Problem 31 - Thomas Calculus

Question

Write inequalities to describe the sets in Exercises \(29-34\) The half-space consisting of the points on and below the \(x y\) -plane

Step-by-Step Solution

Answer

The inequality is \(z \leq 0\).

Step 1: Understand the Concept of Half-Space

To determine the inequality describing a half-space, we first need to recognize the part of the 3-dimensional space we're dealing with. The exercise refers to the points on and below the \(xy\)-plane.

Step 2: Define the Region of Interest

In three-dimensional space, the \(xy\)-plane is characterized by all points where the \(z\)-coordinate is zero. Thus, the half-space \'on and below\' the \(xy\)-plane includes all points where the \(z\)-coordinate is less than or equal to zero.

Step 3: Write the Inequality

The inequality that represents all points on and below the \(xy\)-plane is given by \(z \leq 0\). This captures all points where the \(z\)-value is zero (on the plane) or negative (below the plane).

Key Concepts

Half-Space

In the realm of 3D geometry, half-space represents a division of space by a plane into two infinite sections. Imagine slicing through a cake with a flat knife: you're left with two halves. - In 3D, planes such as the xy-plane act like the knife, creating half-spaces. - Each half-space consists of all points either above or below the plane of interest. For the exercise at hand, we are interested in the half-space below and including the xy-plane. This half-space includes all points whose z-coordinate is less than or equal to zero. Understanding half-spaces helps in visualizing 3D constraints and regions that meet specific conditions defined by inequalities.

XY-plane

The xy-plane is a fundamental concept in 3D coordinates. It extends infinitely along the x and y axes, essentially forming a flat sheet where every point only moves horizontally or vertically. - When we talk about a point being "on the xy-plane," it means the z-coordinate for that point is zero. - Conversely, a point "below the xy-plane" refers to one where z is negative. This plane serves as a reference point, almost like the ground level, in a 3D coordinate system. Recognizing how points relate to this plane is crucial in setting up equations and understanding spatial relationships in exercises involving products of inequalities.

3D Coordinates

3D coordinates consist of three numbers, generally represented as \((x, y, z)\).These describe a unique point in three-dimensional space:- The x-coordinate indicates the horizontal position,- The y-coordinate corresponds to the vertical position,- The z-coordinate shows the depth or elevation. In our exercise, adjusting the z-coordinate allows us to determine if a point lies on, above, or below the xy-plane. In practice, changing any of these coordinates shifts a point in space, reflecting the versatile nature of 3D representations, crucial for visualizing spatial systems.

Inequality Representation

Inequality representation is using inequalities to define a region in space based on certain constraints. For example, the inequality\(z \leq 0\)clearly defines all points "on and below" the xy-plane:- Points meeting this inequality condition will include or lie below the "ground" set by the xy-plane.- Points with a z-coordinate greater than zero would fail to satisfy this inequality, as they lie above the plane.Setting up inequalities allows us to describe specific areas in 3D space comprehensively. These mathematical notations are particularly powerful as they offer a clear method to define and solve geometry problems by marking boundaries and the regions of interest.