Question

Plot the points on the same threedimensional coordinate system. (a) \((3,-2,5)\) (b) \(\left(\frac{3}{2}, 4,-2\right)\)

Short answer

Point (a) is placed 3 units along the X-axis, 2 units in the opposite direction along the Y-axis (due to being negative) and 5 units upward on the Z-axis. Point (b) is placed \(\frac{3}{2}\) units along the X-axis, 4 units upward on the Y-axis and 2 units down the Z-axis (because it's negative).

Step-by-step solution

Understanding the Coordinate System

Firstly, it is crucial to understand the three-dimensional coordinate system. The system has three axes: the X-axis, the Y-axis, and the Z-axis. The X-axis runs horizontally, while the Y-axis runs vertically. The Z-axis, different from a two-dimensional space, is the third axis that represents depth, which runs up and down in a three-dimensional diagram. It's perpendicular to both the X and Y axes.

Plotting Point (a)

The given point (a) is (3,-2,5). This means that it is 3 units along the X-axis, -2 units along the Y-axis (in the opposite direction as it is negative), and 5 units up the Z-axis. Start at the origin (0,0,0), move 3 units right on the X-axis, 2 units left on the Y-axis, and then move 5 units up from that point along the Z-axis to plot the first point.

Plotting Point (b)

The given point (b) is \(\left(\frac{3}{2}, 4,-2\right)\). This means it is \(\frac{3}{2}\) units along the X-axis, 4 units along the Y-axis, and -2 units down the Z-axis (as it is negative). Start at the origin, move \(\frac{3}{2}\) units right on the X-axis, then 4 units up on the Y-axis, and lastly move 2 units down from that point along the Z-axis.

Key concepts

Plotting Points in 3D

Plotting points in a three-dimensional (3D) coordinate system allows us to represent the location of objects in space more accurately than in two dimensions. In 2D, points are determined by two coordinates, usually written as \( (x, y) \) where \( x \) and \( y \) represent positions on the horizontal and vertical axes, respectively. However, real-life spaces are three-dimensional, requiring an additional coordinate.
In 3D, each point is defined by a triplet of coordinates, for example, \( (3, -2, 5) \) as seen in our textbook exercise. These coordinates correspond to the position along the three primary axes of the Cartesian coordinate system: the X-axis, Y-axis, and Z-axis. To plot the point \( (3, -2, 5) \) in the system, you visually trace 3 units along the X-axis, move in the negative direction 2 units along the Y-axis, and then go upwards 5 units along the Z-axis. Understanding this process is key to resolving complex 3D graphing challenges.

Cartesian Coordinate System

The Cartesian coordinate system is the cornerstone of modern geometry and is equally essential in three-dimensional contexts. Named after the French mathematician René Descartes, this system uniquely identifies each point in space using three numerical values representing distances along perpendicular axes. For a 3D Cartesian system, the axes are commonly labeled X, Y, and Z.

It's worth noting that the Cartesian system allows for positive and negative values. In our exercise's context, points with negative coordinates indicate a position in the opposite direction from the positive side of their respective axis. For instance, a negative Y coordinate means movement below the horizontal (X) plane, while a negative Z coordinate suggests depth below the ground or floor level in a vertically oriented space.

X-axis Y-axis Z-axis

The X-axis, Y-axis, and Z-axis are the three axes that compose the Cartesian coordinate system in 3D space. The X-axis typically represents the 'left to right' dimension, while the Y-axis represents the 'up and down' dimension on the plane. Both X and Y-axes are familiar from two-dimensional graphing. The Z-axis, on the other hand, extends 'forward and backward,' introducing the depth aspect unique to 3D graphing.
In this coordinate framework, a point's location is determined by how far it is along each axis. For example, consider the point \( (\frac{3}{2}, 4, -2) \) from the exercise. The number \( \frac{3}{2} \) represents the point's distance along the X-axis, 4 indicates how far up the Y-axis it is, and -2 notes that the point moves backwards, or into the third dimension, along the Z-axis. These axes allow us to navigate and model geometrical figures, relationships, and motion within our three-dimensional world.

Graphing in Three Dimensions

Graphing in three dimensions is an extension of the principles we use in two-dimensional graphing, yet it encompasses a greater level of complexity due to the additional depth component. When we graph points, lines, or shapes in 3D, we must consider all three axes. This approach is instrumental for fields ranging from engineering to physics as it provides a more realistic representation of objects and phenomena.
The steps to plot points such as those in our textbook exercise illustrate the fundamental approach to 3D graphing. Begin by identifying the origin (0,0,0) and move along the axes according to the given coordinates. Although it might initially seem daunting, gaining fluency in 3D graphing opens up a world of possibilities for visualizing and solving real-world problems that can't be confined to the flat geometry of a piece of paper.