Problem 13 - APEX Calculus

Question

Describe the region in space defined by the inequalities. \(x \geq 0, y \geq 0, z \geq 0\)

Step-by-Step Solution

Answer

The region described is the first octant of 3D space, where all coordinates are non-negative.

Step 1: Understand the Inequalities

The given inequalities are:- \( x \geq 0 \)- \( y \geq 0 \)- \( z \geq 0 \)Each inequality indicates that the respective coordinate (\(x\), \(y\), or \(z\)) is non-negative.

Step 2: Visualize the Inequalities in 3D Space

In a 3-dimensional coordinate system, being non-negative for \(x\), \(y\), and \(z\) means the values are on or above the plane of each respective axis. This limits the region to the first octant where all coordinates are non-negative.

Step 3: Define the Enclosed Region

Since all three coordinates must be greater than or equal to zero, the region is the set of all points where \(x\), \(y\), and \(z\) are zero or positive. This is the volume in space known as the first octant.

Key Concepts

First Octant

In three-dimensional space, the first octant is a specific region where all three coordinate values—\(x\), \(y\), and \(z\)—are non-negative. Think of each axis as dividing the space into different parts. When all three of these coordinates are greater than or equal to zero, the point lies in the part of the space we call the first octant.

• The \(x\)-axis is horizontal, going from left to right.
• The \(y\)-axis is also horizontal but goes from front to back.
• The \(z\)-axis is vertical, pointing upwards.

These axes together divide space into eight parts or octants. The first octant, being where all coordinates are positive or zero, is where we find points like (2, 5, 3) or (0, 0, 0). This region helps us visualize and practice working with 3D coordinate systems easily. By focusing on the first octant, we can learn to think spatially without the complication of negative coordinates.

Inequalities in Space

The space in mathematics involving inequalities is more than just numbers on a line. In three-dimensional space, inequalities help define regions by setting limits on where a point can exist relative to its coordinates.

In our case, the inequalities \(x \geq 0\), \(y \geq 0\), and \(z \geq 0\) restrict points to the first octant. These inequalities tell us that every point within this region must be above or on each coordinate plane:

• \(x \geq 0\): points must be on or to the right of the \(yz\)-plane.
• \(y \geq 0\): points must be on or in front of the \(xz\)-plane.
• \(z \geq 0\): points must be on or above the \(xy\)-plane.

By visualizing these planes and their boundaries, we can better understand how points sit in three-dimensional space. Each inequality draws a border, and the first octant is the only section where all three conditions are systematically satisfied. These make 3D inequalities both a practical and visual tool for defining regions and limitations in space.

Non-negative Coordinates

Non-negative coordinates center around the concept that the values of \(x\), \(y\), and \(z\) are equal to or greater than zero. In 3D geometry, this non-negativity confines us to certain areas within the coordinate system.

Consider each axis as a foundation:

• The \(x\)-axis: when \(x\geq0\), we stay on the side that lies to the right of the origin.
• The \(y\)-axis: \(y\geq0\) means staying on the side in front of the origin.
• The \(z\)-axis: \(z\geq0\) keeps us above the origin, akin to standing on the floor and moving upwards.

All coordinates being non-negative is crucial for the first octant. This area is significant not just for theoretical math but also for real-world contexts where negative measures are impossible, like physical dimensions in engineering or architecture. Understanding non-negative coordinates means you comprehend what it means for a point to have positive positioning in space—a basic skill in several fields associated with 3D modeling and simulation.