Problem 44 - Calculus Early Transcendentals: Pearson New International Edition

Question

An object's position \(P\) changes so that its distance from \((1,2,-3)\) always equals its distance from \((2,3,2)\). Find the equation of the plane on which \(P\) lies.

Step-by-Step Solution

Answer

The equation of the plane is \(x + y - 5z = 3\).

Step 1: Understanding the Problem

We need the location of a point, denoted as \(P(x, y, z)\), that is equidistant to two given points, \((1, 2, -3)\) and \((2, 3, 2)\). This requirement describes the locus of all such points, forming a geometrical plane.

Step 2: Apply the Distance Formula

Recall that the distance between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) in 3D space is given by the formula:\[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]Apply this formula to find the distance from \(P(x, y, z)\) to both given points.

Step 3: Set the Distances Equal

For the point \(P\) to be equidistant, set the expressions for the distances from \((1, 2, -3)\) and \((2, 3, 2)\) equal to each other:\[\sqrt{(x - 1)^2 + (y - 2)^2 + (z + 3)^2} = \sqrt{(x - 2)^2 + (y - 3)^2 + (z - 2)^2}\]

Step 4: Square Both Sides

Remove the square roots by squaring both sides of the equation:\[(x - 1)^2 + (y - 2)^2 + (z + 3)^2 = (x - 2)^2 + (y - 3)^2 + (z - 2)^2\] Simplifying both sides will result in a linear equation in terms of \(x\), \(y\), and \(z\).

Step 5: Simplify the Expression

Expand and simplify both sides:\[(x^2 - 2x + 1 + y^2 - 4y + 4 + z^2 + 6z + 9) = (x^2 - 4x + 4 + y^2 - 6y + 9 + z^2 - 4z + 4)\]Cancel out common terms and simplify to find the equation of the plane.

Step 6: Derive the Equation of the Plane

After simplification, the terms reduce to:\[-2x - 4y + 6z + 14 = -4x - 6y - 4z + 17\]Bringing terms to one side gives:\[2x + 2y - 10z - 3 = 0\]Further simplification results in:\[x + y - 5z = 3\] which is the equation of the plane.

Key Concepts

Distance Formula

The distance formula is a crucial tool in mathematics, especially in geometry and algebra. It helps calculate the distance between two points in a coordinate system.
In 3D space, the distance formula is given by:

• \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \)

This equation essentially extends the familiar 2D distance formula by adding an extra dimension, making it possible to calculate distances between points in three-dimensional space.
Each part of the formula corresponds to the difference in each of the x, y, and z coordinates, respectively, squared and summed before taking the square root.
Being comfortable with using this formula allows you to effectively solve problems involving the distances between points in 3D space.

3D Space

3D space, also known as three-dimensional space, is a geometric setting in which three values (dimensions) determine the position of an element.
These dimensions are commonly noted as x, y, and z.
Understanding 3D space is crucial as it extends the concept of geometrical interpretation from two dimensions to an additional third coordinate. This extra dimension adds depth to our visualizations and calculations:

• X-axis: Represents width and moves horizontally.
• Y-axis: Represents height and moves vertically.
• Z-axis: Represents depth, thus adding a layer of complexity and realism.

Familiarity with navigating and calculating in 3D space is key in diverse fields ranging from physics to computer graphics, where spatial awareness goes beyond just flat surfaces.

Geometrical Plane

A geometrical plane in 3D space can be thought of as a flat, two-dimensional surface extending infinitely within three-dimensional space.
Each point on the plane satisfies a specific linear equation of the form \( ax + by + cz = d \), where a, b, and c are the coefficients determining the plane's orientation.
Planes play a fundamental role in geometry, as they serve as a building block for various shapes and figures.

• They can be used to define surfaces found in solid shapes.
• They help explain relationships and intersections between different geometrical shapes.

By understanding and utilizing plane equations, you can solve problems involving intersections, projections, and distances in three-dimensional contexts.

Equidistant Points

Equidistant points are a set of points that maintain equal distance from two or more fixed points.
In a problem involving equidistant points, like the one presented, each point on the resultant plane is equidistant to the two given fixed points.
This concept can be visualized as a balance or a midpoint between objects in space:

• They form a plane, line, or surface based on the number of fixed points involved.
• Solving for equidistant points often requires the use of the distance formula, setting the distances from each fixed point equal.

Recognizing and calculating equidistant points is important in various practical applications, such as network design and architectural layout, where uniform spacing is essential.