What is a Rectangular Coordinate System?

In a rectangular coordinate system, each point based on the pair of coordinates is specified uniquely in the plane. From two fixed perpendicular lines, these coordinates are at a distance in the rectangular coordinate system. Each line is known as the reference axis and axes as plural. The point where they meet is called the origin (0,0).

Any point can be specified by the same principle in 3-dimensional space by three cartesian ordered pairs by putting x value and y value, its distance to 3 mutually orthometric planes. For any dimension n, the n cartesian ordered pair specify the point in an n-dimensional space called the Euclidean space. They are equal in distance from the point to n mutually orthometric planes.
Rectangular coordinate system (RCS) also known as the Cartesian coordinate system is the foundation of analytical geometry and is also used in many other branches of mathematics such as calculus, linear algebra, differential geometry, group theory, etc. It is useful in physics, astronomy, and also in engineering technology. It is an important concept in data processing, computer graphics, and computer geometric design.

Different dimensions

One dimension

For a straight line that is for a one dimensional space choosing a cartesian system involves choosing origin of the line, the length of the line and orientation of the line. With the help of the orientation of the line we can say which half of the line is positive and which half is negative based on the origin of the line. The point P can be specified by its distance from the origin with a + or - sign. It depends on which half of the line contains the point P. A number line is the line with a chosen cartesian system. Each real number has a unique and different location on the number line.

Two dimension

For two-dimension, it is defined by a pair of orthometric axes, a single unit of length for both the axis, and an orientation for both the axis. The point where the two-axis meets are called the origin and each axis is represented as a number line segment. The two axes are x-axis and y-axis. The distance of a point from the x-axis is ordinate and y-axis is known as abscissa. Consider a point P and draw a line through P which is orthometric to each axis and the position where the line meets the axis is interpreted as a number. The two numbers which are chosen are the cartesian ordered pairs of P. In two dimensions they are written as two numbers in parentheses and separated by a comma. For instance (3, -2), (2, 7) or (5, -3). A cartesian plane is a type of Euclidean plane with a chosen cartesian coordinate system. In the Cartesian plane, geometric figures such as unit circle, square, hyperbola, and others can be defined. The two geometrical axes divide the cartesian plane into four right angles called quadrants. The horizontal axis is the x-axis and the vertical axis is the y-axis. The quadrants where all the ordered pairs are positive are called quadrant I. Quadrant II is the quadrant where the x-axis is negative and the y-axis value is positive and for quadrant III, the x-axis value and y-axis value both are negative. For quadrant IV, the value of the x-axis is positive and the value of the y-axis is negative.

The midpoint is the point that bisects a line segment when a line is formed by joining two points that is $(X_{1 ,} Y_1)$ and $(X_2 , Y_2)$. The midpoint of the line segment is given by - ($\frac{X_1+X_2}{2}, \frac{Y_1+ Y_2}{2}$) .

The distance formula in a plane is used to find the distance between two points. To find out the distance we need to join the two points and also form a right triangle and then apply the Pythagorean theorem. Pythagorean theorem states that in a right-angled triangle the square of the hypotenuse is equal to the sum of the square of the other two sides.

$a^2+b^2=c^2$ 

where a and b are the two perpendicular sides of a right-angled triangle and c is the hypotenuse.

Three dimension

For a three-dimensional system, a cartesian ordered pair consists of triplet lines or the axes that have common point called the origin that is perpendicular to each other. Similar to the two-dimensional case, in three-dimensional also the axis becomes a number line segment. A cartesian plane is considered through P which is orthometric to each ordered pair axis. The true statement is that point is interpreted as a number when the cartesian plane cuts the axis. A cartesian plane is defined by each pair of axes. These planes are divided into eight trihedral also called octants. These octants are represented as (x, y, z), (-x, y, z), (x, -y, z), (x, y, -z), ( -x, -y, z), (x, -y, -z), (-x, y, -z) and (-x, -y, -z). The ordered pair is written as three numbers in a bracket separated by commas. The axes are X-axis, Y-axis, and Z-axis. One example of a three-dimensional system is the cartesian grid system.

Notations and conventions

The ordered pair of a point is always written in parentheses separated by commas such as (5, 2) or (3, 5, 2). The origin (0, 0) is often denoted as O. In geometrical interpretations, the unknown ordered pairs are labeled as (x,y) in the two-dimensional plane and (x, y, z) in the three-dimensional plane. In physics and engineering, conventional names are used. For instance how the pressure varies with time is denoted by the x-coordinate of the point and y-coordinate of the point as (p, t). For the n number of ordered pairs in n-dimensional space, the transcript (X₁, X₂, X₃,……X_n) is used. In a computer, these notations are stored in the form of an array.

Orientation and handedness

In two dimension

The y-axis is determined by choosing the x-axis only. The y-axis is always orthometric to the x-axis through the point marked as origin in the x-axis. But we can choose which half of the lines on the perpendicular is positive or negative. A common way of the orientation of these planes is that the positive x-axis points towards the right and a positive y-axis points upwards.

In three dimension

There are two possible orientations for the lines that determines how the z-axis should lie once the x-axis and y-axis are specified. These two possible orientations are called right-handed and left-handed systems. The orientation where the x-y plane is lying horizontally and the z-axis is pointed upwards is called right-handed orientation and where the z-axis is pointed downwards is called left-handed orientation.

Context and application

They have multiple applications in the real world. In engineering projects, the crucial foundation is the agreement on the definition. Knowledge of how to draw an ordered pair system is essential because we cannot assume from where does coordinates come predefined. In scientific and business applications also the knowledge is important.

This topic is significant in the professional exam for undergraduate, Graduate, and Post-graduate courses.
Bachelors in Engineering (Electrical Engineering)
Bachelors in Science
Masters in Engineering (Electrical Engineering)
Masters in Science

Practice Problems

1. What is the other name given to a rectangular coordinate system?

1. Circular coordinate system
2. Cartesian Coordinate system
3. Spherical coordinate system
4. Space coordinate system

Answer: b

Explanation: The other name of a rectangular coordinate system is the Cartesian coordinate system.

2. In which of the following quadrant will the point P(-3,6) lie?

1. I Quadrant
2. II Quadrant
3. III Quadrant
4. IV Quadrant

Answer: d

Explanation: x is negative and y is positive in the fourth quadrant.

3. Can the cartesian ordered pair be related to cylindrical and spherical ordered pairs?

1. Yes
2.  No
3. Maybe
4. None

Answer: a

Explanation: We know that every ordered pair systems are convertible from one system to another and also, all kind of vector operations can be used for this system.

4. Which of the following criteria is used to choose an ordered pair system?

1. Distance
2. Intensity
3. Magnitude
4. Geometry

Answer: d

Explanation: The ordered pair system is chosen based on the geometry.

 5. How are the coordinates in the cartesian system represented?

1. Within brackets and separated by commas.
2. Within brackets and separated by a semicolon.
3. Within brackets and separated by colons.
4. Without brackets and separated by a semicolon.

Answer: a

Explanation: A coordinate system is represented with brackets separated by commas (x, y, z).

Related Concept

• Horizontal and vertical
• Jones Diagram
• Orthogonal coordinates
• Polar coordinate system
• Regular grid
• Spherical coordinate system