Problem 3 - College Algebra and Calculus: An Applied Approach

Question

Plot the points on the same threedimensional coordinate system. (a) \((5,-2,2)\) (b) \((5,-2,-2)\)

Step-by-Step Solution

Answer

The first point \((5,-2,2)\) lies above the x-y plane and the second point \((5,-2,-2)\) lies below the x-y plane.

Step 1: Plotting the first point

Start by plotting the first point \((5,-2,2)\). Starting from the origin, move 5 units in the positive x-direction, 2 units in the negative y-direction, and 2 units in the positive z-direction and mark this point.

Step 2: Plotting the second point

Next, plot the second point \((5,-2,-2)\). Again, starting from the origin, move 5 units in the positive x-direction, 2 units in the negative y-direction, but this time, move 2 units in the negative z-direction. Mark this point.

Step 3: Analysis of the points

Now when comparing the two points, both lie on the same x-axis and y-axis but on different levels on the z-axis. The first point is placed above the x-y plane, while the second point is placed below the x-y plane.

Key Concepts

Plotting Points

Plotting points in a three-dimensional coordinate system involves navigating along three separate axes: the x-axis, y-axis, and z-axis. To successfully plot a point, it's essential to follow an ordered process from the origin, which is the point (0,0,0). Start by moving along the x-axis by the specified units, then proceed along the y-axis, and finally, move along the z-axis. For example, to plot the point (5,-2,2), start at the origin and proceed as follows:

• Move 5 units in the positive x-direction.
• Move 2 units in the negative y-direction.
• Finally, move 2 units in the positive z-direction.

Once you reach the desired coordinates, mark the point clearly. This method of plotting ensures that each coordinate is placed accurately, maintaining consistency when visualizing points in 3D space. Plotting may seem complex, but with practice, it becomes intuitive.

Positive and Negative Directions

Understanding positive and negative directions in a 3D system is crucial for accurate point plotting. Each axis in a 3D coordinate system has two directions: positive and negative.

• The **x-axis** runs horizontally. Moving right from the origin is the positive direction, while moving left is negative.


• The **y-axis** runs vertically. Going upward is the positive direction, and heading downward is the negative direction.


• The **z-axis** runs perpendicular to both x and y. Moving upward from the x-y plane is considered positive, while moving downward is negative.

Comprehending these directions is key for plotting any point accurately, especially in distinguishing whether a point lies above or below the x-y plane, or in any other position in the 3D space. Keeping this framework in mind ensures that you always interpret coordinates correctly and avoid unnecessary plotting errors.

Comparison of Points

When comparing points in a 3D system, look for similarities and differences in their coordinates. The exercise above guides us on comparing points such as (5,-2,2) and (5,-2,-2). To compare these:

• Observe their x-coordinates: Both points share the same x-coordinate, indicating they are aligned along the same vertical plane parallel to the y-axis.


• Examine their y-coordinates: Similarly, both points have identical y-coordinates, showing they also lie along the same vertical plane parallel to the x-axis.


• Check their z-coordinates: Here is where the distinction appears. The first point, (5,-2,2), is situated 2 units above the x-y plane, while the second point, (5,-2,-2), is 2 units below.

By analyzing the coordinates, you can deduce spatial relationships and understand how one point relates to another within the 3D space. This aids in visualization and establishing the relative positions in everyday applications of 3D coordinates.