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Chapter 15: Problem 3

In Exercises \(1-12,\) sketch the graph described by the following cylindrical coordinates in three-dimensional space. $$ z=-1 $$

Short Answer

Expert verified
The graph is a horizontal plane located at \(z = -1\) in 3D space.

Step by step solution

01

Understand the Cylindrical Coordinate System

In the cylindrical coordinate system, a point in space is described by the coordinates \(r, \theta, z\). Here, \(r\) is the radius (distance from the origin in the xy-plane), \(\theta\) is the angle from the positive x-axis in the xy-plane, and \(z\) is the height above the xy-plane.
02

Interpret the Equation

The given equation is \(z = -1\). This implies that for any point on or described by this equation, the z-coordinate is a constant value of -1. The coordinates \(r\) and \(\theta\) are not specified, which means they can take any value.
03

Visualize the Graph in 3D Space

Since \(z = -1\) and \(r\) and \(\theta\) can be any value, all the points that are one unit below the xy-plane form a horizontal plane parallel to the xy-plane itself. This plane is located at \(z = -1\).
04

Sketch the Graph

To sketch the graph, imagine or draw a 3-dimensional coordinate system. Draw a plane that is horizontal and passes through the point where \(z = -1\) in the z-axis. This plane extends infinitely in the x and y directions. It's parallel to the xy-plane and located one unit below it.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

3-dimensional space
Three-dimensional space is the environment where we visualize and analyze 3D objects. It extends the two-dimensional concept of a plane by adding an additional z-axis, allowing us to model the real world more accurately. In 3D space, every point is represented using three coordinates, typically labeled as
  • **x** - The position left or right across the horizontal plane
  • **y** - The position up or down along the vertical plane
  • **z** - The depth or height above or below the xy-plane
Understanding 3D space is essential for many fields, including mathematics, physics, and engineering, as it provides context for how objects interact and relate to one another.
graph sketching
Graph sketching in cylindrical coordinates involves translating the abstract mathematical information into a visual representation. For the given exercise, with the equation \(z = -1\), the task is to sketch a plane in 3D space:
  • Identify the role of each coordinate: Here, only the z-coordinate is specified, meaning the height is constant across all x and y values
  • Visualize how this constant z value impacts the graph: This results in a flat plane extending infinitely in both the x and y directions
  • Finally, represent this visually either by drawing or imagining a plane parallel to the xy-plane, but at a set distance below it
This process helps in developing a deeper understanding of how equations describe shapes and surfaces in three-dimensional space.
cylindrical coordinate system
The cylindrical coordinate system is a three-dimensional counterpart to the two-dimensional polar coordinate system. It uses three dimensions:
  • **r** - The radial distance from the origin, showing how far the point is from the vertical z-axis
  • **\(\theta\)** - The angular coordinate, often measured in radians, indicating the direction of the point from the positive x-axis
  • **z** - The typical height or vertical displacement from the xy-plane
This system is particularly useful for dealing with problems involving symmetry around an axis, like cylinders or rotational patterns. In this exercise, because the z-coordinate is specified as a constant, it defines a plane at that height across all radial and angular positions.
horizontal plane in 3D
A horizontal plane in three-dimensional space is a flat, two-dimensional surface that runs parallel to the xy-plane. In the context of the exercise, the equation \(z = -1\) describes such a plane.
  • This specific horizontal plane is one unit below the xy-plane along the z-axis
  • It extends infinitely in the directions of the x and y axes, forming a sheet-like surface that does not curve or slope
  • The importance of recognizing this plane helps in understanding spatial relationships and positioning in 3D space
Ultimately, horizontal planes are essential for foundational geometry and more advanced applications in physics and engineering, providing critical insights into how various surfaces and structures are aligned or positioned in space.

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