What Is the Cartesian Plane?
The Cartesian plane was introduced by Rene Descartes, a French mathematician. The Cartesian plane is formed by the intersection of two number lines as mentioned earlier. The horizontal number line is known as the x-axis and the vertical number line is known as the y-axis and both these lines meet at a point called the origin. It must be noted that both positive and negative numbers are written on these two number lines. The intersection of the two axes also results in the formation of quadrants on the Cartesian plane that are numbered in an anticlockwise manner. There are four quadrants and each of the quadrants have points, or what is known as an ordered pair that have different signs.
The image represents the Cartesian plane, the quadrants and the two intersecting axes.
We will now move on to defining ordered pairs and how they are represented on the Cartesian or the coordinate plane.
Note: The ‘Cartesian plane’ is also expressed as the ‘coordinate plane’ and both these terms indicate the same concept.
Defining an Ordered Pair
Any point on the Cartesian or the coordinate plane is written as a combination of two values. For example, the point (x, y) when represented on the coordinate plane describes the distance of the point in relation to the origin. So this point (x, y) is called an ordered pair. The point with the coordinates (x, y) is ordered in a specific manner and the value ‘x’ is the distance of the point from the origin on the ‘x-axis’ or the horizontal number line of the plane. The value ‘y’ is the distance of the point from the origin on the ‘y-axis’. There are also specific names given to the values in an ordered pair.
In an ordered pair,(x, y), the first element ‘x’ is called the abscissa and the second element ‘y’ is called the ordinate.
Let us look at a couple of examples of how ordered pairs are represented on the coordinate plane.
In this example there are two ordered pairs represented on the coordinate plane. Do observe the jumps made from the origin.
The first point is A (3,5) – represents a jump of three numbers on the x-axis and a jump of five numbers on the y-axis.
The second point is B (-2, -6) – represents a jump of two numbers to the left of the origin and six numbers down to -6.
Hence, this is the method used to plot ordered pairs on the Cartesian or coordinate plane.
With respect to the quadrants on the plane, the change in the sign of the coordinate points are as follows:
Solved Examples
Example 1: From the coordinate plane given in the image, identify the ordered pairs from A to G as well as identify the quadrants they are placed in.
Solution:
- (-2, 4) – Second quadrant
- (3, 2) – First quadrant
- (6, 4) – First quadrant
- (-4, – 3) – Third quadrant
- (-2, – 1) – Third quadrant
- (1, – 4) – Fourth quadrant
- (4, – 3) – Fourth quadrant
Coordinate geometry is related to the branch of geometry that utilizes the coordinate plane or the Cartesian plane for the analysis of the distance between ordered pairs, the shapes formed and to study complex shapes like curves on the coordinate plane.
Every ordered pair has two elements, one, the distance between the ordered pair from the origin on the x-axis, called the abscissa, and the second, the distance between the ordered pair from the origin on the y-axis, called the ordinate.
The most important advantage of the study of coordinate geometry is that there is a relationship that is created between algebra and geometry through the study of lines or curves formed by ordered pairs on the coordinate plane.