Line of Intersection of Two Planes Calculator – Guide & Formulas
Calculate the line of intersection of two planes in 3D space. Get the direction vector, point on the line, and parametric equations with step-by-step solutions.
Find the line of intersection of two planes in 3D space. Enter the coefficients of both plane equations to get the direction vector, a point on the line, and parametric equations.
Key Takeaway
Use the free Line of Intersection of Two Planes Calculator to calculate the line of intersection of two planes in 3d space. get the direction vector, point on the line, and parametric equations with step-by-step solutions. Get instant results with step-by-step explanations.
How to Use the Line of Intersection of Two Planes Calculator
- Enter the coefficients (a, b, c, d) for the first plane: a₁x + b₁y + c₁z = d₁.
- Enter the coefficients (a, b, c, d) for the second plane: a₂x + b₂y + c₂z = d₂.
- Click Calculate to find the line of intersection.
- Review the direction vector, point on the line, and parametric equations.
The Formula
Variable Definitions
- n₁, n₂: Normal vectors of the two planes
- d: Direction vector of the intersection line
- (a,b,c,d): Coefficients of the plane equation ax + by + cz = d
Intersection of 2x + y - z = 3 and x - y + 2z = 1
Find the line where two planes intersect.
- Step 1: Normal vectors: n₁ = (2, 1, -1), n₂ = (1, -1, 2).
- Step 2: Direction vector: n₁ × n₂ = |i j k; 2 1 -1; 1 -1 2| = (1×2 - (-1)×(-1), (-1)×1 - 2×2, 2×(-1) - 1×1) = (1, -5, -3).
- Step 3: Set z = 0 and solve: 2x + y = 3 and x - y = 1. Adding: 3x = 4, so x = 4/3. Then y = 3 - 2(4/3) = 1/3.
- Step 4: Parametric equations: x = 4/3 + t, y = 1/3 - 5t, z = -3t.
Frequently Asked Questions
When do two planes not intersect?
Two planes are parallel (and distinct) when their normal vectors are proportional but the planes do not coincide. In this case, there is no intersection line.
What if the planes are identical?
If the plane equations are scalar multiples of each other, they represent the same plane. Every point on the plane is an intersection — the "line" is the entire plane.
How do I find a specific point on the intersection line?
Set one variable (usually z) to 0, then solve the resulting 2×2 system of equations for x and y. If that gives no solution, try setting a different variable to 0.
What is the angle between two planes?
The angle between two planes equals the angle between their normal vectors: cos(θ) = (n₁·n₂)/(|n₁|·|n₂|). This is the dihedral angle between the planes.
Can I extend this to 3 or more planes?
Three planes can intersect at a point (unique solution), along a line (infinitely many solutions), or have no common intersection. This requires solving a 3×3 system.
Why use the cross product for the direction?
The cross product of two vectors gives a vector perpendicular to both. Since the normals are perpendicular to their respective planes, the cross product gives a direction along which both plane equations are satisfied.
What are parametric equations?
Parametric equations express x, y, z as functions of a parameter t: x = x₀ + at, y = y₀ + bt, z = z₀ + ct, where (x₀, y₀, z₀) is a point and (a, b, c) is the direction vector.
How do I convert to symmetric form?
From parametric form x = x₀ + at, solve for t: t = (x - x₀)/a. Set equal for all three: (x - x₀)/a = (y - y₀)/b = (z - z₀)/c.