Problem 45 - Thomas Calculus

Question

Find the distance from the plane \(x+2 y+6 z=1\) to the plane \(x+2 y+6 z=10 .\)

Step-by-Step Solution

Answer

The distance between the two planes is \(\frac{9}{\sqrt{41}}\).

Step 1: Understand the Form of Plane Equations

The given planes are both in the form of \( Ax + By + Cz = D \). For both planes, the coefficients \( A = 1 \), \( B = 2 \), and \( C = 6 \) are identical, indicating that the planes are parallel.

Step 2: Use the Formula for Distance Between Parallel Planes

To find the distance between two parallel planes \( Ax + By + Cz = D_1 \) and \( Ax + By + Cz = D_2 \), use the formula:\[\text{Distance} = \frac{|D_2 - D_1|}{\sqrt{A^2 + B^2 + C^2}}\]Here, \( D_1 = 1 \) and \( D_2 = 10 \).

Step 3: Calculate the Numerator

Compute the absolute difference between the constants \( D_2 \) and \( D_1 \):\[|D_2 - D_1| = |10 - 1| = 9\]

Step 4: Calculate the Denominator

Compute the square root of the sum of the squares of the coefficients \( A \), \( B \), and \( C \):\[\sqrt{A^2 + B^2 + C^2} = \sqrt{1^2 + 2^2 + 6^2} = \sqrt{1 + 4 + 36} = \sqrt{41}\]

Step 5: Compute the Distance

Substitute the calculated values into the distance formula:\[\text{Distance} = \frac{9}{\sqrt{41}}\]

Step 6: Simplify the Result (if required)

The distance between the two planes is \( \frac{9}{\sqrt{41}} \). As the problem doesn't specify further simplification or approximation, this is the final result.

Key Concepts

Plane Geometry

In the world of mathematics, plane geometry focuses on flat shapes, like lines, circles, and polygons. A plane itself is a flat, two-dimensional surface that extends infinitely in all directions. It can be visualized as a thin sheet, like an endless piece of paper.

To describe a plane mathematically, we often use an equation in the form of \( Ax + By + Cz = D \), where \( A \), \( B \), \( C \), and \( D \) are constants.

• \( A \), \( B \), and \( C \) are coefficients that determine the plane's orientation in three-dimensional space.
• \( D \) is a constant that positions the plane relative to the origin.

Planes can be parallel, intersecting, or coincidental. Understanding these properties helps us analyze spatial relationships between different planes.

Parallel Planes

When we talk about parallel planes, we're referring to two planes that never intersect. They remain the same distance apart everywhere, much like railway tracks.

This unique property of parallelism arises when the normal vectors (determined by the coefficients \( A \), \( B \), and \( C \) in the plane equation \( Ax + By + Cz = D \)) of the two planes are proportional or identical.

• If the coefficients are precisely the same, as with the equations \( x + 2y + 6z = 1 \) and \( x + 2y + 6z = 10 \), the planes are parallel.

This shows us that parallel planes share the same orientation, only differing by their relative position in space, defined by different constant terms, \( D_1 \) and \( D_2 \). Understanding this underpins many geometrical and real-world applications, such as constructing buildings or bridges.

Distance Formula

Finding the distance between parallel planes is a straightforward application of the distance formula. It's essential in various fields, such as engineering and architecture, where precision is crucial.

For the given parallel planes equation, \( Ax + By + Cz = D_1 \) and \( Ax + By + Cz = D_2 \), the formula is:
\[\text{Distance} = \frac{|D_2 - D_1|}{\sqrt{A^2 + B^2 + C^2}}\]

• The numerator \(|D_2 - D_1|\) is the absolute difference between the constant terms \(D_1\) and \(D_2\).
• The denominator \(\sqrt{A^2 + B^2 + C^2}\) represents the magnitude of the normal vector to the planes.

By substituting in the known values, we can compute the exact distance. The result here was \( \frac{9}{\sqrt{41}} \), which isn’t necessarily a whole number, but it precisely represents the shortest distance between the two planes. Understanding how to use this formula not only helps solve mathematical problems but also equips you with tools useful in real-world applications.