Question

Plot the points on the same threedimensional coordinate system. (a) \((0,4,-5)\) (b) \((4,0,5)\)

Short answer

Points \((0,4,-5)\) and \((4,0,5)\) have been plotted respectively by moving along the x, y, and z axes according to their coordinates.

Step-by-step solution

Understanding 3D Coordinates

Before plotting, it's essential to understand three-dimensional coordinates. They are represented as \((x, y, z)\), where 'x' corresponds to the position on the 'X' axis (left or right on the page), 'y' corresponds to the position on the 'Y' axis (up or down on the page), and 'z' corresponds to the position on the 'Z' axis (depth into or out of the page).

Plotting Point (0,4,-5)

The first point to plot is \((0,4,-5)\). Start at the origin (0,0,0) on the graph. Then, move 0 units along the x-axis (staying at the origin), move 4 units up along the y-axis, and then 5 units into the page - as it is a negative value - along the z-axis. This is where you plot the point.

Plotting Point (4,0,5)

Next, plot the point \((4,0,5)\). Again, start at the origin. Move 4 units to the right along the x-axis, 0 units up along the y-axis, and 5 units out of the page along the z-axis. This is where you place the second point.

Key concepts

Plotting Points

To plot points in a 3D coordinate system, we need to understand the role each of the three coordinates plays. Each point is given as a trio of values:

• The x-coordinate tells us how far to move left or right. Think of this as the horizontal movement.
• The y-coordinate indicates how far to move up or down. This is the vertical movement.
• The z-coordinate determines how far into or out of the page we should move, adding the third dimension of depth.

When plotting a point such as \((0, 4, -5)\), start from the origin, which is the point \((0, 0, 0)\). Move 0 units on the x-axis, 4 units up on the y-axis, and 5 units into the page on the z-axis. Each shift is like stepping through a set of instructions, leading you directly to the targeted spot in 3D space. Remember, the negative z-values indicate moving into the page.

Three-Dimensional Space

Three-dimensional space adds an exciting twist to the usual flat world of two dimensions by introducing depth. Imagine a cube; now, visualize your actions when moving within it:

• Left and right movements are on the x-axis, running horizontally.
• Up and down shifts follow the y-axis, going vertically.
• Forward and backward moves align with the z-axis, giving height.

This added dimension allows us to represent realistic scenarios, such as a point's position in your room. You've probably seen 3D pictures or movies; the concept works similarly, providing a full perspective that includes the otherwise hidden feature of depth.

3D Graphing

3D graphing is a method used to visualize data or geometry in a three-dimensional setting. When plotting in a 3D graph, we see the relationship and intersection of three axes:

• The X-axis, representing lateral movements.
• The Y-axis, capturing vertical shifts.
• The Z-axis, visualizing depth.

Consider how we plotted \((4, 0, 5)\): starting at the origin, moving 4 units to the right on the x-axis, no movement on the y-axis, and rising 5 units out of the page on the z-axis. 3D graphing enables us to see points floating in space, akin to stars in the night sky. Each graph is a window into a deeper understanding of spatial relationships, essential in fields like physics, engineering, and architecture.