# Distance Metrics Explained: Cosine, L2, and Inner Product

To query similarity between vector embeddings, you must choose a distance metric. pgvector supports three primary distance calculations.

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## 1. Cosine Distance (<=> Operator)

Cosine distance measures the difference in **direction or angle** between two vectors, ignoring their scale or magnitude. 

```sql
-- Cosine distance query syntax in pgvector
SELECT * FROM document_sections 
ORDER BY embedding <=> '[0.002, 0.015, ...]' 
LIMIT 5;
```

* **Values**: Ranges from 0 (identical direction) to 2 (opposite directions).
* **Use Case**: This is the default recommendation for text embeddings (such as OpenAI embeddings) because it measures semantic similarity regardless of the text length.

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## 2. L2 Euclidean Distance (<-> Operator)

L2 Euclidean distance measures the **straight-line distance** between two points in a multi-dimensional space.

```sql
-- L2 Euclidean distance query syntax in pgvector
SELECT * FROM document_sections 
ORDER BY embedding <-> '[0.002, 0.015, ...]' 
LIMIT 5;
```

* **Values**: Ranges from 0 (identical points) to infinity.
* **Use Case**: Suitable for image search or cases where vector magnitude is an important feature.

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## 3. Inner Product (<# Operator)

Inner product (dot product) multiplies corresponding coordinates of two vectors and sums the results. 

In pgvector, the `<#` operator represents the negative inner product because PostgreSQL indexes sort in ascending order (smaller values first).

```sql
-- Inner product query syntax in pgvector
SELECT * FROM document_sections 
ORDER BY embedding <# '[0.002, 0.015, ...]' 
LIMIT 5;
```

* **Use Case**: Suitable for vectors that are normalized (where magnitude equals 1), as the dot product calculation is computationally faster than Cosine distance.