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Coordinate Systems

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Coordinate Systems

Being able to describe something's geometric position is fundamental to a useful and robust mathematical system. So how exactly have we as humans taken to doing this? Well, we use coordinate systems of course! From maps to graphs, and the pixels on the screen you are viewing this on, coordinate systems are everywhere in our everyday lives if you look closely enough, and life just wouldn't be the same without them!

Coordinate systems are systems employed to describe the position of points in a defined space using one or multiple numbers.

Coordinate System Meaning

There are many types of coordinate systems, some may be very familiar, such as simple number lines, and others you may not be so familiar with, such as polar coordinate systems. The important thing that all coordinate systems have in common, is that they give us a means of describing where a point is in space, relative to some origin, i.e. the zero point of the system.

The number line below is a very simple example of a coordinate system.

Coordinate systems number line, StudySmarterExample of a number line, StudySmarter Originals

The position of a point on the number line can be expressed as a single number. This number essentially denotes the horizontal distance of that point from the origin, i.e. where the is marked. The position of any point in the coordinate system is made in reference to the origin.

For instance, point on the number line below is 3 units from the origin.

Coordinate systems, number line, StudySmarterPoint A on a number line, StudySmarter Originals

Now, let's take a closer look at some of the main coordinate systems you might come across.

Different Types of Coordinate Systems

There are two primary types of coordinate systems in mathematics: the cartesian coordinate system and the polar coordinate system. The cartesian coordinate system is defined by distances along a set of perpendicular number lines, whereas the polar coordinate system is defined by an angle and radial distance. Let's dig a little deeper into each to get a better understanding.

The Cartesian Coordinate System

By combining one vertical and one horizontal number line, we get the cartesian coordinate system. Each of these number lines is known as an axis, together creating a plane known as the coordinate plane.

Coordinate systems, cartesian coordinate system, StudySmarterCartesian Coordinate System, StudySmarter Originals

Any point on the coordinate plane can be described using two numbers, one to describe the distance along the horizontal axis, also known as the x-axis, and the other to describe the distance up the vertical axis, also known as the y-axis. Each of these numbers is known as a coordinate.

The notation for the coordinates of a point in the cartesian coordinate system is simply a pair of numbers within brackets, the first representing the distance of a point from the origin along the x-axis, and the other representing the distance of a point from the origin along the y-axis.

For instance, the point on the below cartesian coordinate system has coordinates , as it is steps along the -axis, and steps up the -axis.

coordinate systems cartesian coordinates example studysmarterPoint in a cartesian coordinate system, StudySmarter Originals

By extending each axis, and to include numbers less than zero, i.e. negative numbers, we can reveal the full scope of the cartesian coordinate plane.

Coordinate systems, cartesian coordinates example four quadrants, StudySmarterFour quadrant cartesian coordinate plane, StudySmarter

  1. If the point is located in the upper-right quadrant the two coordinates x and y will be positive.

  2. If the point is located in the upper-left quadrant then the x coordinate will be negative and the y coordinate will be positive.

  3. If the point is located in the lower-left quadrant then the two coordinates x and y will be negative.

  4. Finally, if the point is located in the lower-right quadrant then the x coordinate will be positive and the Y coordinate will be negative.

(1)

What are the coordinates of point C in the cartesian coordinate system below?

Coordinate systems finding coordinates, StudySmarterCoordinate in the upper-left quadrant of the cartesian coordinate plane, StudySmarter

Solution:

By noticing that point C is located in the upper-left quadrant, we can assert that it must have a positive coordinate and negative coordinate.

From observation, we can deduce that the x coordinate is, as the perpendicular horizontal distance between point C and the y-axis is 3 units.

On the other hand, the y coordinate is, since the perpendicular vertical distance between point C and the x-axis is 5 units.

Therefore, the cartesian coordinates of point C are

(2)

What are the coordinates of point D in the cartesian coordinate system below?

Coordinate systems, cartesian coordinates example, StudySmarterCoordinate in the lower-left quadrant of the cartesian coordinate plane, StudySmarter

Solution:

By noticing that point D is located in the lower-left quadrant, we can assert that it must have a negative coordinate and a negative coordinate.

From observation, we can deduce that the x coordinate is, as the perpendicular horizontal distance between point D and the y-axis isunits.

On the other hand, the y coordinate is since the perpendicular vertical distance between point C and the x-axis is units.

Therefore, the cartesian coordinates of point C are

It is worth mentioning that 2 special cases will occur when the point is located on the x or the y-axes.

This concept can be better explained by the following two examples:

(3)

What are the coordinates of point E in the cartesian coordinate system below?

Coordinate systems, cartesian coordinates example, StudySmarterCartesian coordinates example,StudySmarter Originals

Solution:

Since the point E actually lies on the y-axis, the distance between it and the origin along the x-axis is in fact therefore the x coordinate is.

On the other hand, since the vertical distance between point E and the origin isunits, it is clear that the y coordinate is.

So, we are able to conclude that the cartesian coordinates of E are

(4)

What are the coordinates of point F in the cartesian coordinate system below?

Coordinate systems, cartesian coordinates example, StudySmarterCartesian coordinates example, StudySmarter Originals

Solution:

Since the point F lies on the x-axis, the distance between it and the origin along the y-axis is therefore the y coordinate is

On the other hand, since the horizontal distance between point F and the origin is units, it is clear that the x coordinate is

So, we are able to conclude that the cartesian coordinates of E are

After the two examples given before, we can conclude the following:

  1. If a point is located on the y-axis, then thecoordinate is

  2. If a point is located on the x-axis, then thecoordinate is

Polar Coordinate System

The polar coordinate system bears similarities to the cartesian coordinate system in that the position of any point can be defined by two numbers. However, rather than these two numbers indicating distance along perpendicular axes, in the case of polar coordinates, the two numbers indicate radial distance and angular distance.

What exactly do we mean by this? Let's take a look at a polar coordinate system to find out!

You can see that the polar coordinate system below, rather than being made up of two perpendicular axes, is in fact made up of many concentric circles, with radial lines outward from their common center indicating angles.

Coordinate systems, polar coordinate system, StudySmarterExample of a polar coordinate system,StudySmarter Originals

Any point on this coordinate system can be found by first moving along the number line at by the desired amount, and then performing a circular rotation. In essence, the two coordinates are a radius, , and angle . This is written as

Let's take point A below as an example. To reach point A, we simply move along the number line units, and then make a rotation through

Coordinate systems, finding polar coordinates, StudySmarterFinding polar coordinates, StudySmarter Originals

We could, therefore, say that the polar coordinates of point A are However, more often coordinates in polar coordinate systems are given in radians rather than degrees.

To convert degrees to radians we simply multiply by Therefore, in radians is

So our new polar coordinates for point A are

Let's take a look at another example to make sure we've got it!

What are the polar coordinates of points B and C in the polar coordinate system below? Angles should be converted to radians.

Coordinate systems, polar coordinate example, StudySmarterPolar coordinates example, - StudySmarter Originals

Solution:

Taking point B first, we can see that it can be reached by moving along units along the number line at and then rotating an angular distance of Therefore, the polar coordinates of point B are

To convert to radians we simply multiply it by and so

Therefore the polar coordinates of point B are .

Now we simply do the same for point C. We see that it can be reached by moving along units along the number line at and then rotating through an angular distance of Therefore the polar coordinates of point C are .

But wait! a rotation of is the same as a rotation of . Therefore the polar coordinates can also be written as By multiplying each angle by we get that point C has polar coordinates or alternatively

How exactly do we convert between coordinates in cartesian and polar coordinate systems though? Let's take a look!

Coordinate System Conversion

Converting between the two coordinate systems is simple if we employ some of our knowledge of trigonometry.

Consider the following right-angled triangle in the cartesian coordinate plane. On this triangle, point A's cartesian and polar coordinates have been marked, and respectively.

Coordinate systems, converting polar cartesian coordinates, StudySmarterConverting polar cartesian coordinates, StudySmarter Originals

Using simple trigonometry, we can note the following equations to be true

and

These equations allow us to convert from polar coordinates to cartesian coordinates.

Considering Pythagoras theorem, we can find the following equation for

Finding angular position, , is then a simple case of employing trigonometry once again

or

Let's take a look at a couple of examples to make sure we've got it!

(1)

Convert the cartesian coordinates, , into polar coordinates.

Solution:

To find the coordinate we simply use Pythagoras' theorem

Now to find the coordinate, we use trigonometry

Therefore the polar coordinates are

If you're struggling to get the correct angular coordinate, remember to set your calculator to radians!

(2)

Convert the following polar coordinates, to cartesian coordinates.

Solution:

To convert from polar coordinates to cartesian, we use the following simple trigonometric equations.

and

Starting with the coordinate

And then the coordinate

Therefore the cartesian coordinates are

Use of Coordinate Systems

Coordinate systems are frequently used to represent the location of a point, but they can also be used to specify the location of more complex shapes like lines, planes, circles, or spheres.

Perhaps the most important and oldest use of coordinate systems is in maps. World maps use a special geographic coordinate system that bears similarities to the cartesian coordinate system. Positions in world maps are made up of a latitude coordinate, and a longitude coordinate.

As mentioned earlier in the explanation, screens use coordinate systems as well. Each pixel in a screen has a coordinate that specifies its place horizontally and vertically. This gives each pixel a unique identifier with which it can be located and controlled. This is all going on in the background of the device you are reading this on now!

A modified version of polar coordinates is used in many forms of navigation. Polar coordinates are ideal for navigation as relative position can be defined by the desired angle of movement and the distance between points. The velocity-heading model is a guidance system used in missiles intercepting a moving target, and is based on the idea of polar coordinates!

Coordinate Systems - Key takeaways

  • Coordinate systems are used to locate the position of a point in the plane.
  • In two-dimensional coordinate systems, each point is represented by two integers.
  • Two common coordinate systems are Cartesian and polar coordinate systems.
  • We can convert between polar and cartesian coordinate systems using a mixture of trigonometry and Pythagoras' theorem

Frequently Asked Questions about Coordinate Systems

Coordinate systems are systems used to define the geometric position of points.

Cartesian and Polar coordinate systems.

The rectangular coordinate system is more properly called the cartesian coordinate system. It defines position by using two perpendicular number lines.

Coordinate systems are used in many aspects of everyday life from maps to tv screens.

Converting between cartesian and polar coordinates is a simple case of employing some trigonometry and Pythagoras' theorem.

Final Coordinate Systems Quiz

Question

In a coordinate system, all the points are identified to a reference point having zero coordinates. Is it true or false?

Show answer

Answer

True

Show question

Question

To locate a point in a 2-D space, the plane will be divided into a horizontal line and a vertical line intersecting at the reference point O. Is it true or false?

Show answer

Answer

True 

Show question

Question

Any point in a plane can be represented by one integer. Is it true or false? 


Show answer

Answer

False

Show question

Question

Show answer

Answer

False

Show question

Question

Show answer

Answer

True

Show question

Question

If a point is in the third quadrant of a plane, then:

Show answer

Answer

x and y will be negative

Show question

Question

If a point is in the fourth quadrant of a plane, then: 

Show answer

Answer

x will be positive and y will be negative. 

Show question

Question

If a point is on the x-axis to the right, then: 

Show answer

Answer

x is positive and y=0

Show question

Question

If a point is on the y-axis and below the center point O, then: 

Show answer

Answer

x=0 and y is negative

Show question

Question

There are 2 types of coordinate systems: 

  1. Cartesian Coordinates Systems.
  2. Polar Coordinates Systems.

Is it true or false?

Show answer

Answer

True

Show question

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