Question

In Exercises 89-92, verify that the two planes are parallel, and find the distance between the planes. $$ \begin{array}{l} x-3 y+4 z=10 \\ x-3 y+4 z=6 \end{array} $$

Short answer

The two planes \(x - 3y + 4z = 10\) and \(x - 3y + 4z = 6\) are parallel and the distance between them is approximately 0.7843.

Step-by-step solution

Verify the Parallelism

First observe the two plane equations: \(x - 3y + 4z = 10\) and \(x - 3y + 4z = 6\). The normals of the planes are given by the coefficients of x, y, and z in each equation. Both planes have the same normal vector, which is (1,-3,4). Since the normal vectors are the same, this shows that the two planes are parallel.

Calculate the Distance

The formula to calculate the distance between parallel planes is \(d = |c1 - c2| \div ||n||\). Here, c1 and c2 are the constants on the right side of the plane equations and n is the magnitude of normal vector. For the given planes the constants are 10 and 6, so |c1 - c2| = |10 - 6| = 4. The magnitude of normal vector ||n|| is \(\sqrt{(1)^2 + (-3)^2 + (4)^2}\) = \(\sqrt{26}\). Finally, plug these values into the formula to obtain the distance as \(d = 4 \div \sqrt{26}\)

Simplify the Result

The distance calculated in the previous step can be simplified further into decimal form for easier comprehension. The decimal form of \( \sqrt{26} \) is approximately 5.1, so the distance \(d = 4 \div 5.1\) , which approximately equals to 0.7843.

Key concepts

Parallel Planes in Calculus

Understanding the concept of parallel planes is foundational in vector calculus. Parallel planes can be thought of as 'floors in a multistory building', never touching or intersecting no matter how far they extend. To verify if two planes are parallel, you need to look at their normal vectors. A normal vector is a three-dimensional vector that is perpendicular to the plane. Parallel planes share the same normal vector, pointing in the same direction, signifying that they never intersect.

When given two equations of planes, you check their parallelism by comparing the coefficients of the variables x, y, and z. In the exercise, the coefficients are identical for both planes, which directly indicates parallelism. It is crucial to recognize these patterns, as they are common indicators of parallel planes in problems across calculus.

Normal Vector of a Plane

In three-dimensional space, the normal vector is a key concept that helps in defining a plane's orientation. It's the 'invisible rod' that sticks out perpendicularly from the plane's surface. The components of the normal vector correspond to the coefficients of x, y, and z in the plane's equation, in the format of Ax + By + Cz + D = 0.

For example, in the given planes from the exercise, the normal vector is determined by the coefficients (A, B, C) or (1, -3, 4). This vector is critical in various operations involving planes, such as determining parallelism and calculating distances between planes. Understanding how to extract and use the normal vector is a fundamental skill in calculus and related fields.

Plane Equations

The equation of a plane is a mathematical representation that describes all the points that lie on that plane. It's a way to map out the 'invisible sheet' that extends infinitely in two dimensions. The general form of a plane equation in three-dimensional space is Ax + By + Cz + D = 0, where A, B, and C are not all zero. These coefficients are directly related to the normal vector of the plane, while D is the constant term.

To compute the distance between two parallel planes, you can use the given plane equations, extracting the constant terms and the normal vector. These elements elucidate the planes’ relative positions in space, allowing for the calculation of the shortest distance between them. Enhancing your understanding of plane equations is crucial when studying three-dimensional geometry and solving real-world spatial problems.