Problem 19 - Precalculus with Calculus Previews

Question

Describe the set of points \(P(x, y,\)z) in 3-space whose coordinates satisfy the given equation. $$ z^{2}-25=0 $$

Step-by-Step Solution

Answer

The set of points is two horizontal planes at \( z = 5 \) and \( z = -5 \).

Step 1: Simplifying the Equation

The given equation is \( z^2 - 25 = 0 \). The first step is to solve for \( z \) by adding 25 to both sides, which gives us \( z^2 = 25 \).

Step 2: Solving for z

Now, solve the equation \( z^2 = 25 \) by taking the square root of both sides. This yields two possible solutions: \( z = 5 \) and \( z = -5 \).

Step 3: Describing the Set of Points

Given the solutions \( z = 5 \) and \( z = -5 \), we can conclude that the set of points is two horizontal planes in 3-space. The first plane is at \( z = 5 \), and the second is at \( z = -5 \). In these planes, the values of \( x \) and \( y \) can be any real number, as there are no restrictions on them in the equation.

Key Concepts

Understanding 3-Space

In geometry, 3-space, also known as three-dimensional coordinate space, is a mathematical setting where we identify locations using three coordinates:

• The x-coordinate, which measures the horizontal distance along the x-axis.
• The y-coordinate, which measures the horizontal distance along the y-axis.
• The z-coordinate, which measures the vertical distance along the z-axis.

Each point in 3-space can be represented as \( P(x, y, z) \), encompassing all three dimensions.
This concept is an extension of the two-dimensional plane, adding depth and creating a 3D environment.
Applications of 3-space are vast, ranging from physics and engineering to computer graphics, helping to model and understand real-world scenarios.

Equation Solving in 3D

Equation solving in three-dimensional space often involves finding or describing sets of points that satisfy given mathematical conditions. When an equation is presented, like the one in this exercise \( z^2 - 25 = 0 \), solving it involves isolating the variable of interest.

• The equation is simplified first by moving constants to one side: \( z^2 = 25 \).
• Solving for \( z \) involves taking the square root of both sides, yielding \( z = 5 \) and \( z = -5 \).

These solutions provide specific conditions on \( z \) in the context of 3-space, influencing the nature of the points that are part of the solution set. The process showcases how equations can be manipulated to uncover relationships between coordinates in three dimensions.

Visualizing Horizontal Planes

In the context of 3-space, horizontal planes are flat, two-dimensional surfaces that extend infinitely in the x and y directions, but are fixed at certain \( z \)-values. For this exercise, the solutions \( z = 5 \) and \( z = -5 \) reveal two distinct horizontal planes.
The horizontal plane located at \( z = 5 \) signifies all points of the form \( P(x, y, 5) \), where both \( x \) and \( y \) remain unrestricted, allowing any real number values.
Similarly, the plane at \( z = -5 \) comprises points \( P(x, y, -5) \), with similar freedom in x and y values. These planes are parallel to the xy-plane and help in visualizing solutions in practical scenarios, like determining specific areas at constant heights.