feat: vector distance functions (#2176)

* add: vector-distance-functions

* add: vector-distance-functions index

Author: BohuTANG

docs/en/guides/51-ai-functions/index.md

@@ -42,7 +42,7 @@ Databend provides built-in AI functions for various natural language processing
 
 - [ai_embedding_vector](/sql/sql-functions/ai-functions/ai-embedding-vector): Generates embeddings for text documents.
 - [ai_text_completion](/sql/sql-functions/ai-functions/ai-text-completion): Generates text completions based on a given prompt.
-- [cosine_distance](/sql/sql-functions/ai-functions/ai-cosine-distance): Calculates the cosine distance between two embeddings.
+- [cosine_distance](/sql/sql-functions/vector-distance-functions/vector-cosine-distance): Calculates the cosine distance between two embeddings.
 
 ## Generating Embeddings
 
@@ -67,7 +67,7 @@ VALUES
 
 ## Calculating Cosine Distance
 
-Now, let's find the documents that are most similar to a given query using the [cosine_distance](/sql/sql-functions/ai-functions/ai-cosine-distance) function:
+Now, let's find the documents that are most similar to a given query using the [cosine_distance](/sql/sql-functions/vector-distance-functions/vector-cosine-distance) function:
 ```sql
 SELECT
     id,

docs/en/sql-reference/20-sql-functions/11-ai-functions/04-ai-cosine-distance.md renamed to docs/en/sql-reference/20-sql-functions/11-vector-distance-functions/00-vector-cosine-distance.md

@@ -0,0 +1,84 @@
+---
+title: 'L2_DISTANCE'
+description: 'Measuring Euclidean distance between vectors in Databend'
+---
+
+Calculates the Euclidean (L2) distance between two vectors, measuring the straight-line distance between them in vector space.
+
+## Syntax
+
+```sql
+L2_DISTANCE(vector1, vector2)
+```
+
+## Arguments
+
+- `vector1`: First vector (ARRAY(FLOAT32 NOT NULL))
+- `vector2`: Second vector (ARRAY(FLOAT32 NOT NULL))
+
+## Returns
+
+Returns a FLOAT value representing the Euclidean (L2) distance between the two vectors. The value is always non-negative:
+- 0: Identical vectors
+- Larger values: Vectors that are farther apart
+
+## Description
+
+The L2 distance, also known as Euclidean distance, measures the straight-line distance between two points in Euclidean space. It is one of the most common metrics used in vector similarity search and machine learning applications.
+
+The function:
+
+1. Verifies that both input vectors have the same length
+2. Computes the sum of squared differences between corresponding elements
+3. Returns the square root of this sum
+
+The mathematical formula implemented is:
+
+```
+L2_distance(v1, v2) = √(Σ(v1ᵢ - v2ᵢ)²)
+```
+
+Where v1ᵢ and v2ᵢ are the elements of the input vectors.
+
+:::info
+- This function performs vector computations within Databend and does not rely on external APIs.
+:::
+
+## Examples
+
+Create a table with vector data:
+
+```sql
+CREATE OR REPLACE TABLE vectors (
+    id INT,
+    vec ARRAY(FLOAT32 NOT NULL)
+);
+
+INSERT INTO vectors VALUES
+    (1, [1.0000, 2.0000, 3.0000]),
+    (2, [1.0000, 2.2000, 3.0000]),
+    (3, [4.0000, 5.0000, 6.0000]);
+```
+
+Find the vector closest to [1, 2, 3] using L2 distance:
+
+```sql
+SELECT 
+    id,
+    vec, 
+    L2_DISTANCE(vec, [1.0000, 2.0000, 3.0000]) AS distance
+FROM 
+    vectors
+ORDER BY 
+    distance ASC;
+```
+
+```
++----+-------------------------+----------+
+| id | vec                     | distance |
++----+-------------------------+----------+
+| 1  | [1.0000,2.0000,3.0000]  | 0.0      |
+| 2  | [1.0000,2.2000,3.0000]  | 0.2      |
+| 3  | [4.0000,5.0000,6.0000]  | 5.196152 |
++----+-------------------------+----------+
+```

docs/en/sql-reference/20-sql-functions/11-vector-distance-functions/01-vector-l2-distance.md

@@ -0,0 +1,3 @@
+{
+  "label": "Vector Distance Functions"
+}

docs/en/sql-reference/20-sql-functions/11-vector-distance-functions/_category_.json

@@ -0,0 +1,15 @@
+---
+title: 'Vector Distance Functions'
+description: 'Vector distance functions in Databend for similarity measurement'
+---
+
+
+# Vector Distance Functions
+
+Databend provides functions for measuring distance or similarity between vectors, essential for vector search and machine learning applications.
+
+## Function Comparison
+| Function | Description | Range | Best For | Use Cases |
+|----------|-------------|-------|----------|-----------|
+| [L2_DISTANCE](01-vector-l2-distance.md) | Euclidean (straight-line) distance | [0, ∞) | When magnitude matters | • Image similarity<br/>• Geographical data<br/>• Anomaly detection<br/>• Feature-based clustering |
+| [COSINE_DISTANCE](00-vector-cosine-distance.md) | Angular distance between vectors | [0, 1] | When direction matters more than magnitude | • Document similarity<br/>• Semantic search<br/>• Recommendation systems<br/>• Text analysis |