Problem 7 - Algebra 2

Question

Graph each point in coordinate space. $$ (10,-2,-5) $$

Step-by-Step Solution

Answer

To graph the point \( (10, -2, -5) \) in a three-dimensional coordinate system, start at the origin, move 10 units along the x-axis, 2 units down along the y-axis, and 5 units into the page/screen along the z-axis. Mark and label the point at the intersection of these movements.

Step 1: Understanding the Coordinate System

Before plotting the point, understand that in a three-dimensional coordinate system, points are represented by coordinates in the form \( (x, y, z) \). The first number corresponds to the position on the x-axis (horizontal), the second number corresponds to the position on the y-axis (vertical), and the third number corresponds to the position on the z-axis, which comes out of or goes into the page/screen.

Step 2: Plotting the Point on the x-axis

Start by locating the point on the x-axis. Since the x-coordinate is 10, move 10 units in the positive direction along the x-axis.

Step 3: Plotting the Point on the y-axis

From the point on the x-axis, move parallel to the y-axis according to the y-coordinate. For the point \( (10, -2, -5) \), move 2 units in the negative direction along the y-axis.

Step 4: Plotting the Point on the z-axis

Lastly, from the location you've stopped at after the y-axis movement, move parallel to the z-axis. Because the z-coordinate is -5, move 5 units in the negative direction along the z-axis. You have now arrived at the position of the point in the three-dimensional coordinated space.

Step 5: Mark the Point

With the position determined in the three-dimensional space, draw a dot or a circle to represent the point. Label it with its coordinates, \( (10, -2, -5) \) for clarification.

Key Concepts

Coordinate System

A coordinate system serves as the foundational framework that allows us to identify the location of points in space relative to each other. Imagine a coordinate system as a map that provides a unique set of numbers or coordinates for each point in space, making it possible to pinpoint the exact position of that point.

In a three-dimensional coordinate system, commonly referred to as 3D, we have three axes: the x-axis, the y-axis, and the z-axis. These axes are typically drawn perpendicular to one another, with the x-axis representing width, the y-axis representing height, and the z-axis representing depth. The point at which all three axes intersect is the origin, denoted as (0,0,0), and serves as the reference point for all measurements in the system.

Understanding this setup is crucial because all instructions for plotting or moving in a 3D coordinate system will refer to movements along these axes. For instance, when given coordinates, they tell us how far to move in each axis's direction to find the point's location.

Three-Dimensional Graphing

The process of three-dimensional graphing involves placing points into a 3D space based upon their coordinates. Unlike two-dimensional graphing where points lie on a flat plane described by just x and y values, three-dimensional graphing incorporates an additional axis (z-axis), bringing depth into play.

To visualize this, you might think of a 3D graph as a room where the x-axis and y-axis form the floor, and the z-axis rises up like the corner of a room. Plotting a point within this 'room' requires following sequential steps:

1. Starting at the origin, move horizontally along the x-axis.
2. From that point, make a vertical movement along the y-axis.
3. Finally, move in or out of the page/screen along the z-axis.

Marking the point after these steps completes the graphing process. This framework not only depicts the location but also enables the visualization of shapes and surfaces within the three-dimensional space.

Cartesian Coordinates

The Cartesian coordinate system is one of the most widely used systems for graphing and represents points using x, y, and z coordinates. Invented by René Descartes, this system is integral for translating geometric shapes and locations onto a graph.

When working with Cartesian coordinates, each coordinate corresponds to a specific position along an axis:

• The x-coordinate indicates the distance from the y-z plane, moving left or right.
• The y-coordinate gives the distance from the x-z plane, moving up or down.
• The z-coordinate tells the distance from the x-y plane, moving forwards or backwards.

Using the point (10, -2, -5) as an example, '10' moves us along the x-axis, '-2' along the y-axis, and '-5' along the z-axis. By plotting these values sequentially, we find the point's exact location in the coordinate space. This method is crucial for disciplines like 3D modeling, physics, and mathematics where precise visualization and computation are necessary.