Coordinate System

🏆Practice coordinate system

What is the coordinate system?

A coordinate system, or more precisely, a Cartesian coordinate system, is a method to precisely present the position of points, whether on a plane (two-dimensional) or in three-dimensional space. In this chapter, we will focus on a coordinate system on a plane, that is, a system made up of two axes (straight lines) that are perpendicular to each other (forming a 90-degree angle between them).

To facilitate the understanding of the material, let's look at the following drawing, which gives an example of a two-dimensional coordinate system:

coordinate system image

As mentioned, as can be seen in the drawing, it is customary to identify the horizontal axis with X X and the vertical axis with Y Y .

The intersection point between the two axes is called the origin and is usually identified with the letter O O .

Along the X X horizontal axis, the numbers increase to the right of the origin and decrease to the left of the origin.

Along the Y Y vertical axis, the numbers increase upwards in relation to the origin and decrease downwards in relation to the origin.

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Test yourself on coordinate system!

einstein

Which point is marked on the graph?

–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777–3–3–3–2–2–2–1–1–1111222333000

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Characterization of a Point in a Cartesian Coordinate System

Identifying a point in the coordinate system is done using two values, X X and Y Y .

In the next step, we will use the drawing below and demonstrate how a point can be characterized on a two-dimensional plane.

Identifying a point using two values, X and Y


Quadrants in a Coordinate System

The two axes divide the plane into four quadrants. As you can see in the illustration, the first quadrant is the upper right part, marked with an I I , and the rest follow in a counterclockwise sequence.

Each of the quadrants is characterized by different values of X X and Y Y , which are detailed below:

  • Quadrant I - positive X X values and positive Y Y values
  • Quadrant II - negative X X values and positive Y Y values
  • Quadrant III - negative X X values and negative Y Y values
  • Quadrant IV - positive X X values and negative Y Y values

Quadrants in a Coordinate System

1 -  Quadrants in a coordinate system


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Characterization of a Point in a Cartesian Coordinate System

Identifying a point in the coordinate system is done using two values, X X and Y Y .

In the next step, we'll use the drawing below and demonstrate how a point can be characterized in a two-dimensional plane.

done using two values, X and Y

In the drawing, we see the point B B . We are interested in characterizing it. To do this, first, we'll draw two vertical lines from the point B B to each of the two axes, as shown in the drawing.

We see that on the horizontal X X axis, the point "meets" the value 5 5 , while on the vertical Y Y axis, the point "meets" the value 2 2 .

It's common to write the values as points in parentheses, where the value of X X appears as the first value on the left, while the value of Y Y appears as the second value, on the right.

Therefore, point B B would be written as follows:

B(5,2) B (5,2)


Characterizing Points Located on One of the Axes

When the point is on one of the coordinates (for example, points E E and G G in the illustration)

Points in a Cartesian Coordinate System

2 points in a cartesian coordinate system

Remember the following rule:

  • When the point is on the X X axis, the Y Y value is 0 0 , which means it has coordinates: E(4,0) E (4,0)
  • When the point is on the Y Y axis, the value of X X is 0 0 , which means it has coordinates: G(0,3) G (0, 3)

Do you know what the answer is?

Examples and Practice on Coordinate Systems

Exercise 1

Look at the following coordinate system and these are the points A,B,C,D A, B, C, D .

coordinate system and these are the points A, B, C, D

Solution: 

We will apply what we've learned and drop verticals from the points A,C,D A, C, D to each of the axes 

It should be noted that point B B is on the X X axis, and therefore, its Y Y value is 0 0

Answer: 

A(2,2) A (2,2)

B(1,0) B (-1,0)

C(2,3) C (-2,3)

D(3,2) D (3,-2)

Exercise 2

Plot the following points on the coordinate system:

k(0,5) k (0,5)

L(1,5) L (-1,5)

M(0,0) M (0,0)

N(4,0) N (-4,0)

P(2,4) P (2,4)

coordinate system image


Solution:

For points L L and P P , you need to find the X X values and the Y Y values where the intersection of these values is indeed the desired point.

The points K K and N N are located on the axes themselves, and therefore the following rule applies:

When the point is on the X X axis, the Y Y value is 0 0

When the point is on the Y Y axis, the X X value is 0 0

The point M is located at the intersection of the two axes.

Answer:

See the following drawing:

the following drawing


If you're interested in more information about "graphs," you can find detailed information in the following articles:

Data Collection and Organization - Statistical Research

Reading Information from Graphs

Graph

Discrete Graph

Continuous Graph

In the Tutorela blog, you will find a variety of articles with interesting explanations about mathematics


Examples and Exercises with Solutions for the Coordinate System

Exercise #1

Which point is marked on the graph?

–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777–3–3–3–2–2–2–1–1–1111222333000

Video Solution

Answer

(6,1) (6,1)

Exercise #2

Which point is marked on the map?

–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777–3–3–3–2–2–2–1–1–1111222333000

Video Solution

Answer

(5,3) (5,3)

Exercise #3

Choose the appropriate drawing where the dots appear:

(5,1),(7,0),(4,1) (-5,1),\lparen7,0),(-4,-1)

Video Solution

Answer

–7–7–7–6–6–6–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888999101010111111121212131313–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555000

Exercise #4

Choose the appropriate drawing where the dots appear:

(1,3),(2,5),(3,3) (1,3),\lparen2,5),(3,3)

Video Solution

Answer

–8–8–8–7–7–7–6–6–6–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888999101010111111–4–4–4–3–3–3–2–2–2–1–1–1111222333444555000

Exercise #5

Choose the appropriate drawing where the dots appear:

(1,0),(2,2),(5,4) (1,0),\lparen-2,-2),(-5,4)

Video Solution

Answer

–8–8–8–7–7–7–6–6–6–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888999101010111111–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555000

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