Line of Intersection of Two Planes Calculator
Enter the coefficients of two planes in the form ax + by + cz = d and this calculator finds the line of intersection: the direction vector (the cross product of the two normal vectors), a specific point on the line, and the full parametric equations. It also detects parallel planes and shows the angle between planes. Results update instantly as you type.
Formula
Worked example
Planes: -2x + 3y + 4z = -1 and 2x - y - 3z = 2. Normal vectors n1 = <-2, 3, 4> and n2 = <2, -1, -3>. Cross product r = n1 x n2: rx = 3(-3) - 4(-1) = -5, ry = 4(2) - (-2)(-3) = 2, rz = (-2)(-1) - 3(2) = -4. So r = <-5, 2, -4>. Fix x = 0: solve 3y + 4z = -1 and -y - 3z = 2, giving y = 1, z = -1. Line: <0, 1, -1> + t<-5, 2, -4>, or x(t) = -5t, y(t) = 1 + 2t, z(t) = -1 - 4t.
What is the line of intersection of two planes?
When two non-parallel planes meet in three-dimensional space, the result is a straight line. This is a direct consequence of dimension: a plane is two-dimensional, so the intersection of two of them drops one dimension to produce a one-dimensional object, namely a line. The line extends infinitely in both directions through all points that satisfy both plane equations simultaneously. If the planes are parallel, no such line exists; if they are the same plane in disguise, every point is a solution.
How to find the intersection line: the cross product method
The standard approach has three parts. First, identify the normal vector of each plane from the coefficients a, b, c in the equation ax + by + cz = d. Second, take the cross product r = n1 x n2 to get the direction vector of the intersection line. This works because any line lying in both planes must be perpendicular to both normals, and the cross product produces exactly that perpendicular direction. Third, find a specific starting point P0 on the line by setting the coordinate corresponding to the largest component of r to zero, which reduces the two plane equations to a manageable 2x2 linear system. The full line is then l: P0 + t * r, where t is a free real number.
Parametric vs. symmetric form
The parametric form writes each coordinate as a separate function of the parameter t: x(t) = x0 + t * rx, y(t) = y0 + t * ry, z(t) = z0 + t * rz. It is the most flexible because it works even when a direction component is zero. The symmetric form eliminates t by equating the ratios: (x - x0)/rx = (y - y0)/ry = (z - z0)/rz. It is more compact but fails if any direction component is zero, because division by zero is undefined. When a component is zero, the variable is fixed and the symmetric form omits it or writes it separately.
Angle between the two planes
The dihedral angle between the two planes equals the angle between their normal vectors (or its supplement, whichever is at most 90 degrees). The formula is theta = arccos(|n1 . n2| / (|n1| * |n2|)), where the dot product n1 . n2 = a1*a2 + b1*b2 + c1*c2. Perpendicular planes give 90 degrees; nearly parallel planes give an angle close to 0 degrees. This angle is a property of the two planes regardless of where exactly the intersection line lies.
Possible relationships between two planes
| Relationship | Direction vector |r| | Normal vectors | Intersection |
|---|---|---|---|
| Intersecting | Non-zero | Not parallel | A unique line |
| Parallel (distinct) | Zero | Parallel, d1 != d2 | No intersection |
| Identical (coincident) | Zero | Parallel, d1 = d2 | The entire plane |
The cross product of the normal vectors determines which case applies.
Frequently asked questions
Can the intersection of two planes be a point?
No. Planes are infinite flat surfaces. If two planes share even one point, they share an entire line through that point. The intersection can only be empty (parallel planes), a line (the usual case), or the whole plane itself (if both equations describe the same plane).
What does the direction vector of the intersection line represent?
The direction vector r = n1 x n2 is a vector that points along the intersection line. Because the line lies in both planes, it must be perpendicular to both normal vectors, and the cross product is the unique direction with that property (up to sign and scaling). Any non-zero scalar multiple of r is an equally valid direction vector for the same line.
How do I know if two planes are parallel?
Two planes are parallel when their normal vectors are proportional: a1/a2 = b1/b2 = c1/c2. Equivalently, the cross product n1 x n2 equals the zero vector, so |r| = 0. If the proportionality holds for the normal vectors but not for the constants d, the planes are parallel and distinct; if it holds for d as well, the planes are identical.
What is the parametric equation of the intersection line?
It is l: P0 + t * r, written component-wise as x(t) = x0 + t*rx, y(t) = y0 + t*ry, z(t) = z0 + t*rz. Here P0 = (x0, y0, z0) is any fixed point on the line, r = <rx, ry, rz> is the direction vector, and t is a free real parameter. Different values of t give different points on the line.
Does it matter which point P0 I choose?
No. Any point that satisfies both plane equations is a valid P0. Choosing a different starting point simply shifts t: all choices produce the same infinite line. This calculator picks the most numerically stable starting point by fixing the coordinate with the largest direction-vector component to zero.
How is the angle between the planes calculated?
The angle is theta = arccos(|n1 . n2| / (|n1| * |n2|)). The absolute value ensures the result is between 0 and 90 degrees (since "the angle between two planes" conventionally refers to the acute dihedral angle). If you want the obtuse supplement, subtract from 180 degrees.
What if one of the plane coefficients is zero?
A zero coefficient just means the plane does not tilt in that direction. For example, z = 3 (with a = 0, b = 0, c = 1, d = 3) is a horizontal plane at height 3. Enter zero for any missing coefficient and this calculator handles it correctly, including when the direction vector has a zero component.