Problem 22 - Thomas Calculus

Question

In Exercises \(17-24,\) describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. $$ \text { a. } x=y, \quad z=0 \quad \text { b. } x=y, \text { no restriction on } z $$

Step-by-Step Solution

Answer

a. A line on the xy-plane where \( x = y \) and \( z = 0 \); b. A vertical plane where \( x = y \) with no restriction on \( z \).

Step 1: Identify the Set for Part a

The problem presents two conditions: \( x = y \) and \( z = 0 \). The set of points satisfying \( x = y \) lies on the plane \( y = x \) in three-dimensional space. The condition \( z = 0 \) restricts this plane to the xy-plane, indicating a line where \( x = y \) and \( z \) is constantly 0.

Step 2: Describe the Set for Part a

In three-dimensional space, the set defined by \( x = y \) and \( z = 0 \) is a line where each point has equal x and y coordinates and is positioned on the xy-plane. This line is essentially the line \( y = x \) that exists entirely on the xy-plane.

Step 3: Describe the Set for Part b

For the set in part b, the only restriction is \( x = y \). There is no specific condition on \( z \), allowing \( z \) to take any real value. Thus, the set of points forms an entire vertical plane in space where each point satisfies \( y = x \), but with unrestricted \( z \) values.

Step 4: Summarize the Results

In summary, part a forms a line on the xy-plane where \( x = y \) and \( z = 0 \). In part b, the set forms a plane parallel to the xz-plane and yz-plane with \( x = y \) and no restriction on \( z \), allowing \( z \) to vary freely.

Key Concepts

Three-Dimensional Coordinates

Three-dimensional coordinates enable us to describe a point's location in space using three values, typically represented as \( (x, y, z) \). Each coordinate specifies the point’s position along one of the three axes: the x-axis, y-axis, and z-axis.

The x-axis runs horizontally, the y-axis runs vertically, and the z-axis runs perpendicular to the xy-plane, extending upwards and downwards. Together, these three coordinates give a complete description of any point's exact location in 3D space.

• The **x-coordinate** tells you how far left or right a point is from the origin.
• The **y-coordinate** specifies the distance above or below the origin.
• The **z-coordinate** indicates the height above or the depth below the xy-plane.

By applying these coordinates, we can explore how equations such as \(x = y\) or \(z = 0\) define specific geometrical figures like lines, planes, or in general, any shape in space.

Inequalities in Space

An inequality involving coordinates describes a set of points in space that satisfy certain conditions. For example, inequalities can define regions in space rather than precise locations, allowing a greater degree of freedom for the values that x, y, and z coordinates can take.

In the context of the exercise, inequalities determine the extent of a set within the three-dimensional space.

• When **x = y**, this defines a plane where the x-coordinate equals the y-coordinate for every point.
• If there are no restrictions on z, then solutions can extend infinitely in the positive and negative z-directions.

This capability to describe areas as well as lines and planes makes inequalities a powerful tool in geometry, allowing us to define and visualize large portions of space easily. Being able to express conditions that must be met by the coordinates helps in understanding the spatial relationships and constraints in the three-dimensional world.

Planes and Lines in 3D

In 3D geometry, planes and lines are fundamental concepts. Lines are represented as a set of points that extend indefinitely in two opposite directions, typically defined by equations such as \(x = y\) and \(z = 0\) for the example of a line in the xy-plane.

Planes, on the other hand, are flat, two-dimensional surfaces that extend indefinitely as well. They are described by equations with no restrictions on one of the variables. For instance, a plane can be formed by an equation like \(x = y\), allowing for any value of z, which means the plane extends vertically.

Understanding these can help in visualizing how different regions and figures exist and interact in three-dimensional space.

• A **line in 3D** typically occurs where two planes intersect or by setting a particular condition such as \(z = 0\).
• A **plane in 3D** is an infinitely large flat surface that can be determined by equations derived from conditions like \(x = y\), without additional restrictions.

These geometrical representations are not bound by physical limits like in the real world, making them endless and idealized in the theoretical realm of mathematics.