Cartesian Coordinates System – Dimensions, Formula and Equation

Let’s use an example to help you understand why you need a coordinate system. Imagine a girl in your class seated in the fourth and sixth rows. Consequently, this stance can be described as (4, 6).

What is an Axis?

Axis is a constant reference line used to measure coordinates in mathematics.

X and Y Axis

A rectangular system’s reference lines, the vertical and perpendicular axes, are used to measure distances. These are how they are acquired:

Take XX’ and YY’, two number lines. Put XX’ in the horizontal position, and then write the numbers as we would in the number line. Similar to how we would write in a number line, position YY’ vertically and then continue writing numbers on it. Combining the two lines will cause them to cross at the zeros or origins of the two lines. The x-axis and y-axis are referred to as the horizontal line XX’ and the vertical line YY’. 

Cartesian Coordinates of the plane

The x-coordinate and y-coordinate axes in the diagram below specify the Cartesian coordinates in the plane. The plane’s Cartesian coordinates point is expressed as (x, y). The distance from the origin in the direction along the x-axis is known as the x-coordinate. The x-coordinate is the distance from the y-axis to the right or left. y is referred to as the y-coordinate since it represents the distance from the origin along the y-axis. The y-coordinate shows the distance above or below the x-axis. 

What Is a Cartesian System?

The Cartesian System is the method that is used to identify points in a plane. The source of the cartesian form is the number line.

Dimensions of the Cartesian System

In the cartesian coordinate system, the bifurcation often begins with a one-dimensional system, followed by a two-dimensional system, and then a three-dimensional system. So let’s get into more depth about these cartesian system dimensions.

One Dimensional Cartesian Coordinate System

A straight line with the origin O, a positive side, and a negative side makes up the cartesian coordinate system for a one-dimensional space. One-dimensional implies that the plane has either a vertical or a horizontal line. When a line is plotted horizontally, the right side is considered positive, and the left side is considered negative. In contrast, if the line is vertical, the upper part of the line is seen as positive, and the lower part as negative.

Two-Dimensional Cartesian Coordinate System

Two perpendicular axes constitute a Cartesian coordinate system on a two-dimensional plane. In other words, every point on a flat surface has a reference grid defined by two lines drawn at right angles. Because the reference axes are orthogonal, this system may be referred to as a rectangle or orthogonal coordinate system.

Three-Dimensional Cartesian Coordinate System

The x, y, and z axes, which are all perpendicular to one another and share the same units of length on all three axes, make up the three-dimensional cartesian coordinate system. Like the two-dimensional coordinate system, these three axes intersect at origin O, dividing the space into eight units called octants.

Cartesian System Formulas

The cartesian coordinate system’s formulas make it simple to demonstrate the many properties of lines, curves, and planes in two- and three-dimensional systems.

Cartesian Coordinates Distance Formula

The distance between two coordinates can be calculated using the Cartesian plane distance formula. For example, to calculate the distance (d), or length of the line segment, between the specified coordinates, use the formula below.

\( d = \sqrt{(x_1 – x_2)^2 + (y_1 – y_2)^2} \)

Slope Formula

The equation \( m = Δy \space Δx = (y_2 – y_1)(x_2 – x_1) \) can be used to determine the slope m of a line that passes through the points P(\( x_1 \), \( y_1 \)) and Q(\( x_2 \), \( y_2 \)), 

where ∆y = variation of the y-coordinates 

and ∆x = variation of the x-coordinates.

Check our Slope Calculator.

Mid-Point Formula

If we have coordinates (\( x_1 \), \( y_1 \)) and (\( x_2 \), \( y_2 \)), then \( \dfrac{x_1 + x_2}{2} \), \( \dfrac{y_1 + y_2}{2} \) will get the midpoint of these coordinates.

Section Formula

To determine the coordinates of a point that splits the line segment connecting the points (x₁,y₁) and (x₂,y₂) in the ratio m:n, apply the section formula. The point separating the two points is available between the two points or on the line beyond the two points and is located on the line connecting the two points.

\( [ \dfrac{(m × x_2) + (n × x_1)}{m + n } \) , \( \dfrac{(m × y_2) + (n × y_1)}{m + n } ] \)

Cartesian Equation of a Line

A straight line’s cartesian equation is y = mx + c, where m denotes the gradient of the line and c denotes its intersection with the y-axis.

Cartesian Equation of a Plane

A plane’s equation takes the cartesian form ax + by + cz = d, where a, b, and c are the direction ratios and d is the distance from the origin.

Convert from Polar to Cartesian Coordinates

Use the following equations to convert from Polar Coordinates (r, θ) to Cartesian Coordinates (x,y). Alternatively, you can use an online polar to Cartesian Coordinates calculator.

• x = r × cos(θ)
• y = r × sin(θ)

Convert from Cartesian Coordinates to Polar Coordinates

To convert from Cartesian Coordinates (x,y) to Polar Coordinates (r,θ), use the following equations:

• \( r = \sqrt{x^2 + y^2 }\)
• θ \( = tan^{-1} ( \dfrac{y}{x} ) \)

Conclusion

Its most crucial characteristic is the precise position of a point on a surface or in three-dimensional space. This can be quantified by applying a coordinate system, like the Cartesian system.

Knowing how coordinate systems function will help you mathematically while making graphs and may also keep you from getting lost if you have a map with you.