Problem 15 - Precalculus with Calculus Previews

Question

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

Step-by-Step Solution

Answer

The solution is the union of the xy-plane, yz-plane, and xz-plane.

Step 1: Understand the equation

We're given the equation \(xyz = 0\) that describes the set of points \((x, y, z)\) in a three-dimensional space. To satisfy this equation, at least one of the coordinates must be zero.

Step 2: Analyzing conditions for x, y, z

There are three possibilities to satisfy \(xyz = 0\): (1) \(x = 0\) and \(y, z\) can be any real number, (2) \(y = 0\) and \(x, z\) can be any real number, or (3) \(z = 0\) and \(x, y\) can be any real number.

Step 3: Visualizing the solutions

The conditions result in three planes in 3-space. These are the planes: \(x = 0\), representing the yz-plane; \(y = 0\), representing the xz-plane; and \(z = 0\), representing the xy-plane.

Step 4: Description of the set of points

Thus, the set of points \((x, y, z)\) that satisfy the equation \(xyz = 0\) is the union of the three coordinate planes in 3-space.

Key Concepts

3-space

Three-dimensional space, often referred to as 3-space, is an extension of the familiar two-dimensional plane. It includes an additional dimension, allowing us to define points with three coordinates \(x, y, z\). This extra dimension makes it possible to explore and describe the position of objects in a more comprehensive way. For example:

• The x-axis represents horizontal distance.
• The y-axis represents vertical distance.
• The z-axis adds depth or height.

We commonly encounter 3-space in geometry when we work with problems involving volume, surface area, or spatial configuration. The ability to express a point using three coordinates is fundamental to navigating and understanding three-dimensional shapes, structures, and equations.

equation solution

Finding solutions to equations in 3-space can involve multiple possibilities and requires understanding their spatial implications. Take our equation \(xyz = 0\):

• At least one of the coordinates \(x, y, z\) must be zero for the equation to hold true.
• There are three separate conditions that satisfy the equation, meaning the point could lie on one of the three planes.

This type of problem showcases how an equation can describe various geometric entities within space. In simpler terms, solutions tell us about specific areas or positions where the equation results are valid. This is crucial in mathematics and helps in designing and solving real-world 3D problems such as engineering structures, computer graphics, and physics simulations.

visualization in 3D

Visualizing equations in 3D is a powerful tool to grasp complex spatial relationships. In our case, visualizing where \(xyz = 0\) helps to identify the resulting geometric shapes: three distinct planes.

• The plane \(x = 0\) can be viewed as the yz-plane. It is the collection of all points where the x-coordinate is zero.
• The plane \(y = 0\) corresponds to the xz-plane, capturing all points with a y-coordinate of zero.
• The plane \(z = 0\) is the xy-plane, for all points having a z-coordinate of zero.

Visualizing these planes helps us understand that the equation covers areas in 3-space rather than just specific points. By mapping out these planes, students can better comprehend both the solution set and the spatial properties of the equation at hand. This ability enhances skills crucial in fields like architecture, computer modeling, and advanced calculus.