Rectangular Coordinate System

In a rectangular coordinate system, two real number lines intersect at right angles. The x-axis is horizontal, and the y-axis is vertical, and on this plane, each point connects to an ordered pair of real numbers (x, y). The x-coordinate is the first, and the y-coordinate is the second number. The intersection of two axes in origin (0, 0).

The ordered pair (x, y) denotes the origin distance. The x-coordinate denotes position to the right or left of the origin, and y-coordinate denotes point above or below the origin. This method uniquely identifies each position (point) in the plane.

René Descartes (1596–1650), a French scientist, invented the cartesian coordinate system. It has four quadrants. Quadrant I has positive coordinates, while quadrant II has negative coordinates. Quadrant III has negative coordinates, whereas Quadrant IV has positive x- and negative y-coordinates.

One dimension

Selecting a Cartesian coordinate system for a one-dimensional space (a straight line) requires choosing an origin, a length unit, and line orientation. An “oriented” line “points” from negative to positive half. With a + or – sign, we may represent the distance from O of each point P on the line.

A number line follows a certain Cartesian system. Every real number has its position on the line. Conversely, each point on the line represents a real number in an ordered continuum.

Two dimensions

In a two-dimensional Cartesian coordinate system, each axis has a length and an orientation. It is on the origin of both axes where they meet, converting them into number lines. We draw a line perpendicular to each axis through P, and the point where it meets the axis reads as a number. P’s Cartesian coordinates are the two integers. It enables determining P given its coordinates.

The axes meet at the origin. The origin is (0, 0), while on the positive half-axes, the points are one unit away, i.e. (1, 0) (0, 1).

The first axis is horizontal and oriented right, whereas the second axis is vertical and oriented upward. The origin is O, and the two coordinates are usually x and y, and the axes are the X- and Y-axes. In the original convention, letters in the latter part of the alphabet indicate unknown values, and the first letters of the alphabet indicate known values.

With a chosen Cartesian coordinate system, a cartesian plane is a Euclidean plane. The unit circle (radius equal to length unit, origin), the unit square (diagonal endpoints at (0, 0) and (1, 1)), and the unit hyperbola are all standard representations of geometric figures on a Cartesian plane.

The plane has four quadrants, and the first quadrant is positive. If a point’s coordinates are (x, y), its distance from the X-axis is |y| and from the Y-axis is |x|.

Three dimensions

Three-dimensional Cartesian coordinates consist of an ordered triplet of perpendicular lines (axes), each with an orientation and a single unit of length. Each axis is a number line, as in two dimensions. A hyperplane is perpendicular to all the coordinate axes for each point P in space and the point where the hyperplane intersects the axis as the number. P’s Cartesian coordinates are the three chosen numbers. The reverse construction determines P Using the three coordinates of P.

Alternatively, the distance between P and the hyperplane formed by the other two axes equals each coordinate of a point P, with the sign determined by the orientation of the corresponding axis.

Each axis pair makes a coordinate hyperplane. These hyperplanes split the space into 8 trihedra called octants.

The octants are: | (+x,+y,+z) | (-x,+y,+z) | (+x,+y,-z) | (-x,+y,-z) | (+x,-y,+z) | (-x,-y,+z) | (+x,-y,-z) | (-x,-y,-z) |

In most cases, three numbers (or algebraic formulae) contained in parentheses and separated by commas represent the coordinates. The origin is at (0, 0, 0), while the axes’ unit points are (1, 0, 0), (0, 1, 0), and (0, 0, 1).

The three axes’ coordinates have no standard names. We can represent the coordinates by the letters X, Y, and Z. The axes are the X, Y, and Z axes, and the coordinate hyperplanes are the XY, YZ, and XZ planes.

The first two axes are usually horizontal in mathematics, physics, and engineering, with the third pointing upwards. So the third coordinate is height. The right-hand rule states that the 90-degree angle between the first and second axes must look counterclockwise when seen from the point (0, 0, 1).

Higher dimensions

Cartesian coordinates are unique and unambiguous; therefore, we can identify points on a Cartesian plane by pairs of real numbers or by Cartesian product R2=R X R. Similarly, we can identify points in any Euclidean space of size n using tuples (lists) of n real numbers, or the Cartesian Product R.

Generalisations

Cartesian coordinates enable non-perpendicular axes and different units along each axis. If so, we may project the point onto an axis parallel to the other to create each coordinate. We must compute distances and angles differently in an oblique coordinate system than in a regular Cartesian system, and many standard formulae do not hold.

Quadrants and octants

A two-dimensional Cartesian system’s axes divide the plane into four infinite sections called quadrants. The two coordinate signs are I (+, +), II (, +), III (, +), and IV (+, +). Numbering begins in the upper right quadrant while drawing axes mathematically.

Based on the coordinates of the points, a Cartesian system divides the space into 8 areas called octants. The standard for naming a specific octant is to specify its signs, such as (+ + +) or (− + −). The orthant is a generalisation of the quadrant and octant to the number of dimensions.

Rectangular coordinate system questions

To which side are the negative values on the horizontal axis scaled?

We can scale the negative values on the horizontal axis to the left side.

Which is the first member in the ordered pair (x,y)?

We consider Abscissa as the first member in the ordered pair (x,y)

Conclusion

The Cartesian coordinate system uses two axes (x and y) to represent a two-dimensional plane, and the axes divide the plane into four quadrants.

The plane’s centre is where the axes intersect. (0,0) is the origin (0,0). On each axis, from the origin, rising positive values go up the x-axis and down the y-axis. The arrowheads indicate that the axes are indefinite.

The x-coordinate (horizontal displacement from the origin) and the y-coordinate (vertical displacement from the origin) identify each point in the plane. The ordered pair (x,y) represents their combined distance from the origin, and an ordered pair has x and y coordinates.