Problem 1 - College Algebra and Calculus: An Applied Approach

Question

Plot the points on the same threedimensional coordinate system. (a) \((2,1,3)\) (b) \((-1,2,1)\)

Step-by-Step Solution

Answer

Point (a) \((2,1,3)\) is plotted by moving 2 units along the positive x-axis, 1 unit along the positive y-axis, and 3 units along the positive z-axis. Point (b) \((-1,2,1)\) is plotted by moving 1 unit along the negative x-axis, 2 units along the positive y-axis, and 1 unit along the positive z-axis.

Step 1: Understand Coordinates

In a 3-dimensional coordinate system, each point is identified by three numbers (x, y, z). They represent the distance from the origin to that point along the x-axis, y-axis, and z-axis, respectively. So for point (a) \((2,1,3)\), it means to move 2 units along the x-axis, 1 unit along the y-axis, and 3 units along the z-axis.

Step 2: Plot Point (a)

Starting from the origin (0,0,0), move 2 units along positive x-axis, then 1 unit along positive y-axis, and finally move 3 units along positive z-axis. Remember, positive x-axis extends from origin towards the right, positive y-axis extends upwards and positive z-axis comes out of the page. The final point, (2, 1, 3) is plotted.

Step 3: Understand the Second Set of Coordinates

For the second point (b) \((-1,2,1)\), it involves a movement along the negative x-axis and then along the positive y and z-axes. This negative sign indicates a movement in the opposite direction along the x-axis.

Step 4: Plot Point (b)

Starting again from the origin, this time move 1 unit along the negative x-axis, 2 units along positive y-axis, and 1 unit along positive z-axis. Remember, negative x-axis extends from origin towards the left. The second point, (-1, 2, 1) is plotted.

Key Concepts

Plotting Points in 3D Space

When plotting points in the 3D coordinate system, you work with three coordinates:

• x-coordinate
• y-coordinate
• z-coordinate

Each of these corresponds to a distance from the origin along the respective axis. This forms a traveling route from the origin to the point's location.
For example, to plot the point (2, 1, 3):

• Start at the origin \( (0,0,0) \).
• Move 2 units along the x-axis (right direction).
• Move 1 unit along the y-axis (upward direction).
• Finally, move 3 units along the z-axis (out of the page).

This sequence lands you at the point \( (2, 1, 3) \) within the 3D space. Similarly, follow these steps for any given point, and remember that the negative signs mean traveling in the opposite direction along their respective axes.

Understanding Three-Dimensional Geometry

Three-dimensional geometry is all about working within a space that has width, height, and depth. This allows us to represent points and shapes in a way that mimics real-world structures.
The 3D coordinate system has three axes:

• The x-axis represents horizontal space.
• The y-axis represents vertical space.
• The z-axis represents depth or the 'coming out at you' direction.

Together, these axes create a 3D grid. This grid helps us visualize points and shapes in a context that allows for a deeper understanding of their spatial relationships.
For example, when plotting the point \( (-1, 2, 1) \), the movement is:

• 1 unit left on the x-axis (negative direction)
• 2 units up on the y-axis
• 1 unit out on the z-axis

It's like traveling along these axes, reaching a specific location in 3D space.

Role of Coordinate Axes

The coordinate axes form the backbone of the three-dimensional coordinate system. They give us a frame of reference where each point is described by its position relative to these axes.
Each axis is a number line that extends infinitely in positive and negative directions:

• The positive x-axis moves to the right from the origin.
• The positive y-axis moves upward.
• The positive z-axis comes out of the page toward the viewer.

These axes intersect at the origin, \( (0,0,0) \), which is the starting point for plotting any position. Understanding the placement and direction of these axes is crucial for plotting points accurately. When you move along these directions while plotting, you navigate through the 3D space, identifying exact positions by considering how far along each axis you've traveled. This approach allows you to map out complex geometric scenarios easily!