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

Point on the yz-plane has x=0; point on z-axis has x=0 and y=0.

Step 1: Identify the General Coordinate System

In a three-dimensional coordinate system, points are represented by coordinates \(x, y, z\). We need to understand how these coordinates change in different planes and along axes.

Step 2: Define the Coordinates on the yz-Plane

Any point on the \(yz\)-plane has an \(x\) coordinate of zero, because the point does not extend in the \(x\) direction. Thus, the coordinates of any point on the \(yz\)-plane are of the form \( (0, y, z) \).

Step 3: Describe the z-Axis

The \(z\)-axis is one of the main axes in the three-dimensional coordinate system. A point on the \(z\)-axis has both \(x\) and \(y\) coordinates equal to zero, as it lies entirely along the \(z\) direction. Hence, the coordinates of a point on the \(z\)-axis are \( (0, 0, z) \).

Key Concepts

yz-plane

In a three-dimensional coordinate system, the "yz-plane" is one of several different planes, each playing a unique role in defining points in space. Imagine a sheet of paper that extends vertically and horizontally without any thickness. This is similar to how the "yz-plane" extends infinitely in the directions of the y and z-axes but has no extension in the x direction.

• The x-coordinate on the yz-plane is always zero, making it unique compared to any other plane in a 3D space.
• Any point on this plane will have coordinates expressed as \((0, y, z)\) where y and z can be any value.

This characteristic makes the yz-plane a crucial reference within the 3D coordinate system for handling problems relating to spatial positioning. Knowing how to identify points in this plane is essential for complex mathematical and physical computations.

z-axis

The "z-axis" is a fundamental part of the three-dimensional coordinate system, playing a pivotal role similar to the x and y-axes. The z-axis defines height or depth in a space, depending on your perspective. Consider a vertical line extending infinitely up and down, with no width or breadth.

• A point located on the z-axis will have coordinates of \((0, 0, z)\), meaning the x and y values are zero.
• Only the z-coordinate can vary, extending the point along the z direction.

This axis is fundamental for understanding concepts of height and vertical positioning within many fields, such as physics and engineering. It's a central feature when visualizing 3D models or graphs, helping to determine parallel and perpendicular relationships with the other axes.

coordinate system

The "coordinate system" is an organizational framework that allows us to describe the position of points, lines, and planes in space. In three-dimensional space, this framework consists of three axes: the x-axis, the y-axis, and the z-axis, each perpendicular to the others. Together, they define a space that can encompass length, width, and height.

• Each point in this system is identified by a set of three numbers, \((x, y, z)\), representing its location along the x, y, and z axes respectively.
• The coordinate system serves as a universal language for mathematicians, engineers, and scientists to describe complex geometric shapes and physical phenomena accurately.

Understanding how these axes interact and define a space is crucial for solving spatial problems. The three-dimensional coordinate system is a foundational concept in geometry and calculus, enabling detailed analysis and interpretation of real-world scenarios across various fields.