Understanding Planes and Their Intersections
A plane in 3D space is defined by a point and a normal vector perpendicular to it. The standard equation is ax + by + cz = d, where a, b, and c form the normal vector ⟨a, b, c⟩.
When two planes intersect, the result is always a line—never a single point. This follows from dimensional reduction: lines (1D) intersect at points (0D), so planes (2D) intersect at lines (1D). The only exception occurs when planes are parallel or coincident, in which case no line of intersection exists.
The intersection line is perpendicular to both normal vectors. Its direction is determined by the cross product r = n₁ × n₂, and a specific point lying on both planes anchors it in space.
Parametric Equation of the Intersection Line
The parametric form expresses every point on the intersection line using a single free parameter λ (lambda). This approach works for all non-parallel planes.
l: ⟨x₀, y₀, z₀⟩ + λ⟨rₓ, rᵧ, rᵤ⟩
where rₓ, rᵧ, rᵤ = n₁ × n₂ (cross product of normal vectors)
x₀, y₀, z₀— Coordinates of any point satisfying both plane equationsrₓ, rᵧ, rᵤ— Components of the direction vector (from the cross product)λ— Free parameter; varies over all real numbers to generate all points on the line
Finding a Common Point and Direction Vector
To build the parametric equation, follow these steps:
- Extract normal vectors: From plane equation
a₁x + b₁y + c₁z = d₁, the normal vector isn₁ = ⟨a₁, b₁, c₁⟩. Similarly for the second plane. - Compute the cross product: The direction vector
r = n₁ × n₂points along the line of intersection. Using the determinant method:r = ⟨b₁c₂ − c₁b₂, c₁a₂ − a₁c₂, a₁b₂ − b₁a₂⟩. - Locate a common point: Set one variable (say
x = 0) and solve the resulting system of two equations in two unknowns. Repeat with different fixed variables if needed.
If the direction vector has common factors, divide them out to simplify.
Symmetric Form of the Line Equation
Alternatively, express the line without a parameter by equating ratios of coordinate differences:
(x − x₀) / rₓ = (y − y₀) / rᵧ = (z − z₀) / rᵤ
This form is convenient when you need to describe the line in a single equation. If any component of the direction vector is zero, omit that ratio or use a different approach. For example, if rₓ = 0, the line lies in a plane perpendicular to the x-axis, so you would write x = x₀ separately.
Deriving the symmetric form involves isolating variables from the plane equations and substituting to eliminate parameters—a more laborious but equivalent process to the parametric method.
Common Pitfalls and Practical Considerations
Pay attention to these key issues when computing the intersection line.
- Parallel planes have no intersection — If the normal vectors are scalar multiples of each other (or their cross product is zero), the planes are parallel and do not intersect. Always verify that the cross product is non-zero before proceeding.
- Simplify the direction vector — The direction vector from the cross product may contain common factors. Divide them out to get the simplest form. This makes calculations easier and the result cleaner.
- Verify the common point — After finding a candidate point, substitute its coordinates into both original plane equations. If both are satisfied, the point is valid. If not, recalculate; errors in solving the linear system are common.
- Handle zero components in direction vector — If one or more components of the direction vector are zero, the line is parallel to one of the coordinate planes. This affects the symmetric form—avoid division by zero by rewriting the equation appropriately.