Problem 4 - Advanced Engineering Mathematics

Question

Graph the given point. Use the same coordinate axes. $$ (6,0,0) $$

Step-by-Step Solution

Answer

The point (6,0,0) is plotted 6 units along the x-axis.

Step 1: Understand the Coordinate System

In a three-dimensional coordinate system, a point is represented by three coordinates: \((x, y, z)\). The point \((6,0,0)\) lies on the three-dimensional plane, where 6 is the x-coordinate, 0 is the y-coordinate, and 0 is the z-coordinate.

Step 2: Identify the Axes

In the 3D coordinate plane, there are three axes: the x-axis, the y-axis, and the z-axis. The x-axis runs horizontally, the y-axis runs vertically, and the z-axis represents depth, coming out of or going into the page or screen.

Step 3: Locate the Position on the x-axis

Since the y-coordinate and z-coordinate are both 0, the point \((6,0,0)\) lies entirely on the x-axis. Move 6 units along the x-axis from the origin (0,0,0).

Step 4: Plot the Point

After moving 6 units along the x-axis, mark the point. There is no movement required along the y-axis or z-axis since both coordinates are zero. Ensure this point is placed on the x-axis, correctly representing its location.

Key Concepts

Plotting Points in the 3D Coordinate System

Plotting points in a three-dimensional coordinate system can seem tricky at first, but it's just an extension of what you already know about two dimensions. In 2D, you use two numbers, like a point to describe a location on a plane. In 3D, you add one more number to indicate depth, using the format \((x, y, z)\).The three numbers represent movements along the x-axis, y-axis, and z-axis, respectively. When plotting points, start at the origin \((0, 0, 0)\), then:

• Move along the x-axis in the first coordinate direction.
• Next, move parallel to the y-axis for the second coordinate.
• Finally, move up or down along the z-axis according to the third number.

With practice, you'll find that plotting in 3D is just a logical step beyond 2D with the added fun of exploring a new dimension.

Understanding Coordinate Axes in 3D

In a three-dimensional space, there are three coordinate axes that help define any point's location. These axes intersect at a point called the origin, labeled as \((0, 0, 0)\). Each axis measures distance from the origin in a different direction:

• The **x-axis** typically runs left to right (or horizontally).
• The **y-axis** usually runs up and down (vertically).
• The **z-axis** extends forwards and backwards, giving the impression of depth.

These three perpendicular axes allow us to place any point precisely in 3D space. Just think of them as ladders that let you climb through space in different directions: back-and-forth, side-to-side, and up-and-down. Understanding these axes is crucial for accurately mapping where points exist in the world of three dimensions.

The Role of x-axis, y-axis, and z-axis

To navigate the 3D coordinate system, grasping the role of each axis is fundamental. Each axis specifies a direction of movement:The **x-axis** is your guide for horizontal movements, left and right. For instance, in the point \((6, 0, 0)\),you start by moving 6 units along this axis.

The **y-axis** handles vertical changes. It would alter your position up or down relative to the grid. Since the y-axis is zero in our point, we don't move along it.

Lastly, the **z-axis** represents depth, where you'd move forward or backward. With a zero z-value in \((6, 0, 0)\),there's no depth movement required.Together, these three axes help us chart a precise path to any point in our 3D world, helping to bring the spatial view into our analysis or designs.