Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

In the xz-plane, y-coordinates are always zero, while on the y-axis, both x and z coordinates are zero.

Step 1: Understand the xz-plane

The xz-plane is the plane formed by the x and z axes. In a 3D coordinate system (x, y, z), the xz-plane occurs where the y-coordinate is zero. Thus, every point on the xz-plane has coordinates of the form (x, 0, z).

Step 2: Analyze the xz-plane coordinates

Given the information that the xz-plane requires the y-coordinate to be zero, we can deduce that all points' coordinates follow the format (x, 0, z). The peculiar feature is that the y-coordinate is always zero.

Step 3: Understand the y-axis

The y-axis in a 3D coordinate system is a line where the x and z coordinates are zero, as it only moves along the vertical direction. Thus, any point on the y-axis has the coordinates of the form (0, y, 0).

Step 4: Analyze the y-axis coordinates

Following the constraints of the y-axis, all points on the y-axis have coordinates (0, y, 0). The peculiar feature of the y-axis is that both the x and z coordinates are always zero.

Key Concepts

xz-plane

The xz-plane is an essential concept when working within a 3D coordinate system. Imagine standing on a flat surface that extends infinitely along the x-axis and z-axis. This surface is known as the xz-plane. In three-dimensional coordinate geometry, planes help us understand how points and lines relate to each other in space.

• Absence of Y-dimension: When discussing the xz-plane, we're looking specifically at a plane where the y-coordinate is always zero. This means every point on the xz-plane has the form \((x, 0, z)\).
• Visual Model: If you think about a 3D graph, the xz-plane would be like a flat piece of paper floating above the ground. If you were to plot a point like \((3, 0, 4)\), it would sit on this imaginary sheet.

Understanding the xz-plane is crucial in coordinate geometry because it shows how two axes can create a 2D plane within 3D space, offering a different perspective of the relationships between points.

y-axis

In the framework of a 3D coordinate system, the y-axis serves as one of the three primary axes that defines spatial relationships. This axis is essentially a straight vertical line, offering a path through space where both x and z are always zero.

• Coordinates Specificity: Any point on the y-axis will have the form \((0, y, 0)\). This indicates that the position of the point depends solely on its y-coordinate.
• Vertical Movement: The y-axis showcases movement upward or downward from a point of origin, without the influence or consideration of the x or z dimensions.

The y-axis is an anchor within coordinate geometry, representing a line through the origin (0,0,0) where independent movement of a y-coordinate can be observed, unaffected by changes in other dimensions.

coordinate geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that links algebra and geometry. It does so by using a coordinate system to define geometric objects. This has vast applications in various fields like architecture, engineering, and computer graphics.

• Coordinates: Any point in a 3D coordinate system is defined by a set of numbers, typically represented as \((x, y, z)\). These coordinates help to determine the exact location of points in space.
• Understanding Space Relationships: With coordinate geometry, you can calculate distances, angles, and even the equations of lines and planes.
• Applications: Using this coordinate system allows us to create and understand complex scenes in 3D modeling, enhance design precision in engineering, and improve navigational systems.

Coordinate geometry empowers us to visualize abstract concepts by using coordinates, making spatial reasoning more tangible and intuitive.