Two Planes Intersect In Exactly One Point

Two Planes Intersect in Exactly One Point: A Deep Dive into Linear Algebra and Geometry

Understanding how planes intersect is fundamental to various fields, from computer graphics and game development to structural engineering and crystallography. While many scenarios involve planes intersecting along a line, the case where two planes intersect at precisely one point is less intuitive but equally important. This article delves deep into the mathematical concepts underpinning this specific intersection, exploring the linear algebra and geometric implications. We'll unravel the conditions that lead to this unique intersection and illustrate with practical examples.

Defining the Problem: Planes in Three-Dimensional Space

Before we tackle the specific case, let's establish a solid foundation. In three-dimensional Euclidean space (ℝ³), a plane is defined by a point and a normal vector. The normal vector, denoted by n, is a vector perpendicular to the plane. A common equation for a plane is given by:

n · (r - r₀) = 0

Where:

• n is the normal vector of the plane.
• r is a position vector representing any point on the plane.
• r₀ is the position vector of a known point on the plane.
• '·' denotes the dot product.

This equation essentially states that the vector connecting a point on the plane (represented by r - r₀) and the known point is orthogonal to the normal vector.

Alternatively, we can express the plane's equation in Cartesian coordinates as:

Ax + By + Cz + D = 0

Where A, B, and C are the components of the normal vector n, and D is a constant.

When Two Planes Intersect at a Single Point

Two planes intersect at a single point under a very specific condition: they are not parallel, and they are not coincident (i.e., they are distinct planes). Let's explore why this is the case.

Parallel Planes: No Intersection or Infinite Intersection

If two planes are parallel, their normal vectors are parallel (or anti-parallel). This means one normal vector is a scalar multiple of the other. In this scenario, there are two possibilities:

• No intersection: The planes are distinct and never meet.
• Infinite intersection: The planes are coincident; they are essentially the same plane.

In neither case do the planes intersect at a single point.

Non-parallel Planes: Line of Intersection

When the normal vectors of two planes are not parallel, they intersect. The intersection is not a point but a line. This line contains all points that satisfy both plane equations simultaneously. Finding the parametric equation of this line involves solving a system of linear equations, usually using methods like Gaussian elimination or matrix reduction.

The Unique Case: The Single Point Intersection

The single point intersection requires a subtle shift in perspective. While two planes typically intersect in a line, consider this: what if we introduce a third plane? This third plane, when not parallel to the line of intersection of the first two, will intersect that line at exactly one point.

Think of it this way: The first two planes define a line. The third plane acts as a "selector," picking out a single point from that line. Therefore, the single-point intersection isn't inherent to just two planes; it arises from the interaction of three planes.

Mathematically, this is represented by a system of three linear equations in three variables (x, y, z). If this system has a unique solution, it represents the coordinates of the single intersection point. If the system has no solution, the three planes do not intersect at a common point. If the system has infinitely many solutions, it implies a line (or a plane) of intersection.

Illustrative Examples

Let's illustrate these concepts with examples.

Example 1: Two planes intersecting in a line

Plane 1: x + y + z = 1 Plane 2: 2x - y + z = 3

These planes are not parallel (their normal vectors (1,1,1) and (2,-1,1) are not parallel). Solving this system of equations will yield a line of intersection, not a single point.

Example 2: Three planes intersecting at a single point

Plane 1: x + y + z = 6 Plane 2: x - y + 2z = 7 Plane 3: 2x + y - z = 3

Solving this system of three equations simultaneously (using substitution, elimination, or matrix methods) will produce a unique solution for x, y, and z, representing the single point of intersection.

Example 3: Three planes with no common intersection

Plane 1: x + y + z = 1 Plane 2: x + y + z = 2 Plane 3: x + 2y + 3z = 4

Notice that Plane 1 and Plane 2 are parallel and distinct – a contradiction. This system has no solution, indicating no common intersection point for all three planes.

Practical Applications

The concept of plane intersections has far-reaching applications in various fields:

• Computer Graphics and Game Development: Determining if a ray intersects a polygon (which can be represented by planes) is crucial for rendering and collision detection.
• Structural Engineering: Analyzing the stability and stress distribution in structures involves understanding the intersection points of various planar elements.
• Crystallography: The arrangement of atoms in a crystal lattice can be described using planes, and their intersections are key to understanding crystal structure and properties.
• Robotics: Path planning for robots often involves determining the intersection points of planes representing obstacles and the robot's workspace.
• Medical Imaging: Three-dimensional medical imaging techniques rely on interpreting intersections of planar slices to reconstruct complex structures.

Conclusion: A Foundation for Advanced Concepts

Understanding the conditions under which two planes intersect at a single point (specifically through the interaction of three planes) is crucial for grasping more advanced concepts in linear algebra, geometry, and various applied fields. While the single point intersection of two planes is not directly possible, the intersection is a fundamental concept with significant implications in various domains, highlighting the power of mathematical modelling in real-world problems. This deep dive provides a solid foundation for tackling more complex spatial reasoning and problem-solving tasks. Further exploration could involve analyzing different methods for solving systems of linear equations and applying these concepts to specific practical applications.