2. Euclidean Distance: The Straight Line

Overview

Euclidean distance is the "straight line" distance between two points. It's the most intuitive metric: if you have two points on a map, the Euclidean distance is the length of the straight line connecting them.

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Formula

In 2D:

distance([x₁, y₁], [x₂, y₂]) = √((x₂ - x₁)² + (y₂ - y₁)²)

In 3D:

distance([x₁, y₁, z₁], [x₂, y₂, z₂]) = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)

In ND (the generalization):

distance(A, B) = √(∑(bᵢ - aᵢ)²)

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Examples

Example 1 (2D):

A = [3, 2]
B = [6, 6]

distance = √((6-3)² + (6-2)²)
         = √(9 + 16)
         = √25 = 5

Example 2 (3D):

A = [1, 2, 3]
B = [4, 6, 8]

distance = √((4-1)² + (6-2)² + (8-3)²)
         = √(9 + 16 + 25)
         = √50 ≈ 7.07

Example 3 (1536D - conceptual):

A = [a₁, a₂, ..., a₁₅₃₆]
B = [b₁, b₂, ..., b₁₅₃₆]

distance = √((b₁-a₁)² + (b₂-a₂)² + ... + (b₁₅₃₆-a₁₅₃₆)²)

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Advantages and disadvantages

Advantages:

• ✅ Intuitive (a straight line)
• ✅ It preserves geometric properties
• ✅ Useful when magnitude matters

Disadvantages:

• ❌ In high dimensions, all the distances are similar
• ❌ Sensitive to magnitude (large vectors → large distances)
• ❌ Not optimal for embeddings

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Next capsule: 03-manhattan-distance.md — A path along a grid.