Cartesian Coordinates: Definition, Systems, Diagrams and Example Problems

Cartesian Coordinates: Definition, System, Diagram and Example Problems – What do you mean by Cartesian coordinates? On this occasion About the knowledge.co.id will discuss about Cartesian Coordinates and the things that surround it. Let's look at the discussion together in the article below to better understand it.

Cartesian Coordinates: Definition, Systems, Diagrams and Example Problems

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Cartesian coordinates a formulation in mathematics that plays an important role in the combination of algebra and geometry so it would produce Descartes, Cartesian coordinates, and which had a major influence on the development of geometry analytic. The use of this system was developed in 1637 in two of his writings which introduced new suggestions for indicating the state or position of the points of an object on a surface.

Cartesian coordinates are also often referred to as square coordinates. The term Cartesius is used to commemorate a French mathematician and philosopher named Rene Descartes. He is an expert who has a big role in combining algebra and geometry.

The results of Descartes' discoveries, Cartesian coordinates were very influential in the development of analytic geometry, calculus, and cartography. The initial rationale for using this system was developed in 1637 in two of Descartes' writings.

In his Descartes Discourse on Method, he introduces a new suggestion for indicating the state or point position of an object on a surface. This method is to use two mutually perpendicular axes in a work of La Géométrie, in the concept that will be developed.

So in Cartesian Coordinates you can jump from the top point if the points have been marked in between

[-3.1], [2.3], [-1.5, -2.5] and [0.0]. as dot [0,0] is also called the origin of the sentence.

Because the two axes are perpendicular to each other in the xy plane which is divided into four parts it is called a quadrant and can be seen at the marked points [-3.1], points [2.3], points [-1.5, -2.5].

By convention they can be sorted in opposite directions starting from the top right in a In quadrant I, and both coordinates (x and y) are positive results.

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Coordinate System

The Cartesian coordinate system in two dimensions will generally be found to be defined by two mutually perpendicular axes and both of which lie in one plane (the xy plane).

In the combination of the horizontal axis labeled x and the vertical axis which will be labeled y with a three-dimensional coordinate system as axes that are orthogonal to each other.

At the intersection of the two axes the origin will generally be designated 0 and has a unit length scale marked in a sort of lattice shape.

Function to describe a certain point in a two-dimensional coordinate system with an x ​​value (abscissa) followed by a y value (ordinate) as the format used (x, y).

The axes that are mutually perpendicular in the xy plane are marked with the numbers I, II, III and IV and will apply to the x coordinates with a negative sign and y is positive.

The position of the Cartesian coordinate point written in pairs on the number (x, y) is.

• x is called the abscissa, as well
• y is called the ordinate

In coordinates to be.

• Point A is at coordinates (1,0), with A(1,0)
• Point B is at coordinates (2,4), with B(2,4)
• Point C is at coordinates (5,7), with C(5,7)
• And point D is at coordinates (6,4) with D(6,4)

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Cartesian coordinate function

In mathematics, the system of Cartesian coordinates is used to determine each point inside plane by using two numbers commonly referred to as the x-coordinate and also the y-coordinate of that point.

The x coordinate is often also referred to as the abscissa, while the y coordinate is often referred to as the ordinate.

To interpret the coordinates, two directed lines are needed that are perpendicular to each other [the x axis and the y axis]. As well as the unit length, for which markings are made on both axes.

Look carefully at the image below:

From the picture above, we can see if there are 4 points that have been marked. Among others: [-3,1], [2,3], [-1.5,-2.5] and [0,0]. The point [0,0] is also called the origin.

From the picture above, we can see that:

Because the two axes are perpendicular to each other, the xy plane will be divided into four parts which are known as quadrants. This can be seen in the figure above marked by points [-3,1], points [2,3], points [-1.5,-2.5].

According to the convention, the four quadrants are ordered starting from the top right [quadrant I], circularly anticlockwise.

In quadrant I, both coordinates (x and y) will be positive.

In quadrant II, the x coordinate will be negative and the y coordinate will be positive.

In quadrant III, both coordinates will be negative.

And in quadrant IV, the x coordinates will be positive and y will be negative.

Point [2,3] is in quadrant I, point [-3,1] is in quadrant II and point [-1.5,-2.5] is in quadrant III.

Or in general, the four quadrants are sorted starting from the top right [quadrant I], circularly anticlockwise.

In quadrant I, both coordinates [x and y] will be positive.

In quadrant II, the x coordinate will be negative and the y coordinate will be positive.

In quadrant III, both coordinates will be negative, and in quadrant IV, the x coordinate will be positive and y negative [note again in the picture above].
Quadrant Value x Value y
I is positive [> 0] is positive [> 0]
II is negative [< 0] positive value [> 0]
II is negative [< 0] is negative [< 0]
IV is positive [> 0] is negative [< 0]

The system of cartesian coordinates in two dimensions is generally defined by using two axes that are mutually perpendicular to one another.

Where the two locations of the axes are in one plane, namely the xy plane. The horizontal axis will be labeled x, while the vertical axis will be labeled y.

The point where the two axes meet, the origin, will generally be labeled 0.

Each axis also has a unit length, and each of these lengths will be marked so that it forms a kind of grid.

To describe a certain point in a two-dimensional coordinate system, the x value is written [abscissa], followed by the y value [ordinate].

That way, the format used will always be [x, y] and the order will not be reversed.

The Cartesian coordinate system can also be used in the higher dimensions.

For example: 3 [three] dimensions, using three axes namely the x-axis, y-axis and z-axis.

If in two dimensions the line is in the xy plane, then in a three-dimensional coordinate system, another axis will be added which is often labeled z.

Where this z-axis is mutually perpendicular to the x-axis and y-axis [in other words, the x-axis, y-axis, and z-axis are mutually perpendicular or orthogonal].

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Determining Points in the Cartesian Coordinate System

The flat plane above is referred to as the coordinate plane formed by the Y vertical line (Y axis) and the X horizontal line (X axis).

The points will intersect between the Y line and the X line, which is called the coordinate center (point O).

These coordinates are known as Cartesian coordinate planes. As explained above, the Cartesian coordinate plane is used to determine the location of a point expressed in pairs of numbers.

Note points A, B, C, and D in the plane. To determine the position, start from point O. Then, move horizontally towards the right (X axis), then move up (Y axis).

The position of the point on the Cartesian coordinate plane is written in the form of a number pair (x, y), where:

x is referred to as the abscissa, as well
y is called the ordinate.

In the coordinate plane, then:

Point A is at coordinates (1,0), written as A(1,0).
Point B is at coordinates (2,4), written B(2,4).
Point C is at coordinates (5,7), written as C(5,7).
And point D is at coordinates (6,4) written as D(6,4).

In the Cartesian coordinate plane we can expand it to be like in the image below:

As an example:

The coordinates of point E are (2,2)
The coordinates of point F, namely (-2,1), are obtained by moving horizontally to the left starting from point O by two units then vertically upwards by one unit.
The coordinates of point G, namely (-3,-3), are obtained by moving horizontally to the left starting from point O by three units then vertically down by three units.

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Cartesian benefits

By using the Cartesian coordinate system, we can describe geometric shapes such as curves using algebraic equations. In this modern era, Cartesian coordinates have been widely used. The following are some of the benefits of Cartesian coordinates, including:

First:

In everyday life we ​​often find floor plans and maps. Where is the function of the map itself to make it easier for us to find a location or place or area. Likewise when we want to send a letter to someone. In sending a letter to someone, we must know the complete and correct address of the destination.

It aims to facilitate the delivery of the letter itself. So, if we include the address correctly and completely, the letter will arrive faster. The map also shows latitude and longitude.

Second:

In everyday life, Cartesian coordinates are absolutely necessary. One of them is in matters of aviation. A pilot can fly his airplane without colliding with each other and can also find out if the plane has reached its destination.

This is because the aircraft has been equipped with sophisticated equipment such as a radar as a detection device, a compass as a direction guide, and also a radio as a means of communication. Therefore a pilot must understand how to read and determine the location of a place in the Cartesian coordinate plane.

Third:

In social studies lessons, we often encounter a map of a province or even a map of a country. We can describe the position of a city, mountain, lake, airfield as a position. To facilitate map reading, the map is equipped with horizontal and vertical guidelines or latitude and longitude lines. The basis for making the line which is the basis of the coordinate plane.

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Cartesian Coordinate Field

In a field can draw something feeling it's easier in the Cartesian coordinate plane with the plane flat in the coordinate plane on the vertical Y line (called the Y axis) and the horizontal X line (called the Y axis). X).

The intersection of the X and Y axes is called the center coordinate or base coordinate, so these coordinate planes are called Cartesian coordinate planes.

Coordinate planes can be used to define positions with specified points in a number pair, for example the x and y axes are divided into x axes. and will get a positive result and a negative y axis.

Quadrant I of the x-axis and y-axis positive results
Quadrant II of the x-axis and y-axis positive results
Quadrant III of the x-axis and y-axis negative results
Quadrant IV of the x-axis and y-axis results are negative

Accept this example!

Point B lies I with positive x – y values
Reach point II on positive and negative x values
Point D is in quadrant III in negative x and y values
Point A is in quadrant IV in positive x and negative values

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Examples of Problems and Discussion of Cartesian Coordinates

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  Problem 1

The ordinate of point A (9, 21) is.

a. -9
b. 9
c. -21
d. 21

Answer:

In general write point = (abscissor, ordain), In the above problem, point A (9, 21) is.

abscissa = 9

Ordinate = 21

The correct answer is D.

• Problem 2

Which quadrant are the points below located in?

(2,3)
(3,3)
(-4,7)
(85,-77)
(-54,2)

Answer

(2,3) Located in quadrant I
(3,3) Located in quadrant I
(-4.7) Located in quadrant II
(85,-77) Located in quadrant IV
(-54.2) Located in quadrant III

• Problem 3

The known points P(3, 2) and Q(15, 13) which will be relative to the point Q with respect to P are called.

a. (12, 11)
b. (12, 9)
c. (18, 11)
d. (18, 13)

Answer:

We can find the relative coordinates from point Q to point P by subtracting the numbers.

a. Abscissa Q minus abscissa P

b. The Q ordinate minus the P ordinate

c. So the Q coordinate is relative to P

d. (15-3, 13-2) = (12, 11)

Correct answer. A

• Problem 4.

The ordinate of point A (9, 21) is…

a. -9
b. 9
c. -21
d. 21

Answer:

In general, writing a point = (abscissa, ordinate). In the problem above point A (9, 21) shows if:

Abscess = 9

Ordinate = 21

The correct answer is D.

• Problem 5.

The points P (3, 2) and Q (15, 13) are known. The relative coordinates of point Q to P are...

a. (12, 11)
b. (12, 9)
c. (18, 11)
d. (18, 13)

Answer:

We can find the relative coordinates of point Q to point P by subtracting:

a. The abscissa of Q minus the abscissa of P

b. The Q ordinate minus the P ordinate

Thus, the relative coordinates of Q to P are:

(15 – 3, 13 – 2) = (12, 11)

So, the correct answer is A.

• Problem 6.

The complement of an angle of 48 degrees is...

a. 42°
b. 52°
c. 68°
d. 138°

Answer:

Complement = 90 – 48 = 42

So, the correct answer is A.

• Problem 7.

Points A (3, 2), B (0, 2) and C (-5, 2) as points crossed by the p line parallel to the p-line, q-line

a. Parallel to the x-axis
b. Parallel to the y-axis
c. Perpendicular to the x-axis
d. Perpendicular to the y axis

Answer: d

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