Question

A point in the three-dimensional coordinate system has coordinates \(\left(x_{0}, y_{0}, z_{0}\right) .\) Describe what each coordinate mea- sures

Short answer

In a three-dimensional coordinate system, \(x_{0}\) is the distance along the x-axis from the origin, \(y_{0}\) is the distance along the y-axis from the origin, and \(z_{0}\) is the distance along the z-axis from the origin. These coordinates uniquely define any point in three-dimensional space.

Step-by-step solution

Understand the three-dimensional coordinate system

The three-dimensional coordinate system is a way to define points in space. This system involves three axes - the x-axis, the y-axis, and the z-axis - which all are perpendicular to each other. Every point in the space can be defined uniquely by a set of three numbers (x, y, z) where each number is the distance from the origin along the respective axis.

Define the x-coordinate

The x-coordinate, denoted \(x_{0}\), measures the horizontal distance from the origin along the x-axis. Positive values of \(x_{0}\) indicate locations to the right of the origin, while negative values indicate locations to the left.

Define the y-coordinate

The y-coordinate, denoted \(y_{0}\), measures the vertical distance from the origin along the y-axis. Positive values of \(y_{0}\) indicate locations above the origin, while negative values indicate locations below.

Define the z-coordinate

The z-coordinate, denoted \(z_{0}\), measures the distance from the origin along the z-axis, which is perpendicular to the x-y plane. Positive values of \(z_{0}\) indicate locations in front of the origin (or above if the z-axis is considered to be vertical), while negative values indicate locations behind (or below).

Key concepts

Understanding the X-Coordinate

The x-coordinate in a three-dimensional coordinate system is akin to the first step in pinpointing a location on a map. To understand its role, visualize the three-dimensional space as a vast expanse where every position is unique and can be identified with precision.

In this spatial grid, the x-coordinate represents the horizontal axis, which can be thought of as the ground beneath our feet extending endlessly to the left and right. It measures how far to the east or west a point is from the central reference point, known as the origin. When we assign a number to the x-coordinate, we're specifying exactly where along this horizontal line our point sits. If we choose a positive x-coordinate, we're venturing to the right of the origin. In contrast, a negative x-coordinate takes us to the left. It's essential to grasp this concept as it lays the foundation for understanding the other coordinates.

Deciphering the Y-Coordinate

The y-coordinate serves as the vertical counterpart to the x-coordinate within the three-dimensional coordinate system. Once the horizontal position is established using the x-coordinate, we employ the y-coordinate to scale the vertical landscape.

Imagine you are in an elevator; the y-coordinate tells us how many floors up or down from the ground floor our point is situated. Positive y-coordinates signify that our point is above the ground floor (the origin), akin to going up the elevator. Conversely, negative y-coordinates indicate a position below the ground floor, akin to descending to a basement level. It's crucial for students to understand this vertical component, as it provides the 'height' aspect necessary for mapping a point's position in three-dimensional space.

Navigating the Z-Coordinate

The z-coordinate is what transforms our understanding from a flat plane to the realm of three dimensions. Unlike the x and y coordinates, which lie on the horizontal plane, the z-coordinate emerges perpendicularly, adding depth to our spatial map.

Consider the z-coordinate as an indicator of depth or front-to-back positioning. It's as if you're walking directly towards or away from a mountain; moving forward increases the positive value of the z-coordinate, while stepping back decreases it, possibly into negative territory. In a real-world context, if we consider ourselves facing north, an object with a positive z-coordinate would be in front of us, while one with a negative z-coordinate would lie behind us. Grasping the z-coordinate's significance is essential, as it completes the trio needed to identify any point's location in a three-dimensional space.