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

On the \(xz\)-plane, points have form \((x, 0, z)\); on the \(y\)-axis, points have form \((0, y, 0)\).

Step 1: Understanding the xz-plane

The \(xz\)-plane is a plane where the \(y\)-coordinate is always equal to zero. This means that for any point \((x, y, z)\) located on the \(xz\)-plane, the \(y\)-coordinate must be zero. Therefore, the coordinates of any point in the \(xz\)-plane can be written as \((x, 0, z)\).

Step 2: Analyzing the y-axis

The \(y\)-axis is a vertical line that goes through the origin of the 3D coordinate system, where \(x\) and \(z\) coordinates are always zero. Therefore, any point on the \(y\)-axis has coordinates of the form \((0, y, 0)\), where \(y\) can be any real number.

Key Concepts

xz-plane

Imagine a flat surface extending in both the x and z directions, without rising or dipping. This is the xz-plane in three-dimensional space. An important thing about the xz-plane is that the y-coordinate is always zero no matter the values of x and z. This means a point on the xz-plane is written as \((x, 0, z)\).

For example, if you have a point \((3, 0, 5)\), you can be certain it lies on the xz-plane because its y coordinate is zero. This flat nature of the xz-plane is similar to a flat floor, extending infinitely left to right and forward to back, but not upward or downward.

y-axis

Visualize a line stretching straight up and down through the origin of the 3D coordinate system. This line is the y-axis. On the y-axis, both the x and z coordinates are always zero because the line does not extend in these directions.

Every point on the y-axis can be expressed as \((0, y, 0)\). Here, the y can be any real number, allowing the point to potentially lie anywhere along the infinite length of the axis. So if you see a coordinate like \((0, 7, 0)\), it is located directly on the y-axis, 7 units above the origin.

coordinates

Coordinates are essential tools in geometry, acting like addresses for points in space. In three-dimensional geometry, each point is identified by three numbers: \((x, y, z)\). These coordinates tell you exactly where a point is located along the three axes: x-axis, y-axis, and z-axis.

• The x-coordinate shows how far left or right the point is.
• The y-coordinate tells you how high up or down the point is.
• The z-coordinate indicates how close or far away the point is from the observer.

Understanding coordinates helps you navigate both real-world spaces and mathematical exercises with confidence. It's like using a map, but in three dimensions!

three-dimensional geometry

Imagine the world around you but without any boundaries in any direction. That's a glimpse into three-dimensional geometry. Unlike two-dimensional geometry, which covers only flat surfaces like paper, 3D geometry includes depth, giving volume and form to objects.

This type of geometry is governed by the x, y, and z coordinates. These coordinates allow us to locate any point within this vast space. Whether you're finding the location of a star in the sky or a hidden treasure underground, understanding three-dimensional geometry is crucial.

In everyday life, 3D is everywhere, from the box you received in the mail to the buildings you walk by. Diving into three-dimensional geometry means exploring how everything fits together in this limitless space, making sense of every object's precise place and dimensions.