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Chapter 8: Problem 22

Describe in words the region of \(\mathbb{R}^{3}\) represented by the equations or inequalities. \(y=-2\)

Short Answer

Expert verified
The region is a plane in \(\mathbb{R}^{3}\) parallel to the xz-plane at \(y = -2\).

Step by step solution

01

Understand the Equation

The equation given is \(y = -2\). This equation is in the form of \(y = b\), where \(b\) is a constant. This represents a specific set of points where the \(y\)-coordinate is always \(-2\).
02

Identify the Relationship in 3D Space

Since this is only an equation in terms of \(y\), there are no restrictions on the \(x\) or \(z\) coordinates. As a result, \(x\) and \(z\) can take any real number values. This indicates we are dealing with a two-dimensional subset of \(\mathbb{R}^{3}\).
03

Describe the Region

In three-dimensional space, the equation \(y = -2\) describes a plane parallel to the \(xz\)-plane (which is the plane where \(y = 0\)) but shifted in the \(y\)-direction to \(y = -2\). This plane extends infinitely in both the \(x\) and \(z\) directions.

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Key Concepts

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

Planes in 3D Space
When we talk about a plane in 3D space, we're discussing a flat, two-dimensional surface that extends infinitely. Unlike lines, which are one-dimensional, planes add another layer of dimensionality. They're like infinite sheets of paper that can float anywhere in three-dimensional space.
The equation \(y = -2\) is a prime example of a plane in action. This particular equation means that every point on this plane has a \(y\)-coordinate of \(-2\). There are no restrictions on the \(x\) or \(z\) coordinates, allowing them to span all real numbers.
  • This plane is parallel to the \(xz\)-plane, which is the plane you get when \(y = 0\).
  • By shifting the plane to \(y = -2\), you've simply moved it two units along the \(y\)-axis without changing its orientation.
Remember, while we can visualize a plane on paper as having edges, in mathematical reality, a plane doesn't stop. It just keeps going!
Coordinate Systems
Understanding coordinate systems is key to mastering 3D geometry. The most common system used in 3D geometry is the Cartesian coordinate system, consisting of three axes: \(x\), \(y\), and \(z\). Each point in space is defined by these three values, forming the structure around which all 3D geometry revolves.
The equation \(y = -2\) is an equation within this system. Here:
  • The \(x\)-axis typically measures horizontal distances.
  • The \(y\)-axis measures vertical distances, although it can be in any direction relative to the problem at hand.
  • The \(z\)-axis often measures depth, adding the third dimension that distinguishes 3D space from 2D planes.
Visualizing this axis-system helps with understanding how planes like \(y = -2\) fit into and extend within the 3D space. This system is also versatile, accommodating transformations such as translations and rotations, allowing us to analyze and adjust positions within the 3D space effectively.
3D Geometry Concepts
3D geometry introduces various concepts that extend those in two dimensions, enhancing our ability to describe the universe. At its core, 3D geometry deals with objects having width, depth, and height, such as spheres, cubes, and, reflecting our original exercise, planes.
In the case of the plane defined by \(y = -2\):
  • We perceive it as a two-dimensional slice of the broader 3D environment.
  • It works alongside other elements determined by equations or inequalities to create or divide spaces within \(\mathbb{R}^3\).
  • Such planes can interact to form or intersect with other shapes and planes.
Analyzing these relationships is central to solving complex spatial problems, and understanding the inherent nature of planes and their roles offers a foundation for more intricate geometrical concepts.

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Most popular questions from this chapter

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