Question

Buildings You can think of your classroom as a model of six planes: the ceiling, the floor, and the four walls. a. Find two planes that intersect. b. Find two planes that do not intersect. c. Is it possible for three planes to intersect? If so, find the intersection.

Short answer

a. Ceiling and wall intersect; b. Ceiling and floor do not intersect; c. Three planes (ceiling and two walls) intersect at a corner.

Step-by-step solution

Identify Intersecting Planes

In a classroom, planes that intersect are those that meet along an edge. For instance, the ceiling and one of the walls intersect along the edge where they meet. Similarly, the floor and one of the walls intersect at the line where they meet.

Identify Non-Intersecting Planes

Two planes that do not intersect are parallel to each other. In the classroom model, an example would be the ceiling and the floor or opposite walls. These planes do not meet at any edge.

Examine Intersection of Three Planes

To find where three planes intersect, consider where a corner of the classroom is formed. Three intersecting planes would be the ceiling, one wall, and an adjacent wall meeting at a corner. The intersection of these three planes forms a single point at the corner.

Key concepts

Intersecting Planes

Imagine standing in a classroom; the walls, floor, and ceiling around you represent planes. Intersecting planes occur when two of these planes meet. An intersection happens when two planes collide at a line. For example, the ceiling plane meets each of the wall planes along a line, and that's where they intersect.
In a classroom, similar intersections occur where each wall meets the floor. At these meeting points, a distinct line forms which is the edge between two planes. It’s like the corner where two sheets of paper touch when held upright beside each other.

• A good way to picture intersecting planes is by visualizing an open book. Each page represents a plane, and where the pages meet is their line of intersection.

Parallel Planes

Parallel planes are a bit like lifelong friends that never meet. In your classroom, parallel planes are the ceiling and the floor, and sometimes even opposing walls. They exist side by side but never intersect or touch.
You might wonder, how is it confirmed that two planes are parallel? Simply put, if they maintain the same distance apart everywhere and will never meet, they are parallel.

• An everyday example can be seen in train tracks, which, if extended to infinity, are designed to never meet. They are parallel precisely because they don't intersect.

Intersection of Planes

Thinking a little more complex, consider what happens when three planes intersect. Instead of forming just a line like two intersecting planes, the intersection of three planes often results in a point. Picture a corner in the room where the ceiling meets two adjoining walls; this point is where all three planes intersect.
This might call to mind the corner of a cube where its edges meet, each edge belonging to one of the three intersecting planes.

• So, while two planes crossing form a line, add a third plane, and you get a point where all three planes meet.

Recognizing the intersections of multiple planes helps in understanding how spaces are constructed, both in theory and real-life architecture.