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 to 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 a 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 the 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 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 a number of coordinates for any pointanin 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 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 the vertical number line is understood as Y-axis and therefore the point of intersection of those two axes is understood as the origin and it's denoted as ‘ O ‘.

Note:

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 the 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 the 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.

2. Polar Coordinate System

Another common coordinate system for the plane is that 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.

Plotting Coordinates on Graph

We always write coordinates in brackets, with a comma separating the two coordinates. The first number represents the point on the X-axis, and the second number represents the position on the Y-axis, as coordinates are ordered pairs of numbers.

When reading or plotting coordinates, we always start at the first and work our way up. This is a good way to remember it: Up the stairs and across the landing. We follow the X-axis until we reach 4 and draw a vertical line at x=4 to plot the points (4,5) in the Cartesian coordinate plane.

Similarly, we construct a horizontal line at y=5 by following the Y-axis until we reach 5. In the Cartesian plane, the intersection of these two lines equals (4,5). This position is 4 units from the Y-axis and 5 units away from the X-axis. As a result, the position of (4,5) is in the Cartesian plane.

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 around 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.

Question 3) In which quadrant do the following points lie?

(a) (3,–8)

(b) (–1,–3)

(c) (2,5)

(d) (–7,3)

Answer: 

(a) The x coordinate is positive, and the coordinate is negative. So, point (3,–8) lies in the IV quadrant

(b) The x coordinate is negative, and the coordinate is negative. So, point (–1,–3) lies in the III quadrant.

(c) The x coordinate is positive, and the coordinate is positive. So point (2,5) lies in the I quadrant.

(d) The x coordinate is negative, and the coordinate is positive. So, point (–7,3) lies in the II quadrant.

Question 4) Plot the following points 

A(2,2),B(–2,2),C(–2,–1),D(2,–1)

A(2,2),B(–2,2),C(–2,–1),D(2,–1) in the Cartesian plane. Discuss the type of the diagram by joining all the points taken in order.

Question 5) Calvin is required to locate the points M (3, 0), N (3,5), and P (3, -2) on a Cartesian system and check if they are collinear. Help Calvin do this.