docs: Update vector index sql examples

docs/en/sql-reference/20-sql-functions/07-aggregate-functions/aggregate-array-agg.md

@@ -3,7 +3,7 @@ title: ARRAY_AGG
 title_includes: LIST
 ---
 
-The ARRAY_AGG function (also known by its alias LIST) transforms all the values, including NULL, of a specific column in a query result into an array.
+The ARRAY_AGG function (also known by its alias LIST) transforms all the values, excluding NULL, of a specific column in a query result into an array.
 
 ## Syntax
 
@@ -68,4 +68,4 @@ GROUP BY movie_title;
 | movie_title |  ratings   |
 |-------------|------------|
 | Inception   | [5, 5, 4]  |
-```
+```

docs/en/sql-reference/20-sql-functions/11-vector-functions/00-vector-cosine-distance.md

@@ -46,13 +46,28 @@ This function performs vector computations within Databend and does not rely on
 
 ## Examples
 
+### Basic Usage
+
+```sql
+-- Calculate cosine distance between two vectors
+SELECT COSINE_DISTANCE([1.0, 2.0, 3.0]::vector(3), [4.0, 5.0, 6.0]::vector(3)) AS distance;
+```
+
+Result:
+```
+╭─────────────╮
+│   distance  │
+├─────────────┤
+│ 0.025368214 │
+╰─────────────╯
+```
+
 Create a table with vector data:
 
 ```sql
 CREATE OR REPLACE TABLE vectors (
     id INT,
-    vec VECTOR(3),
-    VECTOR INDEX idx_vec(vec) distance='cosine'
+    vec VECTOR(3)
 );
 
 INSERT INTO vectors VALUES
@@ -64,20 +79,22 @@ INSERT INTO vectors VALUES
 Find the vector most similar to [1, 2, 3]:
 
 ```sql
-SELECT 
+SELECT
+    id, 
     vec, 
     COSINE_DISTANCE(vec, [1.0000, 2.0000, 3.0000]::VECTOR(3)) AS distance
 FROM 
     vectors
 ORDER BY 
-    distance ASC
-LIMIT 1;
+    distance ASC;
 ```
 
 ```
-+-------------------------+----------+
-| vec                     | distance |
-+-------------------------+----------+
-| [1.0000,2.2000,3.0000]  | 0.0      |
-+-------------------------+----------+
+╭────────────────────────────────────╮
+│ id │    vec    │      distance     │
+├────┼───────────┼───────────────────┤
+│  1 │ [1,2,3]   │ 0.000000059604645 │
+│  2 │ [1,2.2,3] │     0.00096315145 │
+│  3 │ [4,5,6]   │       0.025368214 │
+╰────────────────────────────────────╯
 ```

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

@@ -49,13 +49,28 @@ Where v1ᵢ and v2ᵢ are the elements of the input vectors.
 
 ## Examples
 
+### Basic Usage
+
+```sql
+-- Calculate L2 distance between two vectors
+SELECT L2_DISTANCE([1.0, 2.0, 3.0]::vector(3), [4.0, 5.0, 6.0]::vector(3)) AS distance;
+```
+
+Result:
+```
+╭──────────╮
+│ distance │
+├──────────┤
+│ 5.196152 │
+╰──────────╯
+```
+
 Create a table with vector data:
 
 ```sql
 CREATE OR REPLACE TABLE vectors (
     id INT,
-    vec VECTOR(3),
-    VECTOR INDEX idx_vec(vec) distance='l2'
+    vec VECTOR(3)
 );
 
 INSERT INTO vectors VALUES
@@ -78,11 +93,12 @@ ORDER BY
 ```
 
 ```
-+----+-------------------------+----------+
-| 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 |
-+----+-------------------------+----------+
-```
+╭─────────────────────────────╮
+│ id │    vec    │  distance  │
+├────┼───────────┼────────────┤
+│  1 │ [1,2,3]   │          0 │
+│  2 │ [1,2.2,3] │ 0.20000005 │
+│  3 │ [4,5,6]   │   5.196152 │
+╰─────────────────────────────╯
+```
+

docs/en/sql-reference/20-sql-functions/11-vector-functions/02-vector-l1-distance.md

@@ -37,38 +37,52 @@ Formula: `L1_DISTANCE(a, b) = |a1 - b1| + |a2 - b2| + ... + |an - bn|`
 
 ```sql
 -- Calculate L1 distance between two vectors
-SELECT L1_DISTANCE([1.0, 2.0, 3.0], [4.0, 5.0, 6.0]) AS distance;
+SELECT L1_DISTANCE([1.0, 2.0, 3.0]::vector(3), [4.0, 5.0, 6.0]::vector(3)) AS distance;
 ```
 
 Result:
 ```
-┌──────────┐
+╭──────────╮
 │ distance │
 ├──────────┤
-│ 9.0      │
-└──────────┘
+│        9 │
+╰──────────╯
 ```
 
-### Using with VECTOR Type
+Create a table with vector data:
 
 ```sql
--- Create table with VECTOR columns
-CREATE TABLE products (
+CREATE OR REPLACE TABLE vectors (
     id INT,
-    features VECTOR(3),
-    VECTOR INDEX idx_features(features) distance='l1'
+    vec VECTOR(3)
 );
 
-INSERT INTO products VALUES 
-    (1, [1.0, 2.0, 3.0]::VECTOR(3)),
-    (2, [2.0, 3.0, 4.0]::VECTOR(3));
+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 L1 distance:
 
--- Find products similar to a query vector using L1 distance
-SELECT 
+```sql
+SELECT
     id,
-    features,
-    L1_DISTANCE(features, [1.5, 2.5, 3.5]::VECTOR(3)) AS distance
-FROM products
-ORDER BY distance ASC
-LIMIT 5;
+    vec,
+    L1_DISTANCE(vec, [1.0000, 2.0000, 3.0000]::VECTOR(3)) AS distance
+FROM
+    vectors
+ORDER BY
+    distance ASC;
 ```
+
+```
+╭─────────────────────────────╮
+│ id │    vec    │  distance  │
+├────┼───────────┼────────────┤
+│  1 │ [1,2,3]   │          0 │
+│  2 │ [1,2.2,3] │ 0.20000005 │
+│  3 │ [4,5,6]   │          9 │
+╰─────────────────────────────╯
+```
+

docs/en/sql-reference/20-sql-functions/11-vector-functions/index.md

@@ -9,30 +9,30 @@ This section provides reference information for vector functions in Databend. Th
 
 | Function | Description | Example |
 |----------|-------------|--------|
-| [COSINE_DISTANCE](./00-vector-cosine-distance.md) | Calculates angular distance between vectors (range: 0-1) | `COSINE_DISTANCE([1,2,3], [4,5,6])` |
-| [L1_DISTANCE](./02-vector-l1-distance.md) | Calculates Manhattan (L1) distance between vectors | `L1_DISTANCE([1,2,3], [4,5,6])` |
-| [L2_DISTANCE](./01-vector-l2-distance.md) | Calculates Euclidean (straight-line) distance | `L2_DISTANCE([1,2,3], [4,5,6])` |
+| [COSINE_DISTANCE](./00-vector-cosine-distance.md) | Calculates Cosine distance between vectors (range: 0-1) | `COSINE_DISTANCE([1,2,3]::VECTOR(3), [4,5,6]::VECTOR(3))` |
+| [L1_DISTANCE](./02-vector-l1-distance.md) | Calculates Manhattan (L1) distance between vectors | `L1_DISTANCE([1,2,3]::VECTOR(3), [4,5,6]::VECTOR(3))` |
+| [L2_DISTANCE](./01-vector-l2-distance.md) | Calculates Euclidean (straight-line) distance | `L2_DISTANCE([1,2,3]::VECTOR(3), [4,5,6]::VECTOR(3))` |
+| [INNER_PRODUCT](./03-inner-product.md) | Calculates the inner product (dot product) of two vectors | `INNER_PRODUCT([1,2,3]::VECTOR(3), [4,5,6]::VECTOR(3))` |
 
 ## Vector Analysis Functions
 
 | Function | Description | Example |
 |----------|-------------|--------|
-| [INNER_PRODUCT](./03-inner-product.md) | Calculates the inner product (dot product) of two vectors | `INNER_PRODUCT([1,2,3], [4,5,6])` |
-| [VECTOR_NORM](./05-vector-norm.md) | Calculates the L2 norm (magnitude) of a vector | `VECTOR_NORM([1,2,3])` |
-| [VECTOR_DIMS](./04-vector-dims.md) | Returns the dimensionality of a vector | `VECTOR_DIMS([1,2,3])` |
+| [VECTOR_NORM](./05-vector-norm.md) | Calculates the L2 norm (magnitude) of a vector | `VECTOR_NORM([1,2,3]::VECTOR(3))` |
+| [VECTOR_DIMS](./04-vector-dims.md) | Returns the dimensionality of a vector | `VECTOR_DIMS([1,2,3]::VECTOR(3))` |
 
 ## Distance Functions Comparison
 
 | Function | Description | Range | Best For | Use Cases |
 |----------|-------------|-------|----------|-----------|
-| [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 |
+| [COSINE_DISTANCE](./00-vector-cosine-distance.md) | Cosine distance between vectors | [0, 1] | When direction matters more than magnitude | • Document similarity<br/>• Semantic search<br/>• Recommendation systems<br/>• Text analysis |
 | [L1_DISTANCE](./02-vector-l1-distance.md) | Manhattan (L1) distance between vectors | [0, ∞) | Robust to outliers | • Feature comparison<br/>• Outlier detection<br/>• Grid-based pathfinding<br/>• Clustering algorithms |
-| [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 |
+| [L2_DISTANCE](./01-vector-l2-distance.md) | Euclidean (straight-line) distance | [0, ∞) | When magnitude and absolute differences are important | • Image similarity<br/>• Geographical data<br/>• Anomaly detection<br/>• Feature-based clustering |
+| [INNER_PRODUCT](./03-inner-product.md) | Dot product of two vectors | (-∞, ∞) | When both magnitude and direction are important | • Neural networks<br/>• Machine learning<br/>• Physics calculations<br/>• Vector projections |
 
 ## Vector Analysis Functions Comparison
 
 | Function | Description | Range | Best For | Use Cases |
 |----------|-------------|-------|----------|-----------|
-| [INNER_PRODUCT](./03-inner-product.md) | Dot product of two vectors | (-∞, ∞) | Measuring vector similarity and projections | • Neural networks<br/>• Machine learning<br/>• Physics calculations<br/>• Vector projections |
 | [VECTOR_NORM](./05-vector-norm.md) | L2 norm (magnitude) of a vector | [0, ∞) | Vector normalization and magnitude | • Vector normalization<br/>• Feature scaling<br/>• Magnitude calculations<br/>• Physics applications |
 | [VECTOR_DIMS](./04-vector-dims.md) | Number of vector dimensions | [1, 4096] | Vector validation and processing | • Data validation<br/>• Dynamic processing<br/>• Debugging<br/>• Compatibility checks |