Distance Metrics Comparison

Understanding when to use each metric for optimal similarity search.

Text/Embeddings

Cosine

Clustering

Euclidean

Sparse Data

Manhattan

Normalized

Dot Product

Metric Range Best For Considers Magnitude?
Euclidean (L2) [0, ∞) General purpose, clustering Yes
Cosine Similarity [-1, 1] Text, embeddings, NLP No
Manhattan (L1) [0, ∞) High-dimensional, sparse data Yes
Dot Product (-∞, ∞) Normalized vectors, recommendations Yes
Chebyshev (L∞) [0, ∞) When max deviation matters Yes
Hamming [0, n] Binary vectors, hashes N/A

Euclidean Distance (L2)

d(A, B) = √(Σ(aᵢ - bᵢ)²)

The "straight line" distance between two points in space. Most intuitive metric.

Use When

  • Actual distance matters
  • Clustering algorithms (K-means)
  • Low to medium dimensions
  • Vectors are similarly scaled

Avoid When

  • Very high dimensions (curse of dimensionality)
  • Only direction matters
  • Vectors have different scales

Cosine Similarity

cos(θ) = (A · B) / (|A| × |B|)

Measures the angle between vectors, ignoring their magnitude. Values range from -1 (opposite) to 1 (identical direction).

Use When

  • Text similarity / NLP
  • Embedding comparisons
  • Document similarity
  • Recommendation systems

Avoid When

  • Magnitude is meaningful
  • Physical distance matters
  • Zero vectors present

Manhattan Distance (L1)

d(A, B) = Σ|aᵢ - bᵢ|

Sum of absolute differences. Like walking on a grid (city blocks).

Use When

  • High-dimensional data
  • Sparse vectors
  • Robust to outliers needed
  • Grid-based movement

Avoid When

  • Direct distance is important
  • Rotation invariance needed

Dot Product (Inner Product)

A · B = Σ(aᵢ × bᵢ)

Simple sum of element-wise products. Fast to compute, equals cosine similarity for normalized vectors.

Use When

  • Vectors are normalized
  • Speed is critical
  • Recommendation systems
  • Neural network outputs

Avoid When

  • Vectors are not normalized
  • Bounded results needed

Try It Yourself

Compare all distance metrics for any two vectors using our interactive calculator.

Open Distance Calculator →