Last updated: August 1, 2026
Line of Intersection of Two Planes Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
Enter both planes in ax + by + cz = d form to classify the pair as intersecting, parallel, or coincident. When the planes intersect in a unique line, the calculator returns one point on that line plus a direction vector obtained from the cross product of the plane normals.
Enter both plane equations in ax plus by plus cz equals d form to see whether they intersect, are parallel, or are the same plane, and to get a point and direction vector when a line exists.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
Use the cross product of the plane normals. If n1 × n2 is nonzero, it gives the line direction.
Where:
- n1=first plane normal vector
- n2=second plane normal vector
- d=direction vector of the line
- P=one point on the line
Worked Examples
Two planes meeting in a line
Use x + y + z = 4 and x - y + z = 2.
- 1Compute the cross product of the normals.
- 2Set x = 0 to solve a convenient 2×2 subsystem.
- 3Use the resulting point with the direction vector.
Parallel planes
Compare x + y + z = 2 with 2x + 2y + 2z = 7.
- 1Notice the normals are proportional.
- 2Check the constants.
- 3Conclude the planes are distinct and parallel.
Coincident planes
Compare x + y + z = 2 with 2x + 2y + 2z = 4.
- 1All coefficients scale by the same factor.
- 2Both equations describe the same plane.
Decimal coefficients
Use 0.5x + y + z = 3 and x - z = 1.
- 1Compute the direction from the normals.
- 2Solve for one point on the line.
Introduction
Classify two planes and return a point-direction form when they intersect in a line.
Overview
Classify two planes and return a point-direction form when they intersect in a line.
Intersection relationship
Analytic geometry homework
Engineering intersection checks
3D graphics constraints
Input format
Enter both planes in ax + by + cz = d form.
Separate rows with semicolons
Separate entries with commas or spaces
Brackets are optional
Keep row lengths consistent
Formula and method
Use the cross product of the plane normals. If n1 × n2 is nonzero, it gives the line direction.
- N1 matters:
first plane normal vector.
- N2 matters:
second plane normal vector.
- D matters:
direction vector of the line.
- P matters:
one point on the line.
Validation rules
The calculator uses deterministic validation before it computes the final result.
All eight coefficients must be finite numbers
Parallel or coincident planes return empty point and direction outputs
The solver rejects degenerate 2×2 subsystems
Outputs are rounded to six decimals
Interpreting the outputs
Use the main result first, then the secondary fields to verify the structure and meaning of the answer.
Read relationship first
A nonempty direction vector means the planes intersect in a unique line
Any scalar multiple of the direction vector describes the same line
Any point on the same line is equivalent to the reported point
Common use cases
These are typical reasons to use this calculator in study or applied work.
Analytic geometry homework
Engineering intersection checks
3D graphics constraints
Point-direction line derivation
FAQs
What equation form should I use?
Use ax + by + cz = d for each plane, with all variable terms on the left.
Why is the direction vector a cross product?
Because the intersection line must be perpendicular to both plane normals.
What does parallel mean here?
The plane normals are proportional but the two full equations are not the same.
What does coincident mean?
Both equations describe the same plane, so there is no unique line to report.
Can my point differ from the calculator result?
Yes. Any point on the same line is valid.
Are decimals allowed?
Yes. Any finite real coefficients are supported.