3.3. Summary

To locate the position of an object or a point in a plane, we require two perpendicular lines. One of them is horizontal, and the other is vertical.

In geometry, to describe where a point is situated within a two-dimensional space (referred to as a plane), we need a method of measurement. This is accomplished through the use of two intersecting lines that meet at a 90-degree angle. One line runs left to right (horizontal), called the x-axis, and the other runs up and down (vertical), called the y-axis. Together, these two lines help us pinpoint a specific location on the plane.

Imagine a map of a city where streets run north-south and east-west. The horizontal street represents the x-axis, while the vertical street represents the y-axis. By using the intersection of these streets, you can direct someone to a specific building by specifying which street to take and how far to go.

The plane is called the Cartesian, or coordinate plane, and the lines are called the coordinate axes.

The intersection of the two perpendicular lines forms what is known as the Cartesian plane. In this context, the term 'Cartesian' is derived from the mathematician René Descartes. The horizontal line is referred to as the x-axis, and the vertical line is known as the y-axis. The Cartesian plane allows for a standardized way to determine positions of points using numerical coordinates.

Think of the Cartesian plane as a giant graph grid, like the ones used in games where players move on a map. Each move can be calculated using coordinates, making it easier to describe where players are positioned in relation to each other.

The coordinate axes divide the plane into four parts called quadrants.

When the x-axis and y-axis intersect, they create four distinct areas known as quadrants. Each quadrant has specific characteristics based on the signs of the coordinates. These quadrants allow us to categorize points based on their location concerning the axes. Quadrant I (first quadrant) contains points where both coordinates are positive, showing it is located in the upper right area of the plane.

Imagine a game board that is split into four sections, each section representing a different category like Food, Transportation, Entertainment, and Education. Depending on where you place your marker on the board signifies what category you are referring to, just like how quadrants categorize coordinates in the Cartesian plane.

The point of intersection of the axes is called the origin.

The location where the x-axis and y-axis intersect is known as the origin. The coordinates of the origin are represented as (0, 0), meaning it has no distance from either axis. In many mathematical contexts, the origin serves as the central reference point for measuring the position of all other points in the Cartesian plane.

Think of the origin like the starting line in a race. All measurements of distance (like how far runners go) start from this point, and it defines where the race begins.

The distance of a point from the y-axis is called its x-coordinate, or abscissa, and the distance of the point from the x-axis is called its y-coordinate, or ordinate.

When identifying the precise location of a point in the Cartesian plane, we use two distances: the x-coordinate (or abscissa), which indicates how far the point is from the y-axis, and the y-coordinate (or ordinate), which shows how far the point is from the x-axis. Together, they form ordered pairs (x, y) that specify the exact position of a point.

It's similar to finding a friend's house in a neighborhood. If you say they live on 'Main Street 5 blocks east and 2 blocks north', you're specifying how far along the street you move (x-coordinate) and then how far up you go (y-coordinate).

The coordinates of a point are of the form (+, +) in the first quadrant, (–, +) in the second quadrant, (–, –) in the third quadrant and (+, –) in the fourth quadrant, where + denotes a positive real number and – denotes a negative real number.

Each quadrant in the Cartesian plane has specific signs for coordinates that help to easily identify in which quadrant a point lies. For instance, in the first quadrant, both x and y coordinates are positive, which means any point there is located in the top right section of the plane. As we move counter-clockwise through the quadrants, the signs change according to their relative positions to the axes.

Imagine each quadrant as a part of a treasure map. You could say the treasure is hidden in quadrant one (northeast) or quadrant three (southwest). The signs of the coordinates guide you to the exact location of the treasure.

If x ≠ y, then (x, y) ≠ (y, x), and (x, y) = (y, x) if x = y.

In the Cartesian plane, the order of coordinates is important. If the first and second values are different (that is, x is not equal to y), then switching their places results in a different location on the plane. Conversely, if both coordinates are the same, then switching them has no effect, as both points represent the same location.

Think of it like a pair of shoes: the left shoe (x) is not the same as the right shoe (y). If you swap them, your shoes won’t fit the same way. However, if you have two left shoes, switching them doesn’t change the pair!

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