/*! 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 6 In Exercises \(5-26,\) sketch th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 29: Problem 6

In Exercises \(5-26,\) sketch the graphs of the given equations in the rectangular coordinate system in three dimensions. $$2 x-y-z+6=0$$

Short Answer

Expert verified
Plot the intercepts (-3,0,0), (0,6,0), and (0,0,6) in a 3D space and connect them to sketch the plane.

Step by step solution

01

Rewrite the Equation in Standard Form

Given the equation, rewrite it in a form that's easy to interpret for graphing. Our equation is \(2x - y - z + 6 = 0\). To rewrite it in a more standard form, let's solve for \(z\):\[z = 2x - y + 6\]. This form shows \(z\) as a function of \(x\) and \(y\), which is convenient for graphing in three dimensions.
02

Identify the Intercepts

Finding the intercepts will help in sketching the plane. **x-intercept**: Set \(y = 0\) and \(z = 0\), solve for \(x\):\[2x + 6 = 0\Rightarrow x = -3\]. The x-intercept is \((-3, 0, 0)\). **y-intercept**: Set \(x = 0\) and \(z = 0\), solve for \(y\):\[-y + 6 = 0\Rightarrow y = 6\]. The y-intercept is \((0, 6, 0)\). **z-intercept**: Set \(x = 0\) and \(y = 0\), solve for \(z\):\[-z + 6 = 0\Rightarrow z = 6\]. The z-intercept is \((0, 0, 6)\).
03

Plot the Intercepts in 3D Coordinate System

Using the intercepts found, plot the points \((-3, 0, 0)\), \((0, 6, 0)\), and \((0, 0, 6)\) in a 3D coordinate system. These points are where the plane intersects each of the axes.
04

Sketch the Plane

Connect the intercepts with straight lines to form a triangle. This triangle approximates the section of the plane that passes through these intercepts. Extend the plane through the coordinate system to visualize the infinite plane represented by \(2x - y - z + 6 = 0\).

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.

Rectangular Coordinate System
In mathematics, the rectangular coordinate system is a foundational concept that helps us visualize and locate points in space. It consists of three perpendicular axes that intersect at a point called the origin. Here's how it works:
  • The x-axis runs horizontally and typically represents the width.
  • The y-axis runs vertically and usually represents height.
  • The z-axis comes out perpendicularly from the intersection of the x and y axes and represents depth.
This three-dimensional setup is crucial for 3D graphing, allowing us to plot points and shapes precisely in space. When using this coordinate system, the position of any point can be represented as a set of three numbers \(x, y, z\), corresponding to distances along these axes from the origin.
Plane Equations
A plane in a three-dimensional space can be described using a linear equation. The general form of a plane equation is \(ax + by + cz + d = 0\), where \(a, b,\) and \(c\) are coefficients that form a vector perpendicular to the plane, and \(d\) is a constant. Let's break it down:
  • The coefficients \(a, b,\) and \(c\) determine the orientation of the plane by dictating its normal vector.
  • Solving for \(z\) gives us the equation in a form \(z = mx + ny + c\), where it's clear how \(x\) and \(y\) influence \(z\).
In our specific equation \(2x-y-z+6=0\), rewriting it as \(z = 2x - y + 6\) helps visualize how changes in \(x\) and \(y\) affect \(z\). Understanding plane equations is essential for diving deep into three-dimensional geometry.
Intercepts
Intercepts are points where a graph crosses the axes in the coordinate system. They provide essential reference points when sketching a graph, particularly in three dimensions:
  • x-intercept: This is where the graph cuts the x-axis, found by setting \(y\) and \(z\) to zero.
  • y-intercept: This is where the graph crosses the y-axis, found by setting \(x\) and \(z\) to zero.
  • z-intercept: This is where the graph intersects the z-axis, found by setting \(x\) and \(y\) to zero.
For the plane described by \(2x - y - z + 6 = 0\), the intercepts are \((-3, 0, 0)\), \(0, 6, 0)\), and \(0, 0, 6)\). By plotting these intercepts, you can better visualize where the plane crosses each axis.
Three-Dimensional Geometry
Three-dimensional geometry is the branch of mathematics that studies the properties and relations of points, lines, surfaces, and solids in three dimensions. Here's why it's significant:
  • It introduces the concept of depth, which provides a more realistic representation of space, essential in fields like physics and engineering.
  • With three dimensions, you can explore various geometric shapes and their properties, such as cubes, spheres, and planes.
  • Understanding 3D geometry is crucial for visualizing spatial relationships and modeling real-world situations.
When sketching a 3D plane using a coordinate system, we connect intercepts and visualize the infinite extension of planes. This helps grasp complex spatial concepts, making three-dimensional geometry an exciting and powerful tool in the study of mathematics.

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

Find the indicated volumes by double integration. Evaluate \(A=\int_{-\pi / 6}^{\pi / 6} \int_{\sqrt{2}}^{2 \sqrt{\cos 2 \theta}} r d r d \theta,\) the area outside the circle \(r=\sqrt{2}\) and inside the lemniscate \(r^{2}=4 \cos 2 \theta,\) using polar coordinates.

Find the indicated volumes by double integration. Evaluate the triple integral \(\int_{-2}^{2} \int_{1}^{2} \int_{1}^{e} \frac{x y^{2}}{z} d z d x d y\) by integrating over \(z, x,\) and \(y\) in that order (similar to a double integral).

Solve the given problems. Describe the surface for which the cylindrical coordinate equation is (a) \(\theta=3,\) (b) \(z=r^{2}\).

Sketch the indicated curves and surfaces. The surface of a small hill can be roughly approximated by the equation \(z\left(2 x^{2}+y^{2}+100\right)=1500,\) where the units are meters. Draw the surface of the hill and the contours for \(z=3 \mathrm{m}\) \(z=6 \mathrm{m}, z=9 \mathrm{m}, z=12 \mathrm{m},\) and \(z=15 \mathrm{m}\).

Evaluate the given functions. $$H(p, q)=p-\frac{p-2 q^{2}-5 q}{p+q} ; \text { find } H(p, q+k)$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Decision Maths

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.