/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 Give a geometric description of ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 3

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$y=0, \quad z=0$$

Short Answer

Expert verified
The set of points is the x-axis in three-dimensional space.

Step by step solution

01

Identify the First Equation

The given exercise is made of two equations. The first equation is \( y = 0 \). This equation implies that the y-coordinate of every point in the set is zero, meaning that all the points lie on the plane where \( y = 0 \). This is the xz-plane in three-dimensional space.
02

Analyze the Second Equation

The second equation is \( z = 0 \). This means that the z-coordinate of every point is zero. Thus, all points satisfying this equation lie on the plane where \( z = 0 \), which is the xy-plane in three-dimensional space.
03

Combine Both Conditions

To satisfy both equations \( y = 0 \) and \( z = 0 \), a point must lie at the intersection of the planes \( y = 0 \) and \( z = 0 \). The intersection of these two planes is a line.
04

Describe the Intersection

The line of intersection is where both \( y \) and \( z \) equal zero, focusing solely on the x-axis. Thus, every point on this line has coordinates \( (x, 0, 0) \), for any real number \( x \).
05

Geometric Description

The geometric description of the set of points is the x-axis itself, as all points (x, 0, 0) lie precisely there.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Understanding Planes in 3D Geometry
In three-dimensional space, a plane is a flat, two-dimensional surface that extends infinitely. Each plane can be defined by an equation, like in our original equations: \( y = 0 \) and \( z = 0 \). When you see \( y = 0 \), it means every point on this plane has a zero in their y-coordinate, forming what we call the **xz-plane**.
  • This implies that you can move anywhere along the x and z axes while staying on this plane, but you won't move in the y direction.
  • Similarly, the equation \( z = 0 \) signifies all points that lie within the **xy-plane**, where the z-coordinate is zero.
  • Here, you can move freely along the x and y axes, but the z component remains static.
These planes are fundamental in understanding 3D spaces, helping define boundaries and areas within those spaces.
Lines of Intersection in 3D Space
When two planes intersect, they form a line. To visualize this, imagine two pieces of paper placed at an angle, touching along a shared edge.
  • For the equations we considered, \( y = 0 \) and \( z = 0 \), the planes intersect where both of these conditions are met.
  • Geometrically, their intersection is a line where both the y and z coordinates are zero, which maps to our x-axis in this case.
  • This line, noted as \( (x, 0, 0) \), shows that regardless of the x value, both y and z will remain zero on the intersection.
The concept of lines of intersection helps determine regions and pathways created by overlapping planes, acting as a guide like arrows in 3D navigation.
Role of Coordinate Axes in Geometry
The coordinate axes in three-dimensional space form the foundation for locating points and directions. These axes are the x, y, and z axes. Here, they play distinct roles:
  • The **x-axis** runs horizontally, providing a basis of reference for points where both y and z coordinates are zero.
  • In this problem, the intersection of the planes \( y = 0 \) and \( z = 0 \) aligns perfectly along this x-axis.
  • The significance here is in how these axes can define not just points but entire lines or segments in space, aiding in visualizing how multiple geometrical elements integrate.
By understanding these axes, you learn critical skills for mapping out the spatial relationships that are key to mastering 3D geometry.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use a CAS to plot the surfaces Identify the type of quadric surface from your graph. $$y-\sqrt{4-z^{2}}=0$$

Use a CAS to plot the surfaces Identify the type of quadric surface from your graph. $$\frac{x^{2}}{9}-1=\frac{y^{2}}{16}+\frac{z^{2}}{2}$$

Show that the volume of the segment cut from the paraboloid $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=\frac{z}{c}$$ by the plane \(z=h\) equals half the segment's base times its altitude.

Write inequalities to describe the sets. The upper hemisphere of the sphere of radius 1 centered at the origin

Find the areas of the triangles whose vertices are given in Exercises \(41-47.\) $$A(0,0), \quad B(-2,3), \quad C(3,1)$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.