# 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:

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

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

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

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

## Test yourself on coordinate system!

Which point is marked on the map?

## Characterization of a Point in a Cartesian Coordinate System

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

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

## 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 $I$, and the rest follow in a counterclockwise sequence.

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

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

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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, $X$ and $Y$.

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

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

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

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

Therefore, point $B$ would be written as follows:

$B (5,2)$

## Characterizing Points Located on One of the Axes

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

Points in a Cartesian Coordinate System

Remember the following rule:

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

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 $A, B, C, D$.

Solution:

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

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

$A (2,2)$

$B (-1,0)$

$C (-2,3)$

$D (3,-2)$

### Exercise 2

Plot the following points on the coordinate system:

$k (0,5)$

$L (-1,5)$

$M (0,0)$

$N (-4,0)$

$P (2,4)$

Solution:

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

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

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

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

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

See 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

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 map?

### Video Solution

$(5,3)$

### Exercise #2

Which point is marked on the graph?

### Video Solution

$(6,1)$

### Exercise #3

Choose the appropriate drawing where the dots appear:

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

### Exercise #4

Choose the appropriate drawing where the dots appear:

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

### Video Solution

$(2,0),(2,4),(6,-2)$