Question

Show that the (shortest) distance between two planes \(\mathbf{n} \cdot \mathbf{p}=d_{1}\) and \(\mathbf{n} \cdot \mathbf{p}=d_{2}\) with \(\mathbf{n}\) as normal is \(\frac{\left|d_{2}-d_{1}\right|}{\|\mathbf{n}\|}\).

Short answer

The shortest distance is \( \frac{|d_2 - d_1|}{\|\mathbf{n}\|} \).

Step-by-step solution

Identify the Normal Vector

Both planes share the same normal vector \( \mathbf{n} \), as they are parallel. The equations of the planes are given as \( \mathbf{n} \cdot \mathbf{p} = d_1 \) and \( \mathbf{n} \cdot \mathbf{p} = d_2 \). This normal vector \( \mathbf{n} \) is crucial for finding the shortest distance between the planes.

Find a Point on Each Plane

Let's find a point \( \mathbf{p}_1 \) on the first plane \( \mathbf{n} \cdot \mathbf{p}_1 = d_1 \) and a point \( \mathbf{p}_2 \) on the second plane \( \mathbf{n} \cdot \mathbf{p}_2 = d_2 \). For simplicity, we can assume that \( \mathbf{p}_1 = \frac{d_1}{\|\mathbf{n}\|^2} \mathbf{n} \) and \( \mathbf{p}_2 = \frac{d_2}{\|\mathbf{n}\|^2} \mathbf{n} \), as these satisfy the respective plane equations.

Calculate the Distance Between the Two Points

The vector connecting \( \mathbf{p}_1 \) and \( \mathbf{p}_2 \) is \( \mathbf{p}_2 - \mathbf{p}_1 \). This vector can be written as \[ \mathbf{p}_2 - \mathbf{p}_1 = \frac{d_2}{\|\mathbf{n}\|^2}\mathbf{n} - \frac{d_1}{\|\mathbf{n}\|^2}\mathbf{n} = \frac{d_2 - d_1}{\|\mathbf{n}\|^2}\mathbf{n} \]. The distance between these two points is the magnitude of this vector.

Compute the Magnitude of the Connecting Vector

The magnitude of the vector \( \mathbf{p}_2 - \mathbf{p}_1 \) is \[ \left\| \mathbf{p}_2 - \mathbf{p}_1 \right\| = \left\| \frac{d_2 - d_1}{\|\mathbf{n}\|^2}\mathbf{n} \right\| = \frac{|d_2 - d_1|}{\|\mathbf{n}\|^2} \times \|\mathbf{n}\| = \frac{|d_2 - d_1|}{\|\mathbf{n}\|} \] since the magnitude of \( \mathbf{n} \) \( (\|\mathbf{n}\|) \) cancels appropriately.

Key concepts

Normal Vector

In the world of mathematics and physics, a 'normal vector' is essential for understanding the orientation of planes and surfaces. When we talk about parallel planes, the normal vector, denoted as \( \mathbf{n} \), provides crucial information about their orientation in space. The normal vector is always perpendicular to the plane it describes, like an arrow sticking out from a flat surface.

In the context of the exercise, both planes share the same normal vector, \( \mathbf{n} \), because they are parallel. This fact simplifies calculations for finding the shortest distance between them. Recognizing the normal vector allows us to understand that although the planes are parallel (and thus never intersect), they share a consistent orientation.

Vector Magnitude

The 'vector magnitude', often represented by \( \|\mathbf{n}\| \), measures the length or size of the vector. Think of it as the distance from the starting point of the vector to its endpoint in geometric terms. The magnitude gives us more than just the length though—it plays a vital role in normalizing vectors and in calculations involving distance and direction.

In the solution of this exercise, we rely on knowing the magnitude of the normal vector to determine the shortest distance between planes. Specifically, it's used in the equation \( \frac{|d_2 - d_1|}{\|\mathbf{n}\|} \), where the magnitude helps normalize the distance formula, confirming that the shortest pathway between such parallel planes is orthogonal to their surfaces.

Plane Equations

A plane equation is a mathematical expression that defines a flat, two-dimensional surface within three-dimensional space. These equations are usually written in the form of \( \mathbf{n} \cdot \mathbf{p} = d \). Here, \( \mathbf{n} \) is the normal vector perpendicular to the plane, \( \mathbf{p} \) is any point on the plane, and \( d \) is a constant that provides positional information about the plane's orientation relative to the origin.

In this exercise, we're given two plane equations: \( \mathbf{n} \cdot \mathbf{p} = d_1 \) and \( \mathbf{n} \cdot \mathbf{p} = d_2 \). These represent two parallel planes. The different constants \( d_1 \) and \( d_2 \) indicate how far these planes are from the origin, and comparing them helps find how far apart the planes truly are in space.

Vector Algebra

Vector Algebra is a fundamental part of understanding how vectors behave and interact. This branch of mathematics involves operations such as addition, subtraction, and multiplication with vectors. It helps solve problems involving forces, velocities, or, as in this exercise, planes.

In the context of the problem, we use basic vector algebra to manipulate and understand the relationship between the planes. By finding a vector that connects points on each plane and calculating its magnitude, we uncover the shortest distance between these parallel surfaces. This application demonstrates how vector algebra is crucial for solving geometric problems, aiding in visualizing and navigating three-dimensional space.