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Cartesian System
“Algebra” and “Geometry” were the two main branches of ancient mathematics. Geometry did not use algebraic equations, and algebra did not use geometric figures either. For the first time, René Descartes, the great French mathematician, connected these two schools of thought. In order to better explain both geometry and algebra, he developed the Cartesian plane or frame of reference. Most coordinate systems use two numbers, a coordinate, to identify a condition to some extent. There are a few fixed points of reference, known as the origin, that each of these numbers represents a distance from the aim. If the X value is more than zero, then the point is further to the right or left than it was originally. The second number, known as the Y value, specifies how above or below the point is from the origin. The starting point has a coordinate of (0, 0)
Different Types of Coordinate Systems
Cartesian Coordinate System
The Cartesian System of Coordinates is a good illustration of a coordinate system in action. Points in the plane are picked based on any two perpendicular lines and their signed distances from each other.
3D Cartesian System
The three coordinates of a point in three dimensions are typically the signed distances to each of the orthogonal planes. For any point in an n-dimensional Euclidean space, we can generalise to create n coordinates. The three-dimensional system can either be a right-handed system or a left-handed system depending on the direction that’s left or right and the arrangement of the coordinate axes. This is frequently one of many different systems of coordinates.
Polar Coordinate System
The coordinate system is another widely used plane-coordinate system. Because of the polar axis, a point is selected and a ray is drawn from it. There is one line through the pole whose angle with the polar axis equals for any given angle (can be measured counterclockwise from the axis to the line). Then, for any given value of r, there exists a single point on this line with a signed distance from the origin of r. There is only one point that can be represented by a given set of coordinates (r, ). For example, the polar coordinates of an analogous point are (r,θ ), (r, θ+2π), and (-r, θ+π). For any value of, (0, θ) can be used to represent the pole.
Quadrants of Coordinate System
A quadrant is an area between the x and y axes, hence there are four in a graph. According to the x and y axes, the Cartesian plane has four quadrants that are separated into two. Quadrant I is located in the upper right corner, while the next quadrants are numbered from II through IV clockwise. Four quadrants make up the Cartesian plane. x and y are both positive in Quadrant I; in Quadrant II, the x-coordinate is negative, but the y-coordinate is positive; in Quadrant III, both x and y are negative; and in Quadrant IV, x is positive and y negative.
Cartesian plane
For the sake of simplicity, the signed distances between two fixed perpendicularly oriented lines can be used to establish a Cartesian system in terms of distance measurements in the same unit. Descartes, a French mathematician, defined the application of the Cartesian plane in mathematics, which has two perpendicular number lines: the horizontal x-axis and the vertical y-axis. To characterise each point on the plane, we can use an ordered pair of numbers to do so.
Number Line
A number line is exactly what it sounds like: a horizontal line with even numbers arranged at regular intervals. However, because it is not a ruler, the distance between each number is irrelevant; instead, it is determined by those numbers that are printed on the line itself. A number ladder is just a number line that has been stacked on top of one another.
Application of Cartesian plane
- Describing position: The position of any object in the actual world can be defined using a basic coordinate system. For example, you could define your phone’s position as being 2 metres across from the entrance, 3.5 metres up from the floor, and 4 metres in front of the window.
- Location of Air Transport: If one needs to know where anything is located, a coordinate plane is a highly important resource. Mapping-based applications are prevalent as a result. In a given area, an ATC must be able to pinpoint the precise location of every plane in the sky. Each airborne vehicle is given a unique set of coordinates to help pinpoint its exact location. To avoid this problem, the “air traffic controller” might provide specific coordinates for each “aircraft.” It’s also possible for the “plane” to report to the correct place. As a result, air transportation relies heavily on the use of coordinate systems. If there were no coordinate systems, pilots and others involved in flying would have no way of determining the precise location of their aircraft, increasing the likelihood of an accident.
- Map Projections: Coordinate systems for flat surfaces, such as printed maps or computer screens, are called “projected coordinate systems.” Geographical information can be expressed in x and y coordinates in two dimensions or three dimensions using Cartesian coordinate systems.
Conclusion
Using the cartesian system, a point can be represented in the n-dimensional coordinate plane uniquely. Rene Descartes, a 17th-century French philosopher and mathematician, first suggested the cartesian system theory. The cartesian coordinate system changed the study of mathematics by providing a link between Euclidean geometry and algebra. Analytical geometry relies on the cartesian coordinate system to represent objects in the n-dimensional plane, including lines, curves, and geometric forms.
