Problem 6 - Advanced Engineering Mathematics

Question

Graph the given point. Use the same coordinate axes. $$ (5,-4,3) $$

Step-by-Step Solution

Answer

Plot the point (5,-4,3) in 3D space.

Step 1: Understand the Coordinates

The point given is in a 3D coordinate system, represented as \((x, y, z) = (5, -4, 3)\). The x-coordinate is 5, the y-coordinate is -4, and the z-coordinate is 3.

Step 2: Set Up the 3D Axes

A 3D coordinate system has three axes: the x-axis (horizontal), the y-axis (horizontal, perpendicular to the x-axis), and the z-axis (vertical). Arrange these axes in such a way so that they meet at the origin \((0, 0, 0)\).

Step 3: Locate the X-Coordinate

From the origin, move along the x-axis to the coordinate \(x = 5\). This is 5 units in the positive x-direction.

Step 4: Locate the Y-Coordinate

From the point \((5, 0, 0)\), move parallel to the y-axis to account for the y-coordinate \(y = -4\), which is 4 units in the negative y-direction.

Step 5: Locate the Z-Coordinate

Starting from \((5, -4, 0)\), move along the z-axis to the point \(z = 3\). This means going 3 units upward in the positive z-direction.

Step 6: Plot the Point

Finally, the point \((5, -4, 3)\) is located by combining steps 3, 4, and 5, which gives the point in space.

Key Concepts

Graphing Points in 3D

Graphing points in a 3D coordinate system can be a bit tricky at first, but it's all about understanding the three axes involved. Picture a room where the floor and two walls form a corner. Each corner represents a point in this 3-axis system: the x-axis runs from left to right, the y-axis from front to back, and the z-axis goes up and down. The intersection where these axes meet is called the origin, represented by the point \((0, 0, 0)\).

When graphing a point like \((5, -4, 3)\), you're essentially finding a specific location in this 3D space. Imagine it as following a path:

• Start from the origin.
• Move along the x-axis for the x-coordinate.
• Then move along the y-axis for the y-coordinate.
• Finally, move up or down the z-axis for the z-coordinate.

These steps will guide you to plotting any 3D point effectively.

Understanding the X-Coordinate

The x-coordinate in a 3D system determines the position along the x-axis. This axis is generally visualized as a horizontal line running left to right. When plotting a point, the x-coordinate tells you how far you need to move from the origin in this direction.

For example, in the point \((5, -4, 3)\), the x-coordinate is 5. This means you start by moving 5 units away from the starting point, along the x-axis. Moving in the positive x-direction is like moving to the right.
Imagine standing at the origin and taking 5 steps to the right – that's where you pause for the next step.

Deciphering the Y-Coordinate

The y-coordinate is critical for guiding your movement along the y-axis, which is perpendicular to the x-axis and runs front to back in the 3D space. In 3D graphing, this is your second move after you've established your x-coordinate position.

In the \((5, -4, 3)\) point, the y-coordinate is \(-4\). This indicates you must move 4 units in the negative, or opposite, y-direction from your current location at\((5, 0, 0)\). Visualize this as stepping backward 4 steps.
The y-axis represents depth in 3D plotting, and negative values signify moving away from you, akin to reversing your steps toward the back wall in our room analogy.

Navigating the Z-Coordinate

The z-coordinate adds a vertical dimension to your plotting journey. Think of it like climbing stairs or descending a ramp. The z-axis introduces height or depth, depending on whether you move up or down.

For the point \((5, -4, 3)\), the z-coordinate is 3. This tells you to move up 3 units from \((5, -4, 0)\). In practical terms, it's akin to standing on a step at height 3 above the ground level, giving the point its final vertical position.
Always remember, positive z-coordinates mean moving upwards, like reaching for the ceiling, while negative z-coordinates would have you moving downward, like heading towards a basement. This third dimension completes the 3D plot of any point, giving it an exact spot in space.