How is distance calculated in Cartesian coordinates?

The calculation of distance in Cartesian coordinates is based on the Euclidean distance formula, which is rooted in the Pythagorean theorem. This formula applies to measuring the straight-line distance between two points in a two-dimensional space, where each point is represented by its x and y coordinates.

In the Euclidean distance formula, the distance (d) between two points ((x_1, y_1)) and ((x_2, y_2)) is calculated using the equation:

[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

This expression effectively represents the length of the hypotenuse of a right triangle formed by the horizontal and vertical distances (the differences in the x and y coordinates) between the two points. Because Cartesian coordinates represent a flat, two-dimensional plane, this method of distance calculation is straightforward and highly effective, enabling precise calculations in various applications such as geometry, computer graphics, and geographic information systems.

The other methods mentioned, such as the Haversine formula, are applicable to spherical coordinates and are used to measure distances over the Earth's surface, making them unsuitable for calculations within a Cartesian framework. Similarly,