We have acknowledged the fact that various objects around us are three-dimensional. Even many shapes are two-dimensionally present around us. But, do we know that these objects can be represented on paper by specific knowledge of axes and planes? We draw shapes in a 2D plane, i.e., only in two directions; x and y. What is the use of representing the shapes in such a way? In practical life, we need to know the coordinates of any place or position of a person. Sailors use the coordinate plane concept to track their way back home and draw maps. That way, it is easier to locate the object within no time. Curious to know what coordinates are or what is a coordinate system? Continue reading further to learn more.
What is a coordinate plane: Learning geometry in a new way
As mentioned above, coordinates help us locate an object anywhere in the world. And these coordinates are represented in a coordinate plane. A coordinate plane is a two-dimensional (three-dimensional, while considering 3D objects) plane consisting of vertical and horizontal axes. The horizontal axis is known as the x-axis, and the vertical axis is known as the y-axis. They are perpendicular to each other, meaning they will form a 90-degree angle at their meeting point.
The two-dimensional coordinate plane is the Cartesian plane, named after a well-known French mathematician, Rene Descartes. Rene Descartes was the first to define a rectangular coordinate system by which we can represent each point with the help of numbers, called coordinates. Why a rectangular coordinate system? Because the entire plane looks like a rectangle! Before moving further to understanding the coordinate plane, let us see the parts in a coordinate system.
What makes a coordinate plane?
Have you seen a coordinate plane yet? What does it look like? A coordinate plane consists of axes, origin, and quadrants. The two lines, vertical and horizontal that go never-ending in a coordinate plane are known as axes. The horizontal axis is the x-axis or the abscissa, whereas the vertical axis is the y-axis or ordinate.
These axes meet up at a common point, called origin. Origin is the starting point of any coordinate plane. When these two lines intersect, they make four divisions of the entire coordinate plane. These four divisions are known as quadrants. So, we can say the whole coordinate system consists of four quadrants. The top right division is known as the first quadrant. The top left is known as the second quadrant. The bottom left is the third quadrant, and the bottom right is the fourth quadrant.
Learning to represent coordinates in a coordinate plane
Now, we know the parts in a coordinate plane. We also know the coordinate plane quadrants. But how do you represent the coordinates? Is it that important to learn to represent coordinates? Certainly, it is! The coordinates differ in each of the coordinate plane quadrants mentioned above. Coordinates are a set of numbers that represent the location of an object. Coordinates are written as (x, y), where x is the value from the origin in the direction of the x-axis, and y is the value from the origin in the direction of the y-axis. The standard form of representing the coordinates in a coordinate system is known as an ordered pair.
What are the coordinates of the origin? Since the origin is the starting point, how do we represent a starting point? By the number zero, right? Therefore, the coordinates of the origin are (0, 0).
In coordinate plane quadrants, the sign of the coordinates changes. The universal rule is: that the right of the x-axis is positive, and the left side of the x-axis is negative. In the same way, the top of the y-axis is positive, and the bottom of the y-axis is negative. Now, can we locate the coordinates in coordinate plane quadrants?
Coordinate plane quadrants: Representing coordinates in a coordinate plane
Representing the coordinates in a coordinate plane becomes easier when one has understood the concept of negative and positive axes. Did you follow up on the universal rule mentioned above?
In quadrant 1, the sign of the x-axis is positive, and the y-axis is also positive. Therefore, the coordinates in quadrant 1 are represented as (x, y). In quadrant 2, the x-axis is negative, but the y-axis is positive. Therefore, the coordinates are denoted as (-x, y). Similarly, can you figure out the coordinates for the other two coordinate plane quadrants?
Here is a summary of the representation of coordinate plane quadrants:
Now, we have understood the representation of coordinates. But, can we place these coordinates in a graph? Is it possible to make a coordinate plane graph?
Coordinate plane graph: Representing figures neat and clean
Have you seen a graph before? What are the basic features of a graph? All the blocks are of uniform shape and size in a graph, right? Therefore, representing coordinates in a graph is easier than representing coordinates randomly in a plane. The axes are drawn on the graph, as shown in the figure below. This representation is known as a blank coordinate plane. Only axes, quadrants, and origin can be seen in a blank coordinate plane. With the help of a blank coordinate plane, we can learn to make coordinates in a coordinate plane.
Consider (2, 3) need to be represented in a coordinate plane on the graph. How can we do that? Since both the coordinates are positive, they must be in the first quadrant. We even know that the first coordinate represents the point on the x-axis and the other on the y-axis. Therefore, 2 has to be 2 units away from the origin in the direction of the x-axis, and 3 has to be 3 units away from the origin in the direction of the y-axis. The final coordinate will be (2, 3) as shown.
This rule is followed to represent two-dimensional shapes and figures in a coordinate plane. But then, how to represent a 3D object in a coordinate plane graph and its respective quadrants?
3D representation on a 2D coordinate plane: Here’s how to do it
While dealing with the two-dimensional representation of coordinates in a plane, two directions were covered; left and right, and top and bottom. But while representing a three-dimensional object in a two-dimensional plane, a third direction is also added, i.e., forward and backward. Therefore, in a 3D coordinate plane, three directions come into the picture; left and right, top and bottom, and forward and backward.
Consider the shape representation in the image below. The coordinates of the blue dot are (2, 4, 5), which means the point is 2 units away from the x-axis, 4 units away from the y-axis, and 5 units away from the z-axis. Since all these directions are positive, the object is placed in the first quadrant.
Do you know how many quadrants are there in a 3D coordinate system? Or what are their sign conventions? Here is the summary of the coordinate plane quadrants and their sign conventions in a 3D coordinate system:
|I||(x, y, z)|
|II||(-x, y, z)|
|III||(-x, -y, z)|
|IV||(x, -y, z)|
|V||(x, y, -z)|
|VI||(-x, y, -z)|
|VII||(-x, -y, -z)|
|VIII||(x, -y, -z)|
So, now we know about the coordinate system, its parts, and how to represent objects in two and three dimensions. How about practicing them with a few examples?
Learning what is a coordinate plane with examples
Example 1: How can we represent (-3, 2) in a blank coordinate plane?
Solution: After observing the coordinate points, we can see x coordinate is negative, whereas y is positive. It means this point lies in the 2nd quadrant of the Cartesian plane. Therefore, taking 3 and 2 units in the negative and positive side of the x and y axes, respectively, we get the point in the blank coordinate plane as shown:
Example 2: Represent (-4, -2) in a coordinate plane graph.
Solution: By examining the coordinates (-4, -2), we know both x and y coordinates are negative. Therefore, this coordinate will lie in the 3rd quadrant.
Taking 4 units on the negative side of the x-axis and 2 units on the negative side of the y-axis, we represent (-4, -2) as shown below:
Example 3: Plot A (1,2), B ( 3, -2), C (-3, -3), D ( -4, 2), E (-1, 2), F (3,0), G (0, -3) on a Cartesian plane. Also, tell which points lie in the second and third quadrant?
Solution: From the points, we get to see,
A lies in the 1st quadrant
B in the 4th quadrant
C in 3rd quadrant
D in 2nd quadrant
E in 2nd quadrant
F in 1st quadrant, because even though the, y coordinate is zero, the x coordinate is positive. Hence, 1st quadrant. We consider 0 to be positive while figuring out the coordinate plane quadrants.
G in the 4th quadrant, can you figure out why?