Problem 5 - Algebra 2

Question

Graph each point in coordinate space. $$ (5,0,-2) $$

Step-by-Step Solution

Answer

In three-dimensional space, plot the point \(5,0,-2\) by moving 5 units along the X-axis, 0 units along the Y-axis, and -2 units along the Z-axis (into the page).

Step 1: Understanding the Coordinate Space

Note that the given point, \(5,0,-2\), is in a three-dimensional coordinate space. This means it has an X-coordinate (5), Y-coordinate (0), and Z-coordinate (-2). The X-axis is horizontal, the Y-axis is vertical, and the Z-axis comes out towards or goes in away from us.

Step 2: Plotting the X-coordinate

From the origin, move 5 units in the positive direction along the X-axis, since the X-coordinate is positive.

Step 3: Plotting the Y-coordinate

From the new position, do not move along the Y-axis since the Y-coordinate is 0. Stay at your current position on the X-axis.

Step 4: Plotting the Z-coordinate

Move 2 units in the negative direction along the Z-axis (into the page) since the Z-coordinate is -2. Mark this position as the point \(5,0,-2\).

Step 5: Labeling and Drawing the Point

Label the final position with the point's coordinates \(5,0,-2\) and draw a dot to represent the point on the graph.

Key Concepts

Three-Dimensional Coordinate System

A three-dimensional coordinate system is an extension of the two-dimensional Cartesian grid you might already be familiar with. While a flat map consists of two dimensions, representing width (X-axis) and height (Y-axis), the three-dimensional system adds a new depth dimension (Z-axis).

This extra dimension allows for the representation of any point in 3D space. Imagine you are in an empty room. Points in this room can be identified by how far to the side (X), how far up (Y), and how deep into the room (Z) they are. Coordinates are usually written in the form \(x, y, z\), representing these three directions respectively.

Graphing in 3D

Graphing in 3D can initially seem daunting, but it follows the same principles as 2D graphing, just with an extra step. Once you've identified the coordinates, you plot each one, axis by axis.

First, locate the position on the X-axis, then adjust your position along the Y-axis. Finally, you'll plot the Z-coordinate, which can be visualized as moving in or out of the page or screen on which you’re drawing. Using this systematic approach, plotting points in three-dimensional space one coordinate at a time becomes a manageable task.

Coordinate Axes

In the context of a three-dimensional system, there are three coordinate axes: X, Y, and Z.

• X-axis: usually displayed horizontally.
• Y-axis: displayed vertically.
• Z-axis: represents the depth and is perpendicular to both the X and Y axes.

Remember, positive directions are typically rightward (X-axis), upward (Y-axis), and outwards, towards the viewer (Z-axis), while the negative directions are the opposite. In plotting points, you move along these axes based on the sign and magnitude of each coordinate.

3D Space Visualization

Visualizing 3D space on a flat surface requires imagination and practice. One strategy is to think of viewing a corner of a room from the inside. The corner where the walls and floor meet serves as the origin.

From this vantage point, one wall becomes the X-axis, the other the Y-axis, and the ceiling or floor can represent the Z-axis. By practicing visualizing objects and points in spaces like this, you can enhance your ability to understand and graph complex structures in three-dimensional coordinate space.