Problem 37 - Thomas Calculus

Question

Write inequalities to describe the sets in Exercises \(35-40\) 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 XY-plane

The XY-plane is the two-dimensional plane that contains all the points where the z-coordinate is zero. Think of it as the 'floor' in a three-dimensional space where you usually plot x, y, and z axes.

Step 2: Identify Points on the XY-plane

Points that are on the XY-plane have z-coordinates equal to zero. Therefore, if a point is to lie on the XY-plane, it must satisfy the equation \( z = 0 \).

Step 3: Determine Points Below the XY-plane

Points that are below the XY-plane will have a z-coordinate that is less than zero, meaning \( z < 0 \).

Step 4: Combine the Conditions

The half-space consisting of the points on and below the XY-plane is described by combining the conditions for both on and below: \( z \leq 0 \). This inequality describes all points where z is equal to or less than zero.

Key Concepts

XY-plane

The XY-plane is a fundamental concept in three-dimensional space. It's the flat, two-dimensional surface that you get when the third dimension, represented by the z-coordinate, is zero. Imagine a sheet of paper lying perfectly flat on a tabletop. This sheet represents the XY-plane.

• All points on this plane have coordinates in the form { ( x, y, 0 ) }, where x and y can be any real numbers, but z is always zero.
• This makes it easy to visualize because we're only dealing with two axes – the X-axis and the Y-axis.

Understanding the XY-plane is crucial because it's used as a reference point for plotting and analyzing points and shapes in three-dimensional space. It also helps in identifying other geometric structures by serving as a base.
Thus, when dealing with points on the XY-plane, remember that their z-coordinate won't change, as it's fixed at zero.

three-dimensional space

Three-dimensional space is where all three coordinates (x, y, and z) come into play. It’s what we live in every day, filled with width, height, and depth. This is often referred to as 3D space.

• In mathematics, we use it to provide a coordinate system that locates points within this space, giving a clear understanding of their position relative to each other.
• If you imagine walking along the floor (XY-plane), stepping up or down adds the third dimension, described by the z-coordinate.

In 3D space, you have:

• The X-axis running horizontally from side to side.
• The Y-axis running horizontally from front to back.
• The Z-axis running vertically up and down.

Together, these axes form a three-dimensional coordinate system that helps us describe the position and relationships of points and shapes. It is common in virtual 3D modeling and real-world applications like physics and engineering.

coordinate systems

Coordinate systems are essential for locating points in different spaces, including two and three-dimensional spaces.
In a coordinate system, each point is defined by a set of numbers called coordinates, which represent its position relative to a set of axis lines.

• In a 2D system, like the plane, you have two axes: X (horizontal) and Y (vertical), and a coordinate appears as (x, y).
• In 3D systems, you include a third axis, Z, vertically, so coordinates appear as (x, y, z).

Every point in space can be identified by its unique coordinates, making it easier for us to talk about, manipulate, and calculate positions. Coordinate systems are used extensively in math, physics, engineering, and computer graphics.