How this instrument works
Each plane Ax + By + Cz = D carries a normal vector (A, B, C) — the single direction perpendicular to every line drawn inside that flat surface. When two planes are not parallel, they cross along exactly one straight line, and that line runs inside both planes at once, so its direction must be perpendicular to both normals simultaneously. The cross product of two vectors is defined to be exactly that: a new vector perpendicular to both inputs. Crossing the two normals therefore hands back the intersection line's direction directly, with no equation-solving in between.
The relationship runs backwards from a plane built out of three points, where crossing two vectors that lie flat inside a surface produces the normal sticking straight out of it. Here the roles trade places: crossing two normals produces a direction that lies flat inside both surfaces instead. Notice, too, what never enters the arithmetic — each plane's constant term D. Slide either plane parallel to itself by changing D and the intersection line moves to a new location in space, but its direction never changes, because D only sets how far the plane sits from the origin, not which way it tilts.
A result of (0, 0, 0) is not a bug — it means the two normal vectors point along the same line, so the planes themselves are parallel. Parallel planes with different constant terms never meet at all, and parallel planes with the identical equation are the same plane occupying every one of infinitely many shared points, so neither case has a single line to describe. Watch for this whenever a computed direction collapses to zero: it is the calculator's way of reporting that the two input planes were never going to cross to begin with.
- Enter the first plane's normal coefficients into Plane 1: A, Plane 1: B, and Plane 1: C — the numbers in Ax + By + Cz = D.
- Enter the second plane's coefficients the same way into Plane 2: A, Plane 2: B, and Plane 2: C.
- Read Direction vector: x, Direction vector: y, and Direction vector: z together as one arrow's three components, not three separate answers.
- A result of (0, 0, 0) means the two planes are parallel and share no single line — check that neither plane's coefficients are a multiple of the other's.
- Double every coefficient in Plane 2 and confirm every direction-vector component doubles too, while the line it points along stays exactly the same.
Worked example — the planes x+y+z=1 and x−y+z=1
Set Plane 1 to A=1, B=1, C=1 and Plane 2 to A=1, B=−1, C=1 — the coefficients behind x+y+z=1 and x−y+z=1. Following the formulas: dx = B1C2 − C1B2 = 1(1) − 1(−1) = 2; dy = C1A2 − A1C2 = 1(1) − 1(1) = 0; dz = A1B2 − B1A2 = 1(−1) − 1(1) = −2. The sheet reports the direction vector (2, 0, −2).
Both dot products confirm the answer: (1,1,1)·(2,0,−2) = 2+0−2 = 0, and (1,−1,1)·(2,0,−2) = 2+0−2 = 0, so the direction sits perpendicular to each normal, exactly as required. Setting z=0 and solving x+y=1 alongside x−y=1 gives the point (1,0,0) on both planes, so the full line is (1,0,0) + t(2,0,−2) — the direction vector this sheet returns plus one point found by hand.
Questions
What does the direction vector this calculator returns actually represent?
It is the direction the line of intersection runs, not the line's location. Picture an infinite arrow lying inside both planes at once, pointing the way the crossing line travels; scaling it up or down, or reversing its sign, still describes the same line, since only the direction matters, not the magnitude.
How does crossing two normal vectors differ from crossing two ordinary vectors?
The arithmetic is identical, but the geometric roles are opposite. Crossing two vectors lying flat inside a surface, as when building a plane from three points, produces the normal sticking out of it; crossing two normals, as this calculator does, produces a direction lying flat inside both planes instead — the intersection line's own direction.
Why does the calculator return (0, 0, 0) for some pairs of planes?
A zero result means the two normal vectors are parallel, so the planes themselves are parallel — either they never meet, or they are the same plane written two different ways. Either way there is no single line to describe, so the direction vector collapses to zero rather than pointing anywhere.
Does each plane's constant term D affect the answer?
No — only the A, B, and C coefficients enter the formulas. D sets how far a plane sits from the origin without changing its tilt, so sliding either plane parallel to itself by editing D moves the intersection line to a new location in space but never changes the direction this calculator reports.
How do I find an actual point on the line, not just its direction?
Pick a value for one coordinate, commonly z=0, and solve the two plane equations as a pair of simultaneous equations in the remaining two variables. That yields one point the line passes through; combined with the direction vector from this calculator, the point gives the full parametric line — a step that needs each plane's D, which this sheet does not collect since direction alone never depends on it.
Why must the intersection line's direction be perpendicular to both normals?
Because the line lies entirely inside both planes, and every direction inside a plane is, by definition, perpendicular to that plane's normal vector. A single direction perpendicular to two different normals at once is exactly what a cross product is built to produce, which is why n1 × n2 always lands on the intersection line's direction whenever the planes actually cross.