Problem 13 - Thomas Calculus

Question

In Exercises \(13-18\) , describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. $$ (a)x \geq 0, \quad y \geq 0, \quad z=0 \quad \text { b. } x \geq 0, \quad y \leq 0, \quad z=0 $$

Step-by-Step Solution

Answer

Part (a) is the first quadrant on the XY-plane, and part (b) is the fourth quadrant on the XY-plane.

Step 1: Understand the meaning of the inequalities and equation

The exercise provides inequalities and an equation involving coordinates \((x, y, z)\). We need to understand what the values of \(x\), \(y\), and \(z\) represent in three-dimensional space and how the given conditions affect the set of points. The condition \(z = 0\) indicates that the points lie on the XY-plane.

Step 2: Analyze part (a) – Determine the region for x ≥ 0, y ≥ 0, z = 0

For part (a), \((x \geq 0, y \geq 0, z = 0)\), we are looking at the first quadrant of the XY-plane where \(x\) and \(y\) are non-negative. This region is part of the XY-plane where both coordinates are positive or zero, and \(z\) is always zero.

Step 3: Describe the set of points for part (a)

The set of points that satisfy the conditions of part (a) form a subset on the XY-plane. The subset is known as the first quadrant, including the positive x-axis and y-axis, i.e., \((x, y, 0)\) such that \((x \geq 0, y \geq 0)\).

Step 4: Analyze part (b) – Determine the region for x ≥ 0, y ≤ 0, z = 0

For part (b), \((x \geq 0, y \leq 0, z = 0)\), we consider the fourth quadrant of the XY-plane where \(x\) is non-negative and \(y\) is non-positive. This specifies a region of the XY-plane where \(x\) is positive or zero, \(y\) is zero or negative, and \(z\) equals zero.

Step 5: Describe the set of points for part (b)

The set of points that meet part (b)'s conditions form a portion of the XY-plane, specifically the fourth quadrant. This includes the positive x-axis and the negative y-axis, expressed as \((x, y, 0)\) such that \((x \geq 0, y \leq 0)\).

Key Concepts

Three-Dimensional Space

Three-dimensional space is where we live! It's the space all around us that can be defined using three coordinates: \(x\), \(y\), and \(z\). These coordinates let us pinpoint any location in this space by giving us directions along three different lines:

• The \(x\)-axis: this is like moving side to side.
• The \(y\)-axis: think about moving forward and backward.
• The \(z\)-axis: this is the up and down direction.

When points are defined in three-dimensional space, they have all three of these values. For instance, the point \((3, 5, -2)\) shows a spot that's three steps right, five steps forward, and two steps down from the origin (0, 0, 0).

It's important because it helps us map and visualize objects, structures, and spaces in our everyday world.

Inequalities in Space

When we talk about inequalities in space, we are setting rules or regions where our points can be. These might say that a point can only exist in a place where certain conditions are met, like where \(x \geq 0\) or \(y \leq 4\).

For example, the inequality \(x \geq 0\) tells us that points must exist in a region where \(x\) is not negative. This means we're looking at the whole right side of the \(y\)-axis in a two-dimensional graph or half of the space, including it, in three-dimensional worlds.

Similarly, if \(z = 0\), all points must lie on the flat \(xy\)-plane, like a piece of paper resting flat on a table. Understanding these inequalities lets us draw and interpret shapes and volumes, creating regions that satisfy the conditions provided.

Quadrants in XY-plane

The XY-plane is a flat surface formed by the \(x\)-axis and \(y\)-axis. We can divide this plane into four main parts called quadrants. Each quadrant has different properties based on the signs (positive or negative) of \(x\) and \(y\):

• First Quadrant: Here, both \(x\) and \(y\) values are positive. We find it in the top-right corner.
• Second Quadrant: \(x\) is negative, but \(y\) is positive, situated in the top-left corner.
• Third Quadrant: Both \(x\) and \(y\) are negative, seen in the bottom-left.
• Fourth Quadrant: \(x\) is positive, and \(y\) is negative, located at the bottom-right.

In the exercise provided, when they mention the XY-plane specifically with \(z = 0\), we're only working on this flat plane where points are constrained by these quadrants. Each quadrant helps us identify where a point is or belongs based on its \(x\), \(y\), and sometimes \(z\) values. It's like a map guiding us to the right location, helping us understand and solve problems related to coordinates and locations on graphs.