Concise Answer
Yes, when two distinct, non-parallel planes intersect, their intersection is a straight line.
Understanding Plane Intersection
In three-dimensional geometry, a plane is a flat, two-dimensional surface that extends infinitely. When two such planes are not parallel to each other and are distinct, they will invariably cross paths.
Imagine two sheets of paper extending endlessly in all directions. If they are not held parallel, they will meet along a seam or a fold.
The Resulting Shape
The collection of all points that lie on both intersecting planes forms their intersection. Based on fundamental geometric principles, this intersection takes a specific shape.
As stated in geometric definitions: Intersection of two planes is always a straight line.
Why is it a Line?
The reason the intersection is a line stems from the properties of points and lines in a plane.
- Any point that lies on both intersecting planes must satisfy the equations defining both planes simultaneously.
- If two distinct points are found that satisfy both equations (i.e., lie on both planes), they must both lie on the intersection.
- A key principle in geometry is that exactly one line can be passed through any two given distinct points.
- Since the intersection contains at least two distinct points (for non-parallel planes) and any point on the intersection must lie on the line defined by these two points, the entire intersection must form a straight line.
The provided reference supports this: "Two points in a plane determine exactly one segment. Exactly one line can be passed through the two given points in a plane." While this talks about points in a single plane, the principle applies to points common to both planes, which define the unique line of intersection.
Illustrative Comparison
Consider the intersection of geometric objects:
| Geometric Objects | Typical Intersection Type |
|---|---|
| Two Distinct, Non-Parallel Lines | A single Point |
| A Line and a Plane | A single Point (or the Line itself if contained, or no intersection if parallel) |
| Two Distinct, Non-Parallel Planes | A Straight Line |
This table highlights that the intersection of two planes yields a line, a unique characteristic of their interaction in 3D space.
Practical Example
A common real-world example of the intersection of two planes is the line where two walls meet in a room. Each wall represents a plane, and the seam or corner where they join forms a straight line. Similarly, the spine of a book where two covers meet is another visual representation of this concept.
The interaction of two non-parallel planes in three dimensions fundamentally defines a straight line that extends infinitely, containing all points common to both planes.
