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Chapter 12: Problem 55

Explain what is wrong with the statement. The \(x y\) -plane has equation \(x y=0\)

Short Answer

Expert verified
The XY-plane should be defined by \(z=0\), not \(xy=0\); \(xy=0\) implies the x and y axes, not a plane.

Step by step solution

01

Understanding the XY-plane

The statement asks to describe the equation of the XY-plane. The XY-plane in a three-dimensional space is defined by all points where the z-coordinate is zero (i.e., \(z = 0\)). It encompasses all possible combinations of x and y coordinates.
02

Analyzing the Given Equation

The equation \(xy=0\) suggests locations where either the x-coordinate or the y-coordinate is zero. This implies two sets of linear arrays through higher-dimensional space: the x-axis (where y = 0) and the y-axis (where x = 0). These lines in isolation do not fill the entire two-dimensional XY-plane.
03

Identifying the Error

The statement mistakes the conditions for lines through the origin for a plane. The plane is defined by z = 0, not the intersections of lines described by \(xy=0\). \(xy=0\) rather describes the union of the x-axis and y-axis in the XY-plane.

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Key Concepts

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

Coordinate Planes
Coordinate planes are fundamental components in the study of multivariable calculus. In a three-dimensional space, there are three primary coordinate planes: the XY-plane, YZ-plane, and XZ-plane. Each coordinate plane is defined by holding one of the three coordinates constant.
  • XY-plane: Here, the z-coordinate is constant at zero, meaning the equation of the XY-plane is given by \( z = 0 \). This plane includes all points where any combination of the x and y coordinates are possible, effectively forming a flat surface that extends infinitely in both the x and y directions.

  • YZ-plane: This plane has the x-coordinate constant at zero, described by the equation \( x = 0 \). It represents all possible combinations of y and z coordinates.

  • XZ-plane: Defined by the equation \( y = 0 \), this plane fixes the y-coordinate at zero, allowing x and z coordinates to vary.

Understanding these planes is crucial, as they provide reference surfaces from which we can describe and analyze space and objects within it.
Equations of Planes
Equations of planes in three-dimensional space are key in understanding how different surfaces are represented mathematically. The general equation of a plane is given as \( ax + by + cz = d \), where \( a, b, \) and \( c \) are coefficients that determine the orientation of the plane in space.
  • The normal vector: A vector perpendicular to the plane, often denoted as \( \langle a, b, c \rangle \), helps define the plane's orientation.

  • Plane equation example: The equation \( z = 0 \) is a simple plane equation where only the z-component is fixed. This specific equation represents the XY-plane.

  • Misinterpretation example: The equation \( xy = 0 \) incorrectly describes a plane. Instead, it defines lines where either \( x = 0 \) or \( y = 0 \), which are individual lines on the XY-plane rather than the full extent of the plane itself.

Creating accurate equations for planes is essential in solving problems involving intersections, angles, and distances in three-dimensional space.
Three-Dimensional Space
Three-dimensional space is one of the most captivating areas explored in multivariable calculus. It introduces a new level of complexity, involving length, width, and height. Each point in this space can be represented by three coordinates: (x, y, z).
  • Axes of reference: The x, y, and z axes intersect perpendicularly and are crucial for defining the three-dimensional Cartesian coordinate system.

  • Understanding planes: A plane can be visualized as a flat surface extending infinitely in two directions within this space, crucial for segmentation and modeling of geometrical shapes.

  • Intersections and objects: Intersections between planes, lines, and other objects are common tasks, requiring comprehension of spatial geometry and the role each element plays.

Mastering three-dimensional space is essential for various fields, including physics, engineering, and computer graphics, where spatial awareness and manipulation are required.

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Most popular questions from this chapter

Give an example of: A linear function whose contours are lines with slope 2.

Does the line parallel to the \(y\) -axis through the point (2,1,4) intersect the plane \(y=5 ?\) If so, where?

Explain what is wrong with the statement. The distance from (2,3,4) to the \(x\) -axis is 2

Could the table of values represent a linear function? If so, find a possible formula for the function. If not, give a reason why not. $$\begin{array}{|c|c|c|c|} \hline & 1 & 2 & 3 \\ \hline 1 & 1 & 2 & 3 \\ \hline 2 & 4 & 5 & 6 \\ \hline 4 & 7 & 8 & 9 \\ \hline \end{array}$$

Are the statements true or false? Give reasons for your answer. If \(f(x, y)\) is continuous and \(f(1,0)=2,\) then $$\lim _{(x, y) \rightarrow(1,0)} f(x, y)=2.$$
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