How lines and planes meet
Finding an intersection means finding the points that satisfy all the equations at once: solving a system of equations.
Two lines in 2D
| Case | Intersection |
|---|---|
| different slopes | one point |
| parallel, distinct | none |
| coincident (same line) | infinitely many |
Two lines in 3D
Lines in space have one extra possibility:
| Case | Description |
|---|---|
| intersecting | meet at one point |
| parallel | same direction, never meet |
| coincident | the same line |
| skew lines | not parallel, never meet |
Example: Do these lines intersect?
- L₁: [x, y, z] = [1, 0, 2] + t[1, 1, 0]
- L₂: [x, y, z] = [3, 1, 0] + s[0, 1, 1]
Set equal:
- x: 1 + t = 3 → t = 2
- z: 2 = s → s = 2
- y check: 0 + t = 2 and 1 + s = 3 → 2 ≠ 3 ✗
Directions [1, 1, 0] and [0, 1, 1] aren’t parallel, and there’s no common point → skew lines.
A line and a plane
Substitute the line’s parametric equations into the plane equation and solve for the parameter.
Example: Line x = 1 + t, y = 2 − t, z = 3t and plane 2x + y + z = 9.
- 2(1 + t) + (2 − t) + 3t = 9 → 4 + 4t = 9 → t = 1.25
- Point: (2.25, 0.75, 3.75)
| Result | Meaning |
|---|---|
| one value of t | one point |
| no solution (e.g., 4 = 9) | line is parallel to the plane |
| true for all t (e.g., 4 = 4) | line lies in the plane |
Quick check: if the line’s direction vector · the plane’s normal = 0, the line is parallel to (or in) the plane.
Two planes
| Case | Intersection |
|---|---|
| normals not parallel | a line of intersection |
| parallel, distinct | none |
| coincident | the whole plane |
Example: x + y + z = 6 and x − y + z = 2
- Subtract: 2y = 4 → y = 2
- Then x + z = 4. Let z = t → x = 4 − t
- Line: [x, y, z] = [4, 2, 0] + t[−1, 0, 1]
The direction is also n₁ × n₂.
Three planes
Three planes can meet at a single point, along a line, or not at all (several configurations). Solve the 3 × 3 system with elimination.
Example: x + y + z = 6, x − y = 0, z = 2 → x = y = 2 → point (2, 2, 2).
Worked examples
Example 1: In 2D, intersect y = 2x + 1 and y = −x + 4. → (1, 3).
Example 2: Is the line [x, y, z] = t[1, 1, 1] parallel to x − y = 5? [1, 1, 1] · [1, −1, 0] = 0 → parallel; (0, 0, 0) gives 0 ≠ 5 → parallel, no intersection.
Example 3: Do the planes 2x − y + z = 1 and 4x − 2y + 2z = 2 coincide? Second = 2 × first → yes.
Common mistakes
- Using the same parameter for both lines.
- Forgetting to check the third equation in 3D.
- Assuming non-parallel lines in 3D must meet.
Try it yourself
- Intersect x = t, y = 2t, z = 3t with x + y + z = 12.
- Do [1, 2, 3] + t[1, 0, 0] and [0, 2, 3] + s[0, 1, 0] intersect?
- How do the planes x + 2y = 4 and 2x + 4y = 3 relate?
- Solve: x + y + z = 3, y + z = 2, z = 1.
Answers: 1) (2, 4, 6) 2) yes, at (0, 2, 3) 3) parallel and distinct (no intersection) 4) (1, 1, 1)
Quick recap
- An intersection solves a system of equations.
- 3D lines can intersect, be parallel, be coincident or be skew lines.
- Two non-parallel planes meet in a line of intersection.