Differentiate between Euclidean distance and Manhattan distance.*

To differentiate between Euclidean distance and Manhattan distance, we need to understand their definitions and calculations.

1. Euclidean Distance:

   • Euclidean distance is a measure of the straight-line distance between two points in a Euclidean space.
   • It is calculated using the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.
   • In a two-dimensional space, the Euclidean distance between two points (x1, y1) and (x2, y2) is given by the formula: distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
   • In a higher-dimensional space, the formula extends to include the differences between the corresponding coordinates in each dimension.

2. Manhattan Distance:

   • Manhattan distance, also known as taxicab distance or city block distance, is a measure of the distance between two points in a grid-like path.
   • It is calculated by summing the absolute differences between the coordinates of the two points.
   • In a two-dimensional space, the Manhattan distance between two points (x1, y1) and (x2, y2) is given by the formula: distance = |x2 - x1| + |y2 - y1|
   • In a higher-dimensional space, the formula extends to include the absolute differences between the corresponding coordinates in each dimension.

In summary, the main differences between Euclidean distance and Manhattan distance are:

• Euclidean distance measures the straight-line distance between two points, while Manhattan distance measures the distance along a grid-like path.
• Euclidean distance takes into account the differences in all dimensions, while Manhattan distance only considers the differences in each dimension independently.
• Euclidean distance is suitable for continuous data, while Manhattan distance is often used for discrete data or when movement is restricted to grid-like paths.