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Chapter 6: Problem 273

For the following exercises, find parametric descriptions for the following surfaces. Plane \(3 x-2 y+z=2\)

Short Answer

Expert verified
The parametric equations are \(x = \frac{2 + 2s - t}{3}, y = s, z = t\).

Step by step solution

01

Understand the Plane Equation

The given plane equation is \(3x - 2y + z = 2\). This equation represents a surface in three-dimensional space. To write this plane in parametric form, we need to express the coordinates \(x\), \(y\), and \(z\) as functions of two parameters.
02

Choose Free Parameters

For a plane, we can express two variables as free parameters. Let \(y = s\) and \(z = t\), where \(s\) and \(t\) are parameters. Thus, \(y\) and \(z\) vary freely over their respective ranges.
03

Solve for the Third Variable

Substitute \(y = s\) and \(z = t\) into the plane equation to solve for \(x\):\[3x - 2s + t = 2\]Isolate \(x\):\[3x = 2 + 2s - t \x = \frac{2 + 2s - t}{3}\]
04

Combine to Form Parametric Equations

Now, write the parametric equations using the parameters \(s\) and \(t\):\[x = \frac{2 + 2s - t}{3}, \quad y = s, \quad z = t\]These equations describe every point on the plane with parameters \(s\) and \(t\) varying over all real numbers.

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Key Concepts

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

Understanding the Plane Equation
In the realm of geometry, a plane equation is a mathematical representation used to describe flat surfaces in three-dimensional space. The standard form of a plane equation is given by \[ ax + by + cz = d \] where \(a\), \(b\), \(c\), and \(d\) are constants. Each plane equation uniquely corresponds to a specific plane in space.
  • The coefficients \(a\), \(b\), and \(c\) determine the orientation of the plane.
  • The constant \(d\) affects the plane's position in space relative to the origin.
For the plane described by the equation \(3x - 2y + z = 2\), these coefficients define a surface where every point satisfies the equation. The task is to express this relationship in parametric form, which allows a more versatile description through varying parameters.
Exploring Three-Dimensional Space
Three-dimensional space, often abbreviated as 3D space, is the environment where every point can be described using three coordinates: \(x\), \(y\), and \(z\). These coordinates correspond to the spatial dimensions: length, width, and height. Planes, like the one from our exercise, are crucial components that segment this vast space.
  • Planes extend infinitely and have no thickness, limiting them to just two dimensions within the three-dimensional framework.
  • They serve as crucial elements in graphics, physics, engineering, and different branches of mathematics.
Visualizing a plane involves imagining a flat sheet stretching infinitely in both the \(x\) and \(y\) directions within the 3D world. Parametric equations are tools that let us "draw" or define points on this sheet by altering parameters, a more flexible method than traditional Cartesian coordinates.
The Role of Free Parameters
Free parameters play a fundamental role in expressing surfaces like planes using parametric equations. A plane within three-dimensional space can be described by two parameters because its dimension is two-dimensional. In our exercise, this involved:
  • Choosing \(y = s\) and \(z = t\) as the two free parameters.
  • Solving for \(x\) using these parameters, thereby obtaining a parametric representation.
By assigning \(s\) and \(t\) as freely varying over all real numbers, every point on the plane can be captured. This method offers a dynamic and comprehensive way to visualize and analyze surfaces, providing a deeper understanding of their structure and behavior. Being proficient with using free parameters allows for greater flexibility in mathematical modeling and simplifies dealing with complex surfaces.

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Most popular questions from this chapter

For the following exercises, evaluate the line integrals. Let \(C\) be the line segment from point \((0,1,1)\) to point \((2,2,3) .\) Evaluate line integral \(\int_{C} y d s .\)

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