Intersections in Space
When lines cross lines and planes meet planes, the result is always a simpler object — a point or a line.
Introduction
Intersections are everywhere — a street crossing, where a wall meets the floor, where two walls meet each other. Geometry tells us exactly what kind of object results from each type of intersection.
Past Knowledge
Points, lines, planes (1.1.1). Segments and rays (1.1.2).
Today's Goal
Determine the intersection of lines and planes, and identify when objects don't intersect.
Future Success
Transversal intersections (Unit 3) and 3D cross-sections (Unit 12) depend on this.
Key Concepts
Intersection Rules
| Object A | Object B | Intersection |
|---|---|---|
| Line | Line | A point (or ∅ if parallel) |
| Line | Plane | A point (or ∅ / the whole line) |
| Plane | Plane | A line (or ∅ if parallel) |
Key Insight
The intersection always drops one dimension: 2D planes → 1D line; 1D lines → 0D point.
Worked Examples
Two Lines Crossing
Lines and cross at point . What is their intersection?
Two non-parallel coplanar lines intersect at exactly one point.
Answer: Point .
Wall Meets Floor
What is the intersection of a wall and the floor?
Two non-parallel planes intersect along a line — the baseboard edge.
Answer: A line.
Three Planes at a Corner
Two walls and the floor all meet at a room corner. What is the intersection of all three?
Each pair forms a line; all three lines converge at a single point — the corner.
Answer: A point (the corner).
Common Pitfalls
Forgetting Parallel Objects Don't Intersect
Parallel lines or parallel planes never meet — their intersection is the empty set.
Real-Life Applications
Architecture & Room Corners
Every room is defined by wall-floor and wall-wall intersections. The line where a wall meets the ceiling, the point where two walls meet the floor — understanding intersections in 3D space is essential for construction and structural engineering.
Practice Quiz
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