Problem 1 - Advanced Engineering Mathematics

Question

Graph the given point. Use the same coordinate axes. $$ (1,1,5) $$

Step-by-Step Solution

Answer

Plot (1,1,5) by moving one unit along x and y, then five units up along z.

Step 1: Understanding the Coordinate System

To graph the point (1,1,5), we need to understand that it is given in a three-dimensional Cartesian coordinate system consisting of three axes: the x-axis, y-axis, and z-axis.

Step 2: Identify the Coordinates

The given point is (1,1,5). This tells us that the point lies at x = 1, y = 1, and z = 5 in the 3D space.

Step 3: Plot the Point on 3D Coordinate Plane

Start at the origin (0,0,0). Move one unit along the x-axis, one unit along the y-axis to get to (1,1,0). Then move five units upwards along the z-axis to reach the point (1,1,5).

Step 4: Draw Reference Lines

To aid in visualizing the point, draw dashed lines parallel to the axes from (1,1,5) to the planes. Draw a line from (1,1,0) up to (1,1,5), and lines along (1,0,5) and (0,1,5) to (1,1,5).

Key Concepts

Graphing Points

When we talk about graphing points, we're referring to placing specific locations on a graph. These locations, or points, are determined by coordinates. In 3D space, points are denoted by three numbers, like (1,1,5). These numbers describe where the point lies on each of the three axes: the x-axis, y-axis, and z-axis.

• The first number determines the position on the x-axis.
• The second number tells us the position on the y-axis.
• The third number specifies the height on the z-axis.

To graph the point, you start from the origin, which is (0,0,0). You then move along these axes according to each coordinate value. In our example, starting at (0,0,0), you move 1 unit along the x-axis, 1 unit along the y-axis, and then 5 units up along the z-axis. This gives us the point (1,1,5) on the graph.

Coordinate Axes

The coordinate axes in a 3D Cartesian system form the backbone of graphing points. There are three axes:

• The x-axis, which runs horizontally.
• The y-axis, which runs vertically, perpendicular to the x-axis when looking at it in a 2D plane.
• The z-axis, which adds depth by extending perpendicularly out of the x-y plane.

All three axes meet at a single point called the origin, represented as (0,0,0). When graphing points, it's helpful to remember that each axis represents a direction.
By moving along these axes, you can navigate through the three-dimensional space. Each axis acts like a number line, helping you measure distance in one direction. This measurement allows you to pinpoint exact locations in 3D space.

Three-Dimensional Space

Three-dimensional space refers to a "world" that has volume, unlike two-dimensional planes that only have length and width. In a 3D space, we add a third dimension—depth—which is captured by the z-axis. To visualize this, think of a box where the x and y axes form the base, and the z-axis shows the height.
Here are key points about three-dimensional spaces:

• Objects in 3D have depth, giving them shape and volume.
• Navigation in this space requires understanding directions along all three axes.
• Perspective changes how we view points; they can appear closer or further apart based on depth.
• Points in this space are often described using reference lines for better visual comprehension.

Using a combination of the x, y, and z axes, you can represent any location within this 3D space. It's like plotting a course on a map but with an added dimension of movement, allowing for a richer understanding of positioning and location.