Intersections of Planes

Understanding how planes intersect is crucial in architecture, engineering, and mathematics. Large structures are built in three dimensions, making spatial relationships and intersections important for stability and aesthetics.

Investigate: Intersections of Planes

• Use templates or models to visualize how planes intersect.
• Consider different ways planes can intersect:
  • Single intersection: Two planes meet in a line.
  • Coincident planes: Two planes completely overlap.
  • Parallel planes: Two planes do not intersect.

Key Concepts

• Two planes can intersect in a line, be coincident, or be parallel.
• Three planes can intersect in a point, a line, be coincident, or have no solution.
• To determine the intersection of planes, analyze the normal vectors and solve the corresponding system of equations.

Example

Determine the intersection of the planes:

• π1: \(2x + y + z = 1\)
• π2: \(x + y + z = 6\)

Here's how we can find the solution:

1. Find the normal vectors: \( n_1 = [2,1,1] \) and \( n_2 = [1,1,1] \).
2. Since the normals are not parallel, the planes intersect in a line.
3. Use elimination to solve the system and find the parametric equations for the line of intersection.

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Summary

• If planes have non-parallel normal vectors, they intersect in a line.
• If planes have parallel normal vectors but different constants, they are distinct and do not intersect.
• If planes have parallel normal vectors and the same equation, they are coincident.