Understanding Euclidean Distance

Euclidean distance is a fundamental concept in mathematics, often used to measure the straight-line distance between two points in Euclidean space. It's widely applied in various fields, including geometry, physics, and machine learning.

What is Euclidean Distance?

Euclidean distance, also known as L2 distance, is the most common way to measure the distance between two points. In a two-dimensional plane, it's simply the length of the line segment connecting the two points. This concept extends to higher dimensions, where it represents the shortest distance between two points in a multi-dimensional space.

The Euclidean Distance Formula

For two points P(x₁, y₁) and Q(x₂, y₂) in a 2D plane, the Euclidean distance (d) is calculated using the Pythagorean theorem:

d = √((x₂ - x₁)² + (y₂ - y₁)² )

For points in 3D space P(x₁, y₁, z₁) and Q(x₂, y₂, z₂), the formula extends to:

d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)² )

Step-by-Step Example

Let's calculate the Euclidean distance between two points: P(1, 2) and Q(4, 6).

1. Identify the coordinates:
   x₁ = 1, y₁ = 2
   x₂ = 4, y₂ = 6
2. Calculate the differences:
   (x₂ - x₁) = (4 - 1) = 3
   (y₂ - y₁) = (6 - 2) = 4
3. Square the differences:
   (3)² = 9
   (4)² = 16
4. Sum the squared differences:
   9 + 16 = 25
5. Take the square root of the sum:
   √25 = 5

Therefore, the Euclidean distance between P(1, 2) and Q(4, 6) is 5 units.

Frequently Asked Questions (FAQs)

Q: What is the difference between Euclidean distance and Manhattan distance?
A: Euclidean distance is the shortest straight-line path between two points, while Manhattan distance (or Taxicab distance) is the sum of the absolute differences of their Cartesian coordinates, representing movement along a grid.

Q: Can Euclidean distance be negative?
A: No, distance is always a non-negative value. The square root operation in the formula ensures the result is always positive or zero (if the points are identical).

Q: Where is Euclidean distance used in real life?
A: It's used in GPS navigation (calculating distance between two locations), computer graphics (determining object proximity), machine learning (clustering algorithms like K-means), and many engineering applications.