by hand, you can test inputs immediately, inspect intermediate values, and confirm final answers before submitting work or using numbers in downstream calculations. It surfaces key outputs like Direction Vector, Unit Direction, Point on Line in one pass.
Direction: d = n₁ × n₂. Point: solve the 2×3 system by setting one variable to 0. Angle: cos(θ) = |n₁·n₂| / (|n₁||n₂|). Parallel if d = 0; coincident if also d₁/d₂ matches the normal ratio.
Result: Computed from the entered values
Using a1=1, b1=1, c1=1, d1=1, the calculator returns Computed from the entered values. This example mirrors the calculator's live computation flow and is useful for checking manual steps and unit handling.
This calculator is tailored to line of intersection of two planes calculator workflows, including common input modes, unit handling, and special-case behavior. It is designed for fast checking during homework, exam preparation, technical drafting, and coding tasks where trigonometric consistency matters.
Use the primary result together with supporting outputs to verify direction, magnitude, and validity. Cross-check against known identities or geometric constraints, and confirm that angle ranges, sign conventions, and domain restrictions are satisfied before using the numbers elsewhere.
A reliable way to improve is to solve once manually, then verify with the calculator and explain any mismatch. Repeat this on varied examples and edge cases. The built-in preset scenarios for quick trials, comparison tables for side-by-side validation, visual cues that make trends and quadrants easier to read help you build pattern recognition and reduce sign or conversion errors over time.
Two non-parallel planes in 3D space always intersect along a straight line. The direction of this line is perpendicular to both plane normals, computed as their cross product.
Parallel planes have proportional normal vectors but non-proportional constants, so their cross product is the zero vector. They never intersect.
Coincident planes are identical — every point on one plane satisfies the other. Their equations are scalar multiples of each other, including the constant term.
By the cross product: d = n₁ × n₂ = ⟨b₁c₂−c₁b₂, c₁a₂−a₁c₂, a₁b₂−b₁a₂⟩. This vector is perpendicular to both normals.
Set one variable (e.g., z = 0) and solve the resulting 2×2 system. If that system is singular, try setting x = 0 or y = 0 instead.
The dihedral angle is the angle between two planes, computed as arccos(|n₁·n₂| / (|n₁||n₂|)). It ranges from 0° (parallel) to 90° (perpendicular).