---
title: Distance Functions reference
meta:
    description: Functions for calculating distances.
headingMaxLevels: 2
---

# Distance functions

The following functions are used to calculate distances between two points or to normalize vectors.

## L1Norm

Calculates the L1-norm (Manhattan distance or taxicab geometry) of a vector, which is the sum of the absolute values of its components.

### Syntax

```sql
L1Norm(vector)
```

Alias: `normL1`.

### Arguments

- `vector`: [Tuple](../data-types/tuple) or [Array](../data-types/array). The input vector.

### Returns

The L1-norm of the vector. [UInt](../data-types/int-uint), [Float](../data-types/float) or [Decimal](../data-types/decimal).

### Example

```sql
SELECT L1Norm((1, 2))
```

Result:

```result
┌─L1Norm((1, 2))─┐
│              3 │
└────────────────┘
```

## L2Norm

Calculates the L2-norm (Euclidean norm) of a vector, which is the square root of the sum of the squares of its components.

### Syntax

```sql
L2Norm(vector)
```

Alias: `normL2`.

### Arguments

- `vector`: [Tuple](../data-types/tuple) or [Array](../data-types/array). The input vector.

### Returns

The L2-norm of the vector. [Float](../data-types/float).

### Example

```sql
SELECT L2Norm((1, 2))
```

Result:

```result
┌───L2Norm((1, 2))─┐
│ 2.23606797749979 │
└──────────────────┘
```

## L2SquaredNorm

Calculates the squared L2-norm of a vector, which is the sum of the squares of its components. This is equivalent to the square of the Euclidean distance from the origin.

### Syntax

```sql
L2SquaredNorm(vector)
```

Alias: `normL2Squared`.

### Arguments

- `vector`: [Tuple](../data-types/tuple) or [Array](../data-types/array). The input vector.

### Returns

The squared L2-norm of the vector. [Float](../data-types/float).

### Example

```sql
SELECT L2SquaredNorm((1, 2))
```

Result:

```result
┌─L2SquaredNorm((1, 2))─┐
│                     5 │
└───────────────────────┘
```

## LinfNorm

Calculates the L-infinity norm (maximum norm) of a vector, which is the maximum absolute value among its components.

### Syntax

```sql
LinfNorm(vector)
```

Alias: `normLinf`.

### Arguments

- `vector`: [Tuple](../data-types/tuple) or [Array](../data-types/array). The input vector.

### Returns

The L-infinity norm of the vector. [Float](../data-types/float).

### Example

```sql
SELECT LinfNorm((1, -2))
```

Result:

```result
┌─LinfNorm((1, -2))─┐
│                 2 │
└───────────────────┘
```

## LpNorm

Calculates the Lp-norm of a vector, which is a generalization of the L1 and L2 norms. It is the p-th root of the sum of the p-th powers of the absolute values of the vector's components.

### Syntax

```sql
LpNorm(vector, p)
```

Alias: `normLp`.

### Arguments

- `vector`: [Tuple](../data-types/tuple) or [Array](../data-types/array). The input vector.
- `p`: [UInt](../data-types/int-uint) or [Float](../data-types/float). The power `p`, where `p` must be a real number in the range `[1; inf)`.

### Returns

The Lp-norm of the vector. [Float](../data-types/float).

### Example

```sql
SELECT LpNorm((1, -2), 2)
```

Result:

```result
┌─LpNorm((1, -2), 2)─┐
│   2.23606797749979 │
└────────────────────┘
```

## L1Distance

Calculates the L1 distance (Manhattan distance or taxicab geometry) between two vectors. This is the sum of the absolute differences between their corresponding components.

### Syntax

```sql
L1Distance(vector1, vector2)
```

Alias: `distanceL1`.

### Arguments

- `vector1`: [Tuple](../data-types/tuple) or [Array](../data-types/array). The first vector.
- `vector2`: [Tuple](../data-types/tuple) or [Array](../data-types/array). The second vector.

### Returns

The L1 distance between the two vectors. [Float](../data-types/float).

### Example

```sql
SELECT L1Distance((1, 2), (2, 3))
```

Result:

```result
┌─L1Distance((1, 2), (2, 3))─┐
│                          2 │
└────────────────────────────┘
```

## L2Distance

Calculates the L2 distance (Euclidean distance) between two vectors. This is the square root of the sum of the squared differences between their corresponding components.

### Syntax

```sql
L2Distance(vector1, vector2)
```

Alias: `distanceL2`.

### Arguments

- `vector1`: [Tuple](../data-types/tuple) or [Array](../data-types/array). The first vector.
- `vector2`: [Tuple](../data-types/tuple) or [Array](../data-types/array). The second vector.

### Returns

The L2 distance between the two vectors. [Float](../data-types/float).

### Example

```sql
SELECT L2Distance((1, 2), (2, 3))
```

Result:

```result
┌─L2Distance((1, 2), (2, 3))─┐
│         1.4142135623730951 │
└────────────────────────────┘
```

## L2SquaredDistance

Calculates the squared L2 distance between two vectors. This is the sum of the squared differences between their corresponding components.

### Syntax

```sql
L2SquaredDistance(vector1, vector2)
```

Alias: `distanceL2Squared`.

### Arguments

- `vector1`: [Tuple](../data-types/tuple) or [Array](../data-types/array). The first vector.
- `vector2`: [Tuple](../data-types/tuple) or [Array](../data-types/array). The second vector.

### Returns

The squared L2 distance between the two vectors. [Float](../data-types/float).

### Example

```sql
SELECT L2SquaredDistance([1, 2, 3], [0, 0, 0])
```

Result:

```result
┌─L2SquaredDistance([1, 2, 3], [0, 0, 0])─┐
│                                      14 │
└─────────────────────────────────────────┘
```

## LinfDistance

Calculates the L-infinity distance (Chebyshev distance or maximum norm distance) between two vectors. This is the maximum absolute difference between any pair of corresponding components.

### Syntax

```sql
LinfDistance(vector1, vector2)
```

Alias: `distanceLinf`.

### Arguments

- `vector1`: [Tuple](../data-types/tuple) or [Array](../data-types/array). The first vector.
- `vector2`: [Tuple](../data-types/tuple) or [Array](../data-types/array). The second vector.

### Returns

The L-infinity distance between the two vectors. [Float](../data-types/float).

### Example

```sql
SELECT LinfDistance((1, 2), (2, 3))
```

Result:

```result
┌─LinfDistance((1, 2), (2, 3))─┐
│                            1 │
└──────────────────────────────┘
```

## LpDistance

Calculates the Lp distance between two vectors, a generalized distance metric. It is the p-th root of the sum of the p-th powers of the absolute differences between their corresponding components.

### Syntax

```sql
LpDistance(vector1, vector2, p)
```

Alias: `distanceLp`.

### Arguments

- `vector1`: [Tuple](../data-types/tuple) or [Array](../data-types/array). The first vector.
- `vector2`: [Tuple](../data-types/tuple) or [Array](../data-types/array). The second vector.
- `p`: [UInt](../data-types/int-uint) or [Float](../data-types/float). The power `p`, where `p` must be a real number in the range `[1; inf)`.

### Returns

The Lp distance between the two vectors. [Float](../data-types/float).

### Example

```sql
SELECT LpDistance((1, 2), (2, 3), 3)
```

Result:

```result
┌─LpDistance((1, 2), (2, 3), 3)─┐
│            1.2599210498948732 │
└───────────────────────────────┘
```

## L1Normalize

Normalizes a vector to have an L1-norm of 1. This means scaling the vector such that the sum of the absolute values of its components equals 1.

### Syntax

```sql
L1Normalize(tuple)
```

Alias: `normalizeL1`.

### Arguments

- `tuple`: [Tuple](../data-types/tuple). The input vector as a tuple.

### Returns

A unit vector with an L1-norm of 1. [Tuple](../data-types/tuple) of [Float](../data-types/float).

### Example

```sql
SELECT L1Normalize((1, 2))
```

Result:

```result
┌─L1Normalize((1, 2))─────────────────────┐
│ (0.3333333333333333,0.6666666666666666) │
└─────────────────────────────────────────┘
```

## L2Normalize

Normalizes a vector to have an L2-norm (Euclidean norm) of 1. This means scaling the vector such that the square root of the sum of the squares of its components equals 1.

### Syntax

```sql
L2Normalize(tuple)
```

Alias: `normalizeL1`.

### Arguments

- `tuple`: [Tuple](../data-types/tuple). The input vector as a tuple.

### Returns

A unit vector with an L2-norm of 1. [Tuple](../data-types/tuple) of [Float](../data-types/float).

### Example

```sql
SELECT L2Normalize((3, 4))
```

Result:

```result
┌─L2Normalize((3, 4))─┐
│ (0.6,0.8)           │
└─────────────────────┘
```

## LinfNormalize

Normalizes a vector to have an L-infinity norm of 1. This means scaling the vector such that the maximum absolute value among its components equals 1.

### Syntax

```sql
LinfNormalize(tuple)
```

Alias: `normalizeLinf`.

### Arguments

- `tuple`: [Tuple](../data-types/tuple). The input vector as a tuple.

### Returns

A unit vector with an L-infinity norm of 1. [Tuple](../data-types/tuple) of [Float](../data-types/float).

### Example

```sql
SELECT LinfNormalize((3, 4))
```

Result:

```result
┌─LinfNormalize((3, 4))─┐
│ (0.75,1)              │
└───────────────────────┘
```

## LpNormalize

Normalizes a vector to have an Lp-norm of 1. This means scaling the vector such that its Lp-norm equals 1.

### Syntax

```sql
LpNormalize(tuple, p)
```

Alias: `normalizeLp`.

### Arguments

- `tuple`: [Tuple](../data-types/tuple). The input vector as a tuple.
- `p`: [UInt](../data-types/int-uint) or [Float](../data-types/float). The power `p`, where `p` must be a real number in the range `[1; inf)`.

### Returns

A unit vector with an Lp-norm of 1. [Tuple](../data-types/tuple) of [Float](../data-types/float).

### Example

```sql
SELECT LpNormalize((3, 4),5)
```

Result:

```result
┌─LpNormalize((3, 4), 5)──────────────────┐
│ (0.7187302630182624,0.9583070173576831) │
└─────────────────────────────────────────┘
```

## cosineDistance

Calculates the cosine distance between two vectors. This metric measures the angular separation between two vectors, indicating their similarity regardless of magnitude. A smaller value indicates greater similarity.

### Syntax

```sql
cosineDistance(vector1, vector2)
```

### Arguments

- `vector1`: [Tuple](../data-types/tuple) or [Array](../data-types/array). The first vector.
- `vector2`: [Tuple](../data-types/tuple) or [Array](../data-types/array). The second vector.

### Returns

The cosine distance, which is `1 - cosine_similarity`. [Float](../data-types/float).

### Example

```sql
SELECT cosineDistance((1, 2), (2, 3))
```

Result:

```result
┌─cosineDistance((1, 2), (2, 3))─┐
│           0.007722123286332261 │
└────────────────────────────────┘
```