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Chapter 12: Problem 17

In Exercises \(17-24\) , describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. \begin{equation} \text {a. }x \geq 0, \quad y \geq 0, \quad z=0 \quad \text { b. } x \geq 0, \quad y \leq 0, \quad z=0 \end{equation}

Short Answer

Expert verified
a: First quadrant on XY-plane; b: Fourth quadrant on XY-plane.

Step by step solution

01

Understanding Inequality a

The inequalities for part (a) are \( x \geq 0 \), \( y \geq 0 \), and \( z = 0 \). This represents all the points in the \(xy\)-plane (where \(z = 0\)) that lie in the first quadrant. In the Cartesian coordinate system, this includes all points where the x-coordinate and y-coordinate are non-negative, and the z-coordinate is fixed at zero.
02

Plotting the Region for a

Visualize the XY-plane where \( z = 0 \). The inequality \( x \geq 0 \) means start from the y-axis to the right; \( y \geq 0 \) means start from the x-axis upwards. Together, these represent all points in the first quadrant of the XY-plane.
03

Describing the Set for a

The set is all points on or above the x-axis and to the right of the y-axis in the plane where \( z=0 \). Geometrically, it's the entire first quadrant of the XY-plane when projected onto the Z-axis.
04

Understanding Inequality b

The inequalities for part (b) are \( x \geq 0 \), \( y \leq 0 \), and \( z = 0 \). This represents all the points in the \(xy\)-plane that lie in the fourth quadrant. This plane represents all points in space where the x-coordinate is non-negative and the y-coordinate is non-positive.
05

Plotting the Region for b

Visualize again the XY-plane where \( z = 0 \). The inequality \( x \geq 0 \) still means include the positive x-values, but \( y \leq 0 \) means to include all values of y that are zero or negative. This defines the fourth quadrant.
06

Describing the Set for b

The set is all points on or below the x-axis and to the right side of the y-axis in the plane where \( z=0 \). Geometrically, it's the entire fourth quadrant of the XY-plane when projected onto the Z-axis.

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Key Concepts

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

Coordinate System
To understand points and shapes in three-dimensional space, we must grasp the concept of the coordinate system. The Cartesian coordinate system is the most common method to describe the position of a point using three coordinates: x, y, and z.
  • The x-axis runs horizontally.
  • The y-axis runs vertically.
  • The z-axis adds depth, running front to back.
Each point in space can be represented by a set of three numbers (x, y, z), defining a unique position. The intersection of these axes forms the origin, noted as (0, 0, 0). This system simplifies understanding geometric shapes and relationships in three-dimensional space.
Inequalities in Geometry
Inequalities in geometry define a range of values that coordinates can take. In this context, an inequality specifies the regions in a space where points can exist.
When you see an inequality such as \( x \geq 0 \), it tells us that points are on or to the right of the y-z plane, while \( y \leq 0 \) indicates the points lie on or below the x-z plane.
  • Inequalities like \( z = 0 \) specify an exact plane rather than a range.
  • These inequalities help us define specific sections or regions in 3D space.
Understanding these is critical for describing geometric areas and translating mathematical information into visual representations of space.
Quadrants in 3D Space
In three-dimensional space, quadrants become octants due to the addition of the z-axis. The xy-plane alone divides the space into four quadrants similar to the two-dimensional plane.
  • The first quadrant in 2D is where both x and y are positive; in three dimensions, it extends into regions where x, y, and z values can create more complex combinations.
  • For example, with \( z = 0 \), we look at a 2D slice of 3D space, simplifying the analysis to familiar 2D quadrants.
In the exercises, we are essentially working within the planes where \( z = 0 \), analyzing two-dimensional quadrants within this slice. The challenge is to visualize these segments from the larger 3D perspective, imagining them laid out as on a flat surface.
Cartesian Plane Visualization
Visualizing equations or inequalities on a Cartesian plane involves plotting them correctly to see the shape or area they describe. For equations like \( z = 0 \), visualization becomes about understanding a flat plane (XY) in three-dimensional space.
  • Visualize the first quadrant by focusing on non-negative x and y.
  • The fourth quadrant becomes clear when visualizing non-negative x and non-positive y.
By imagining a cut through the 3D space where the plane \( z = 0 \) resides, we create a two-dimensional space similar to graph paper but one embedded in a larger 3D context. This visualization skill translates abstract algebraic statements into pictorial forms that can be easily understood and analyzed.

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Most popular questions from this chapter

Find an equation for the set of all points equidistant from the planes \(y=3\) and \(y=-1\)

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