Coordinate System

What is A Coordinate System?

Mathematics in ancient days was divided into two branches namely ‘Algebra’ and ‘Geometry’. Algebraic equations weren't utilized in geometry and geometrical figures weren't utilized in algebra. But these two branches were put together by the French mathematician Rene Descartes for the primary time. He introduced the concept of the plane or frame of reference to elucidate geometry and algebra together.Most coordinate systems use two numbers, a coordinate, to spot the situation of some extent . Each of those numbers indicates the distance between the purpose and a few fixed points of reference , known as the origin. The primary number, referred to as the X value, indicates how left or right the purpose is from the origin. The second number, referred to as the Y value, indicates how far above or below the purpose is from the origin. The origin features a coordinate of 0, 0.

What is Number Line?

The simplest example of a coordinate system can be the identification of points on a line with real numbers using the concept of number line. In the system of number lines, an arbitrary point O (the origin) is chosen on a given line. The coordinate of a point namely P is defined as the signed distance from the point O to point P, where the signed distance is the distance taken as positive(+ve)  or negative (-ve) depending on which side of the line named P lies. Each point on the number line is assigned a unique coordinate and each real number is the coordinate of a unique point on the number line.

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Different Types of Coordinate Systems

  1. Cartesian Coordinate System-

The prototypical example of a coordinate system can be the Cartesian Coordinate System. In the plane, any two perpendicular lines are chosen and then the coordinates of a point are taken to be the signed distances to the lines.

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In three dimensions, we generally need to choose mutually orthogonal planes and the three coordinates of a point are generally the signed distances to each of the planes. We can generalize to create n number of coordinates for any point in a n-dimensional Euclidean space.

Depending on the direction that's left or right and therefore the order of the coordinate axes, the three-dimensional system could also be a right-handed system or a left-handed system. This is often one among the various coordinate systems.

Let’s Know What is Right-Handed vs. left-Handed

In a right-handed coordinate system the direction in which your hand closes to make a fist can be defined as the direction of a positive rotation around any axis that can be represented by the extended right-hand thumb.

In a left-handed coordinate system the direction when your hand closes to form a fist is that the direction of a positive rotation around any axis which will be represented by the extended left-hand thumb.

A coordinate system or frame of reference is used to locate the position of any point which points are often plotted as an ordered pair (x, y) referred to as Coordinates. The horizontal number line is named as X- axis and vertical number line is understood as Y-axis and therefore the point of intersection of those two axes is understood as origin and it's denoted as ‘ O ‘.

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1. The coordinate plane is also known as a Two- dimensional plane.

2. X-axis can be named as XX’ and Y -axis can be named as YY’

Quadrants of Coordinate System

For quadrants of coordinate system the Coordinate axes named XX’ and YY’ divides the cartesian plane into basically four quadrants. In the fig. 3 shown below, see the following:

  1. The region XOY is known as the first quadrant.

  2. The region X’OY is known as the second quadrant.

  3. The region X’OY’ is known as the third quadrant.

  4. The region Y’OX is known as the fourth quadrant.

These are the quadrants of coordinate system.

Sign Convention

The ray named OX on the X-axis is taken as positive, and the ray named OX’ as negative X-axis, OY on Y-axis as positive and OY’ as negative.

Accordingly, the distance measured along OX will be taken as positive and along OX’ will be taken as negative.

  1. I- quadrant (Positive, Positive) (+,+)

  2. II-quadrant (Negative, Positive) (-,+)

  3. III-quadrant(Negative,Negative) (-,-)

  4. IV-quadrant(Positive, Negative) (+,-)

Similarly, the distance along OY can be taken as positive and along OY’ can be taken negatively.

Polar Coordinate System

Another common coordinate system for the plane is that the coordinate system.A point is chosen because the pole and a ray from now are taken because of the polar axis. For any given angle suppose θ, there's one line through the pole whose angle with the polar axis is θ (can be measured counterclockwise from the axis to the line). Then there's a singular point on this line whose signed distance from the origin is r for given number r. For a given pair of coordinates (r, θ) there's one point, but any point is represented by many pairs of coordinates. For instance , (r, θ), (r, θ+2π) and (−r, θ+π) are all polar coordinates for an equivalent point. The pole can be represented by (0, θ) for any value of θ.

Fun Facts

The Coordinate System has its use in Geography too.

Longitude and latitude are a special kind of coordinate system, known as a spherical coordinate system, since they identify points on a sphere or globe.  However, there are many other coordinate systems utilized in different places round the world to spot locations in the world . All of those coordinate systems place a grid of vertical and horizontal lines over a flat map of some part of the world .

 A complete definition of a coordinate system requires the following points given below:

  • It is the projection that can be used to draw the earth on a flat map.

  • Location of the origin.

  • The units that can be used to measure the distance from the origin.

Questions to be solved

Question 1) In which quadrant is the point with coordinates (–3, –21)?

Answer ) The point (-3, -21)in the Third Quadrant.

Question 2) In which quadrant is the point with coordinates (–3, 21)?

Answer ) The point (-3, 21) lies in the Second Quadrant.

FAQ (Frequently Asked Questions)

1. What are the different types of coordinate systems?

The different types of coordinate systems are :- Horizontal coordinate systems locate data across the surface of the world , and vertical coordinate systems locate the relative height or depth of knowledge . Horizontal coordinate systems are often of three types: geographic, projected, and local.

2. What is a coordinate system in physics?

Coordinate systems can be used to describe the position of an object generally in space. A coordinate system can be defined as an artificial mathematical tool that we can construct in order to describe the position of a real object.

3. What are the foremost common coordinate systems?

A geographic frame of reference uses longitude and latitude expressed in decimal degrees. For instance , WGS 1984 and NAD 1983 are the foremost common datums today. Before 1983, NAD27 was the foremost common datum.

4. What is the standard coordinate system called?

A Cartesian coordinate system in two dimensions (also known as a rectangular coordinate system or an orthogonal coordinate system) can be defined by an ordered pair of perpendicular lines (axes), a single unit of length for both axes, and an orientation for each axis.

5. Why do we use a coordinate system?

Coordinate systems enable geographic datasets to use common locations for integration. A frame of reference may be a coordinate system wont to represent the locations of geographic features, imagery, and observations, like Global Positioning System (GPS) locations, within a standard geographic framework.