Line of Intersection of Two Planes Calculator
Given two planes a₁x + b₁y + c₁z = d₁ and a₂x + b₂y + c₂z = d₂, find the line where they intersect expressed as r(t) = P₀ + t·D — the direction vector D and a specific point P₀ on the line.
Length of the direction vector of the intersection line
- 1
Dₓ = B₁C₂ − C₁B₂
2 × 3 − -1 × -1 = 5 - 2
Dᵧ = C₁A₂ − A₁C₂
-1 × 2 − 1 × 3 = -5 - 3
D_z = A₁B₂ − B₁A₂
1 × -1 − 2 × 2 = -5 - 4
|D| = √(Dₓ² + Dᵧ² + D_z²)
√(5² + -5² + -5²) = 8.6603The magnitude of the cross product n₁ × n₂.
How does this calculator work?
The intersection of a₁x + b₁y + c₁z = d₁ and a₂x + b₂y + c₂z = d₂ is a line with direction D = n₁ × n₂. A point P₀ on the line is found by setting one coordinate to zero and solving the 2×2 sub-system. The full parametric line is r(t) = P₀ + t·D, t ∈ ℝ.
Formula
How this is calculated
Two non-parallel planes in ℝ³ intersect in exactly one line. Each plane is defined by its normal vector n = (a, b, c) and the scalar d. The direction of the intersection line must be perpendicular to both normals — that is precisely what the cross product D = n₁ × n₂ gives: Dₓ = b₁c₂ − c₁b₂, Dᵧ = c₁a₂ − a₁c₂, D_z = a₁b₂ − b₁a₂. If all three components are zero the planes are parallel and there is no intersection line.
To find a specific point on the line, the calculator sets one coordinate to zero and solves the resulting 2×2 linear system using Cramer's rule. It tries z = 0 first, then y = 0, then x = 0, choosing the first non-singular case (determinant above 1e-12). Once a point P₀ is located, the full parametric line is r(t) = P₀ + t·D for all t ∈ ℝ.
The method works for any real coefficients. The results are floating-point and carry rounding error; for exactly integer inputs the output is exact to at least 10 significant figures. Degenerate cases — identical planes or planes where all three 2×2 sub-systems are singular — return the "parallel or coincident" message.
Frequently asked questions
Substitute the point into both plane equations and verify that both are satisfied. Alternatively, check that the point equals P₀ + t·D for some real t: solve for t from any non-zero component of D and confirm the same t satisfies the other two parametric equations.
Identical planes have proportional normal vectors AND proportional d values. The cross product D = n₁ × n₂ is zero just as in the truly parallel case, so the calculator reports "parallel or coincident." Identical planes overlap in an entire plane, not a single line, so no unique intersection line exists.
If the calculator chose z = 0 to find P₀, then P₀ already lies on the xy-plane. To find where the line crosses z = k, evaluate r(t) with t = (k − P₀_z) / D_z (valid only when D_z ≠ 0). Similarly for x = k or y = k.
Also known as
TG we-Calculate Editorial Team. (2026). Line of Intersection of Two Planes Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/line-of-intersection-of-two-planes-calculator
TG we-Calculate Editorial Team. "Line of Intersection of Two Planes Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/line-of-intersection-of-two-planes-calculator.
TG we-Calculate Editorial Team, "Line of Intersection of Two Planes Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/line-of-intersection-of-two-planes-calculator
@misc{wecalculate_line_of_intersection_of_two_planes_calculator, title = {Line of Intersection of Two Planes Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/line-of-intersection-of-two-planes-calculator}}, year = {2026}, note = {TG we-Calculate} }
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