Problem 17 - 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. $$ \text { b. } x \geq 0, \quad y \geq 0, \quad z=0 \quad \text { b. } x \geq 0, \quad y \leq 0, \quad z=0 $$

Step-by-Step Solution

Answer

Region 1: First quadrant of xy-plane; Region 2: Fourth quadrant of xy-plane.

Step 1: Understanding Inequalities for Region 1

The inequalities for the first set are given as \(x \geq 0\), \(y \geq 0\), and \(z = 0\). These describe points in space where \(x\) is non-negative, \(y\) is non-negative, and \(z\) is exactly 0. Essentially, this is the first quadrant of the xy-plane.

Step 2: Interpreting the Geometric Set for Region 1

Since \(z = 0\), the points lie on the xy-plane. Because \(x \) and \(y \) are both non-negative, the region for these points is the first quadrant of the plane \(z=0\) (the xy-plane).

Step 3: Understanding Inequalities for Region 2

For the second set, the inequalities are \(x \geq 0\), \(y \leq 0\), \(z = 0\). They describe points where \(x\) is non-negative, \(y\) is non-positive, with \(z\) again exactly 0. This represents the fourth quadrant of the xy-plane.

Step 4: Interpreting the Geometric Set for Region 2

Here, with \(z = 0\), the points also lie on the xy-plane. The conditions \(x \geq 0\) and \(y \leq 0\) define the fourth quadrant of this plane.

Key Concepts

Quadrants in 3D Space

Understanding quadrants in three-dimensional space can be quite helpful for visualizing and solving various types of problems involving geometry and algebra. In 3D space, just like in 2D geometry, quadrants are a way to segment space. However, instead of just four quadrants like on a flat plane, 3D space is divided into eight octants. This is because you have an additional axis, the z-axis, to consider.

The octants in 3D are designated based on the signs of the coordinates \(x, y, z\).

• In the first octant, all coordinates are positive.
• The second octant typically has a positive x and z, and a negative y.
• Each subsequent octant continues to alter the signs between positive and negative for \(x, y, z\).

The cartesian 3D coordinate system enables us to break down complex structures by understanding in which segment (octant) a point or object lies.

Unlike planes, where you have 2D quadrants, the presence of the z-axis adds complexity. This means while a point might have two positive coordinates, the third could be negative. Understanding this helps differentiate locations and establish relationships in space.

Inequalities in Coordinate Systems

Inequalities are fundamental in defining regions or spaces, especially in coordinate systems like the carteisan plane in 2D and space in 3D. By applying inequalities, we can determine specific areas where a set of points exist. In the context of coordinate systems, inequalities help us establish boundaries.

• A simple inequality like \(x \geq 0\) denotes that x lies on or to the right of the yz-plane.
• Similarly, \(y \geq 0\) implies y is on or above the xz-plane.
• The condition \(z = 0\) fixes all points to the xy-plane, ignoring any elevation.

For example, given the inequalities \(x \geq 0\), \(y \geq 0\), and \(z = 0\), this translates to the first quadrant of the xy-plane where both x and y are non-negative.

Inequalities like these are not just restricting values but creating well-defined spaces. Such definitions are essential for visualizing problems, as in physics or engineering, where they describe fields or boundaries in three-dimensional structures.

Geometric Interpretation of Equations

Equations and inequalities provide more than just algebraic constraints—they often have a meaningful geometric interpretation in space. When working with equations such as \(z = 0\), it's essentially describing an entire plane parallel to the xy-plane. This equation implies any variation along the z-axis is non-existent.

When combined with inequalities, equations can define very specific regions, shapes, or planes:

• The combination \(x \geq 0\), \(y \geq 0\), \(z = 0\) describes a flat, 2D region situated in the 3D space—specifically, the first quadrant of the xy-plane.
• Conversely, \(x \geq 0\), \(y \leq 0\), \(z = 0\) defines the fourth quadrant of the xy-plane.

Understanding these spatial descriptions helps in visualizing data, interpreting 3D models, and solving geometric problems more effectively.

Interpreting equations geometrically allows one to move beyond mere numbers, visualizing the algebraic relationships as actual segments or sections of 3D space—critical in fields like computer graphics, spatial analysis, and physics.