Question

Explain how to plot the point (3,-2,1) in \(\mathrm{R}^{3}\)

Short answer

Question: Explain the steps involved in plotting the point (3, -2, 1) in three-dimensional space.

Step-by-step solution

Understand the Point's Coordinates

The given point is (3, -2, 1), which means its coordinates in 3-dimensional space are \((x, y, z) = (3, -2, 1)\). These coordinates represent the position of the point in the \(\mathrm{R}^{3}\) space.

Setting Up the Axes

When plotting a point in \(\mathrm{R}^{3}\), we need to set up three axes: the x-axis, the y-axis, and the z-axis. These axes are mutually perpendicular to each other and form the basis of the 3D space. Set up the axes in such a way that the x-axis represents the horizontal direction, the y-axis represents the vertical direction, and the z-axis represents the depth direction.

Plot the x-coordinate

Starting from the origin (0, 0, 0), move 3 units along the positive x-axis. The x-coordinate of the given point is 3, so we need to move in the positive direction of the x-axis.

Plot the y-coordinate

Next, we need to plot the y-coordinate, which is -2. From the current position, move 2 units in the negative direction of the y-axis. This will finalise the position in the xy-plane.

Plot the z-coordinate

Finally, we need to plot the z-coordinate, which is 1. From the current position in the xy-plane, move 1 unit in the positive direction of the z-axis. This will place us at the point (3, -2, 1) in \(\mathrm{R}^{3}\).

Mark the Point

Once you have reached the position specified by the coordinates (3, -2, 1) in the \(\mathrm{R}^{3}\) space, draw a small dot or mark to represent the point in the 3D space. Now, the point (3, -2, 1) is successfully plotted in \(\mathrm{R}^{3}\).

Key concepts

Plotting Points in 3D Space

When plotting points in a 3D coordinate system like \(R^{3}\), you're essentially locating a specific spot in a three-dimensional environment. Think of it as marking the exact whereabouts of something within a box. In this context, each point is represented by a set of three numbers or coordinates: \(x, y, z\).

For example, the point \(3, -2, 1\) suggests:

• Move to 3 units parallel to the x-axis.
• Then, shift downwards 2 units along the y-axis, indicating a negative direction.
• Lastly, adjust vertically upwards 1 unit parallel to the z-axis.

Afterwards, mark this position with a dot or similar notation in the 3D space to signify the point has been plotted. Each coordinate describes the position along its respective axis, making plotting accurate and consistent. Understanding how to plot points helps visualize complex spaces around us, effectively bridging abstract concepts with reality.

Understanding the 3D Axes

Visualizing a three-dimensional space begins with understanding the 3D axes. These three perpendicular lines are the essential framework of \(R^{3}\). Imagine the axes as three directions that stretch out infinitely, guiding us where to plot points. Here's each axis explained:

• The **x-axis** runs horizontally from left to right, establishing the width direction.
• The **y-axis** moves vertically, akin to height, going up and down.
• The **z-axis**, often trickier to grasp, represents depth, extending forward and backward.

When set up correctly, the x, y, and z axes form a coordinate system where everything can be precisely located. Coordination among these axes allows for consistent measurement and plotting, forming the backbone of 3D spaces used in everything from engineering designs to video game development. Always remember to keep the axes orthogonal, meaning they should meet at right angles, ensuring accurate spatial representation for plotting.

Exploring Cartesian Coordinates in 3D

The Cartesian coordinate system is a numerical framework used to pinpoint locations in 3D space. Named after René Descartes, this system deploys three numbers: \(x, y, z\), known as Cartesian coordinates, to specify exact places along the orthogonal axes.

Here's how it works:

• The **x-coordinate** defines horizontal placement, tethered to the x-axis.
• The **y-coordinate** outlines vertical position along the y-axis.
• The **z-coordinate** depicts depth, securing alignment with the z-axis.

Using this systematic set of rules, every point can be assigned a unique location within a three-dimensional framework. This makes it integral to fields like physics, engineering, and computer graphics, where precise spatial representation is crucial. Understanding Cartesian coordinates allows us to break down complex spaces into manageable measurements, making the abstract more tangible.