What is the primary purpose of the formula in a Cartesian coordinate system?

In a Cartesian coordinate system, the primary purpose of the formula is indeed to calculate both distance and direction. This system employs a grid defined by perpendicular axes, typically labeled as the x-axis and y-axis, where each point in the space is specified by a pair of coordinates (x, y).

Using the Pythagorean theorem, one can easily determine the distance between any two points by applying the formula:

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

This enables analysts to understand relationships between various points in a two-dimensional space, including how far apart they are and the specific direction from one point to another.

The other choices focus on different aspects of data handling and representation. While defining multi-dimensional data and converting coordinates to geographic data are important in their own contexts, they do not encapsulate the fundamental role of calculating distance and direction in the Cartesian system. Similarly, displaying data on a map involves additional processes, including projection and visualization, which extend beyond the basic function of the coordinate formula. Thus, the essence of a Cartesian system lies primarily in its ability to facilitate the calculation of distance and direction.