Distance Metrics for Vector Search

Common Distance Metrics

Cosine Similarity:

• Measures angle between vectors
• Range: [-1, 1] (or [0, 1] for positive)
• Ignores magnitude
• Best for: Text embeddings, semantic similarity
• Formula: cos(θ) = (A·B) / (||A|| ||B||)

Euclidean Distance (L2):

• Straight-line distance
• Considers magnitude
• Range: [0, ∞)
• Best for: Spatial data, image embeddings
• Formula: ||A - B|| = sqrt(Σ(ai - bi)²)

Dot Product:

• Inner product of vectors
• Fast to compute
• Range: (-∞, ∞)
• Best for: Normalized vectors, efficiency-critical
• Formula: A·B = Σ(ai × bi)

Manhattan Distance (L1):

• Sum of absolute differences
• More robust to outliers
• Range: [0, ∞)
• Best for: High-dimensional sparse data
• Formula: Σ|ai - bi|

When to Use Each

Cosine Similarity: Default for text/semantic search

• Sentence transformers
• Document similarity
• Semantic search

Euclidean: When magnitude matters

• Image embeddings
• Facial recognition
• Spatial coordinates

Dot Product: For speed

• Normalized embeddings
• Real-time systems
• Simple, fast computation

Manhattan: Special cases

• Sparse high-dimensional data
• Outlier robustness needed

Normalization Impact

With Normalized Vectors:

• Cosine ≈ Dot Product
• Euclidean ≈ Angular distance
• Most models normalize internally

Without Normalization:

• Metrics behave differently
• Cosine ignores magnitude
• Euclidean considers it

Performance Characteristics

Speed (fastest to slowest):

1. Dot Product
2. Cosine (with pre-normalized)
3. Manhattan
4. Euclidean

Most Common in Production:

• Text: Cosine (90%)
• Images: Euclidean or Cosine
• Hybrid: Dot product on normalized

Database Support

Database	Cosine	Euclidean	Dot Product	Manhattan
Pinecone	✓	✓	✓	-
Weaviate	✓	✓	✓	✓
Qdrant	✓	✓	✓	✓
Milvus	✓	✓	✓	✓
pgvector	✓	✓	✓	-

Best Practices

1. Use what embedding model recommends
2. Cosine for text (safe default)
3. Normalize if using dot product
4. Test on your data
5. Stay consistent across pipeline
6. Consider computation cost

Common Mistakes

• Mixing metrics in same system
• Not normalizing for dot product
• Using Euclidean for text
• Ignoring model recommendations
• Not testing alternatives