Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

What is peculiar to the coordinates of all points in the \(y z\)-plane? On the \(z\)-axis?

Step-by-Step Solution

Answer

In the \( yz \)-plane, the \( x \)-coordinate is zero; on the \( z \)-axis, both \( x \) and \( y \)-coordinates are zero.

Step 1: Understanding the Problem

We need to find out what is peculiar about the coordinates of points located in the \( yz \)-plane, as well as points on the \( z \)-axis, in a three-dimensional coordinate system.

Step 2: Analyzing the yz-plane

In a 3D coordinate system, the \( yz \)-plane is the plane where all the \( x \)-coordinates are zero. Thus, any point \((x, y, z)\) in the \( yz \)-plane has the form \((0, y, z)\).

Step 3: Analyzing the z-axis

The \( z \)-axis is the line where both \( x \) and \( y \) coordinates are zero. Therefore, any point on the \( z \)-axis can be written as \((0, 0, z)\), where \( z \) is any real number.

Step 4: Summarizing the Findings

For points in the \( yz \)-plane, the peculiar characteristic is that the \( x \)-coordinate is always zero. For points on the \( z \)-axis, both the \( x \)- and \( y \)-coordinates are zero, leaving only the \( z \)-coordinate.

Key Concepts

yz-plane

In three-dimensional geometry, the yz-plane is a key concept. It represents one of the primary planes altogether with the xy-plane and xz-plane.
When visualizing the yz-plane, think of it as a flat surface where all x-coordinates are equal to zero. This means if you have a point written as \( (x, y, z) \), in the yz-plane, it will appear as \( (0, y, z) \).

• The yz-plane divides space into two halves, one with positive x-values and one with negative.
• It is always perpendicular to the x-axis.
• It is parallel to the z-axis and y-axis.

Understanding this plane helps in visualizing how points and lines behave in 3D space, aiding in various mathematical and scientific applications.

z-axis

The z-axis is one of the three axes in a three-dimensional coordinate system, along with the x-axis and y-axis. It is crucial for locating points in this space. Visualize it as a vertical line that goes upwards and downwards through the origin.
In any point denoted as \( (x, y, z) \), the z-coordinate determines the length along this axis. For a point on the z-axis, both the x and y are zero, simplifying the point to \( (0, 0, z) \).

• The z-axis is perpendicular to both the xy-plane and yz-plane.
• Points along this axis show vertical displacement in 3D space.
• It plays a key role in graphing functions and analyzing movement.

The simplicity of its coordinates makes locating points intuitive when exclusively considering elevation or depth.

coordinates

Coordinates in a 3D system are like addresses, telling you where a point can be found in space. Each point is represented by an ordered triplet (x, y, z).
Here is how each is defined:

• x-coordinate: It shows the position along the x-axis.
• y-coordinate: It determines the position on the y-axis.
• z-coordinate: This indicates the vertical position relative to the plane you are considering.

In specific scenarios:

• If x = 0, the point lies somewhere on the yz-plane.
• If y = 0, the point is on the xz-plane.
• If z = 0, the point is on the xy-plane.

Understanding how these coordinates interact is fundamental to navigating and interpreting 3D environments effectively.

three-dimensional geometry

Three-dimensional geometry expands the concepts of traditional 2D geometry into an extra dimension. This allows for more complex shapes and representations of real-world structures.
In this space, you have three axes: x, y, and z. These axes intersect at a point called the origin \((0, 0, 0)\). Each dimension corresponds to a different direction:

• The x-axis usually represents the width.
• The y-axis typically stands for the height.
• The z-axis signifies depth, making the geometry 3D.

In this framework:

• Objects can be rotated, mirrored, and translated in richer ways.
• Shapes like spheres, cubes, and cones can be accurately represented.
• It is crucial for fields such as physics, engineering, and computer graphics.

Embracing the full nature of 3D geometry lets you solve complex problems that wouldn't exist in a simpler 2D world.