---
title: "array_cosine_distance"
slug: "dev-database-reference-sql-functions-vector-array-cosine-distance"
updated: 2026-10-30T18:18:51Z
published: 2026-10-30T18:18:51Z
canonical: "docs.vastdata.com/dev-database-reference-sql-functions-vector-array-cosine-distance"
---

> ## Documentation Index
> Fetch the complete documentation index at: https://docs.vastdata.com/llms.txt
> Use this file to discover all available pages before exploring further.

# array_cosine_distance

*Applies to VAST Cluster 5.5.*

*[Functions](/develop/docs/dev-database-reference-sql-functions-index) › [Vector search SQL](/develop/docs/dev-database-reference-sql-vector-search) › Distance functions.* Returns the cosine distance between two vectors: 1 minus the cosine of the angle between them. It compares directions and ignores length: `0` means the same direction, `1` a right angle, `2` opposite directions. A smaller result means more similar.

## Syntax

```
array_cosine_distance(a, b)
```

| Argument | Type | Description |
| --- | --- | --- |
| `a`, `b` | `FLOAT[n]`, `DOUBLE[n]` or `HALF[n]` fixed-size arrays of the same `n`, or a `VECTOR(FLOAT, n)` column | The two vectors. |

**Returns:** `FLOAT`; `DOUBLE` if either argument is a `DOUBLE` array.

## Set up the examples

The examples connect with the ADBC driver, as in [VAST Query Engine (ADBC)](/develop/docs/dev-database-integrations-adbc), and work in a schema called `docs_sql_fn_vector_array_cosine_distance`. Set `QE_ENDPOINT`, `AWS_ACCESS_KEY_ID`, `AWS_SECRET_ACCESS_KEY` and `VASTDB_BUCKET` ([SQL reference](/develop/docs/dev-database-reference-sql-index#example-connect-and-use-a-transaction)), and run the blocks in order in one Python session. Below each block is the output you should see.

This block connects, defines the `run()` and `plan()` helpers, and creates a table of six documents, each with a 3-dimensional embedding in a `VECTOR(FLOAT, 3)` column and the same values in a plain list column, `raw`. Document 2's embedding is all zeros, and document 3 has none:

```
import os
import re

import adbc_driver_manager
import adbc_driver_manager.dbapi as dbapi
import adbc_driver_vastdb

BUCKET = os.environ["VASTDB_BUCKET"]
SCHEMA = f'"{BUCKET}/docs_sql_fn_vector_array_cosine_distance"'

conn = dbapi.connect(
    driver=adbc_driver_vastdb.get_driver_path(),
    db_kwargs={
        "vast.db.endpoint": os.environ["QE_ENDPOINT"],
        "vast.db.access_key": os.environ["AWS_ACCESS_KEY_ID"],
        "vast.db.secret_key": os.environ["AWS_SECRET_ACCESS_KEY"],
    },
    autocommit=True,
)

def server_message(e):
    """The server's message in a driver error, without the request ID or the echoed statement."""
    text = re.sub(r"\[req=0x[0-9a-f]+\]: ", "", str(e))
    found = (re.search(r'message: "(.*?)", request_id', text, re.S)
             or re.search(r"description: (.*)\)", text, re.S))
    message = found.group(1) if found else text.strip().splitlines()[-1]
    message = re.sub(r"^Plan query: '.*?' failed with: ", "", message, flags=re.S)
    message = re.split(r"\\n|\n", message)[0].replace('\\"', '"')
    return message.replace(BUCKET, "<bucket>")

def run(sql):
    """Run one statement. Print the rows it returns, or the error."""
    try:
        with conn.cursor() as cur:
            cur.execute(sql)
            rows = cur.fetchall() if cur.description else []
    except (adbc_driver_manager.Error, OSError) as e:
        print(f"{type(e).__name__}: {server_message(e)}")
    else:
        print(rows)

def plan(sql):
    """Print the plan without its ~N Rows lines: the table's row count, which can still be 0 just after an insert."""
    with conn.cursor() as cur:
        cur.execute("EXPLAIN " + sql)
        steps = cur.fetchall()[0][1]
    print("\n".join(line for line in steps.splitlines() if "Rows" not in line).replace(BUCKET, "<bucket>"))

TABLE = f"{SCHEMA}.docs"
run(f"CREATE SCHEMA IF NOT EXISTS {SCHEMA}")
run(f"CREATE TABLE IF NOT EXISTS {TABLE} (id BIGINT, title VARCHAR, e VECTOR(FLOAT, 3), raw FLOAT[])")
run(f"INSERT INTO {TABLE} VALUES "
    "(1, 'same', [1.0, 2.0, 3.0]::FLOAT[3], [1.0, 2.0, 3.0]), "
    "(2, 'zero', [0.0, 0.0, 0.0]::FLOAT[3], [0.0, 0.0, 0.0]), "
    "(3, 'none', NULL, NULL), "
    "(4, 'opposite', [-1.0, -2.0, -3.0]::FLOAT[3], [-1.0, -2.0, -3.0]), "
    "(5, 'axis', [1.0, 0.0, 0.0]::FLOAT[3], [1.0, 0.0, 0.0]), "
    "(6, 'three-four', [3.0, 4.0, 0.0]::FLOAT[3], [3.0, 4.0, 0.0])")
```

```
[]
[]
[(6,)]
```

## What it does

`array_cosine_distance(a, b)` is 1 minus (a · b) / (|a| |b|): 1 minus the dot product divided by the product of the two lengths. Only the direction of each vector counts, so a vector and a multiple of it are `0.0` apart:

| Call | Result |
| --- | --- |
| `array_cosine_distance([1.0, 2.0, 3.0]::FLOAT[3], [2.0, 4.0, 6.0]::FLOAT[3])` | `0.0` (same direction) |
| `array_cosine_distance([1.0, 0.0]::FLOAT[2], [0.0, 1.0]::FLOAT[2])` | `1.0` (a right angle) |
| `array_cosine_distance([1.0, 2.0, 3.0]::FLOAT[3], [-1.0, -2.0, -3.0]::FLOAT[3])` | `2.0` (opposite) |

Both arguments must be fixed-size arrays of floating-point values with the same number of elements. A `VECTOR(FLOAT, n)` column is one already, so pass it as it is. Write a query vector as a list literal cast to the array type, `[1.0, 1.0, 1.0]::FLOAT[3]`. A list column (`FLOAT[]`) must be cast to `FLOAT[n]` first.

With two `FLOAT` arrays the result is a `FLOAT`. If either argument is a `DOUBLE` array, the result is a `DOUBLE`. A `HALF` array gives a `FLOAT`.

**Zero vectors.** A vector of zeros has no direction. `array_cosine_distance` returns `2.0` when either vector is all zeros, the same as for opposite vectors, so a zero vector is never close to anything, not even to itself.

## NULLs and edge cases

| Input | Result |
| --- | --- |
| A NULL vector (either argument) | `NULL` |
| Two equal vectors | `0.0` |
| A zero vector, such as `[3.0, 4.0]` against `[0.0, 0.0]`, or two zero vectors | `2.0` |
| One-element vectors `[1.0]` and `[2.0]` | `0.0` (same direction) |
| A NULL element, such as `[1.0, NULL, 3.0]::FLOAT[3]` | Error: `array_cosine_distance: left argument can not contain NULL values` (or `right argument`) |
| Arrays of different sizes | Error: `Binder Error: array_cosine_distance: Array arguments must be of the same size` |
| An uncast list literal, such as `[1.0, 2.0, 3.0]` | Error: `No function matches the given name and argument types &apos;array_cosine_distance(DECIMAL(2,1)[], DECIMAL(2,1)[])&apos;` |
| `array_cosine_distance(NULL, NULL)` | Error: `Could not choose a best candidate function`; give the NULL a type, such as `NULL::FLOAT[3]` |

The same cases, as the cluster returns them. They use the table and helpers from [Set up the examples](/develop/docs/dev-database-reference-sql-functions-vector-array-cosine-distance#set-up-the-examples):

```
run("SELECT array_cosine_distance([1.0, 2.0, 3.0]::FLOAT[3], NULL), array_cosine_distance([1.0, 2.0, 3.0]::FLOAT[3], [1.0, 2.0, 3.0]::FLOAT[3]), "
    "array_cosine_distance([3.0, 4.0]::FLOAT[2], [0.0, 0.0]::FLOAT[2]), array_cosine_distance([0.0, 0.0]::FLOAT[2], [0.0, 0.0]::FLOAT[2]), "
    "array_cosine_distance([1.0]::FLOAT[1], [2.0]::FLOAT[1])")
run("SELECT array_cosine_distance([1.0, NULL, 3.0]::FLOAT[3], [1.0, 2.0, 3.0]::FLOAT[3])")
run("SELECT array_cosine_distance([1.0, 2.0, 3.0]::FLOAT[3], [1.0, 2.0, 3.0, 4.0]::FLOAT[4])")
run("SELECT array_cosine_distance([1.0, 2.0, 3.0], [1.0, 2.0, 3.0])")
run("SELECT array_cosine_distance(NULL, NULL)")
```

```
[(None, 0.0, 2.0, 2.0, 0.0)]
OSError: array_cosine_distance: left argument can not contain NULL values
InternalError: Binder Error: array_cosine_distance: Array arguments must be of the same size
InternalError: Binder Error: No function matches the given name and argument types 'array_cosine_distance(DECIMAL(2,1)[], DECIMAL(2,1)[])'. You might need to add explicit type casts.
InternalError: Binder Error: Could not choose a best candidate function for the function call "array_cosine_distance("NULL", "NULL")". In order to select one, please add explicit type casts.
```

## Examples

The cosine distance of each document's embedding from a query vector. Document 2 (all zeros) gets `2.0`, and document 3 (no embedding) gets NULL:

```
run(f"SELECT id, title, array_cosine_distance(e, [1.0, 1.0, 1.0]::FLOAT[3]) FROM {TABLE} ORDER BY id LIMIT 10")
```

```
[(1, 'same', 0.07417988777160645), (2, 'zero', 2.0), (3, 'none', None), (4, 'opposite', 1.9258201122283936), (5, 'axis', 0.4226497411727905), (6, 'three-four', 0.19170963764190674)]
```

The three documents that point most nearly the same way as the query vector, most similar first. `ORDER BY` on the distance needs a `LIMIT`:

```
run(f"SELECT id, title, array_cosine_distance(e::FLOAT[3], [1.0, 1.0, 1.0]::FLOAT[3]) AS dist "
    f"FROM {TABLE} WHERE e IS NOT NULL ORDER BY dist LIMIT 3")
```

```
[(1, 'same', 0.07417988777160645), (6, 'three-four', 0.19170963764190674), (5, 'axis', 0.4226497411727905)]
```

The documents within a cosine distance of 0.5 of the query vector:

```
run(f"SELECT id, title FROM {TABLE} WHERE array_cosine_distance(e, [1.0, 1.0, 1.0]::FLOAT[3]) < 0.5 ORDER BY id LIMIT 10")
```

```
[(1, 'same'), (5, 'axis'), (6, 'three-four')]
```

The same distance from the list column, cast to a fixed-size array, as `FLOAT` and as `DOUBLE`:

```
run(f"SELECT id, array_cosine_distance(raw::FLOAT[3], [1.0, 1.0, 1.0]::FLOAT[3]), "
    f"array_cosine_distance(raw::DOUBLE[3], [1.0, 1.0, 1.0]::DOUBLE[3]) FROM {TABLE} ORDER BY id LIMIT 10")
```

```
[(1, 0.07417988777160645, 0.07417990022744858), (2, 2.0, 2.0), (3, None, None), (4, 1.9258201122283936, 1.9258200997725514), (5, 0.4226497411727905, 0.42264973081037416), (6, 0.19170963764190674, 0.191709623134524)]
```

## How it runs in VAST

The Query Engine runs on the cluster's CNodes, and a query on a table starts with a scan in VAST DataBase (`VAST_SCAN`). VAST DataBase can evaluate a comparison of a column with a constant, but not a function of a column, so `array_cosine_distance()` is always computed by the Query Engine:

- **In the `SELECT` list,** the scan reads the vector column and the Query Engine computes the distance for each row it returns.
- **In `WHERE`,** the scan sends the vector column for every row, and a `FILTER` step computes the distance and keeps the rows that match.
- **In `ORDER BY ... LIMIT k`,** a `TOP_N` step computes the distance for the rows the scan sends and keeps the `k` smallest.

The plan of the most-similar example shows the `TOP_N` step above the scan, which reads `id` and `e`:

```
plan(f"SELECT id, array_cosine_distance(e::FLOAT[3], [1.0, 1.0, 1.0]::FLOAT[3]) AS dist FROM {TABLE} ORDER BY dist LIMIT 3")
```

```
┌───────────────────────────┐
│           TOP_N           │
│    ────────────────────   │
│           Top: 3          │
│                           │
│         Order By:         │
│ array_cosine_distance(CAST│
│      (vastdb."<bucket>      │
│/docs_sql_fn_vector_array_c│
│ osine_distance".docs.e AS │
│    FLOAT[3]), CAST(main   │
│ .list_value(1.0, 1.0, 1.0)│
│      AS FLOAT[3])) ASC    │
└─────────────┬─────────────┘
┌─────────────┴─────────────┐
│         PROJECTION        │
│    ────────────────────   │
│             id            │
│            dist           │
│                           │
└─────────────┬─────────────┘
┌─────────────┴─────────────┐
│         VAST_SCAN         │
│    ────────────────────   │
│    Function: VAST_SCAN    │
│                           │
│        Projections:       │
│             id            │
│             e             │
│                           │
└───────────────────────────┘
```

For tables with a vector index: which index metric goes with `array_cosine_distance`, and the query shape that uses the index, are in the VAST Vector Search section: [distance functions](/develop/docs/dev-vector-search-reference-sql#distance-functions) and [query shape that uses the index](/develop/docs/dev-vector-search-reference-sql#query-shape-that-uses-the-index).

## Make it faster

- **Search large tables through a vector index.** Create the table with a vector index and write the query in the shape the index uses; see [query design](/develop/docs/dev-vector-search-guides-query-design).
- **Add pushed-down conditions.** A comparison of a column with a constant, such as `title &lt;&gt; &apos;zero&apos;`, can be evaluated in VAST DataBase ([pushdowns](/develop/docs/dev-database-reference-pushdowns)), so only the rows that pass it reach the distance computation. The plan has no `FILTER` step for it:

```
run(f"SELECT id, array_cosine_distance(e::FLOAT[3], [1.0, 1.0, 1.0]::FLOAT[3]) AS dist "
    f"FROM {TABLE} WHERE title <> 'zero' ORDER BY dist LIMIT 3")
plan(f"SELECT id, array_cosine_distance(e::FLOAT[3], [1.0, 1.0, 1.0]::FLOAT[3]) AS dist "
     f"FROM {TABLE} WHERE title <> 'zero' ORDER BY dist LIMIT 3")
```

```
[(1, 0.07417988777160645), (6, 0.19170963764190674), (5, 0.4226497411727905)]
┌───────────────────────────┐
│           TOP_N           │
│    ────────────────────   │
│           Top: 3          │
│                           │
│         Order By:         │
│ array_cosine_distance(CAST│
│      (vastdb."<bucket>      │
│/docs_sql_fn_vector_array_c│
│ osine_distance".docs.e AS │
│    FLOAT[3]), CAST(main   │
│ .list_value(1.0, 1.0, 1.0)│
│      AS FLOAT[3])) ASC    │
└─────────────┬─────────────┘
┌─────────────┴─────────────┐
│         PROJECTION        │
│    ────────────────────   │
│             id            │
│            dist           │
│                           │
└─────────────┬─────────────┘
┌─────────────┴─────────────┐
│         VAST_SCAN         │
│    ────────────────────   │
│    Function: VAST_SCAN    │
│                           │
│        Projections:       │
│             id            │
│             e             │
│                           │
└───────────────────────────┘
```

## Limitations and errors

- **Fixed-size arrays only.** A list, such as a `FLOAT[]` column or an uncast literal, fails with `No function matches the given name and argument types`. Cast it: `raw::FLOAT[3]`.
- **The dimension must match.** Arrays of different sizes fail, and so does casting a vector to another size.
- **No NULL elements.** A vector with a NULL element fails; a NULL vector gives `NULL`.
- **Floating-point arrays only.** Two integer arrays fail with `Binder Error: array_cosine_distance: Arguments must be arrays of FLOAT16, FLOAT or DOUBLE`.
- **Zero vectors give `2.0`.** See [What it does](/develop/docs/dev-database-reference-sql-functions-vector-array-cosine-distance#what-it-does). With `FLOAT` arrays, very small values, such as `1e-30`, gave `2.0` too.
- **`ORDER BY` needs a `LIMIT`.** Without one it fails with `The operator &apos;ORDER_BY&apos; type=1 is currently not supported for this usage.`
- **Other SQL dialects' names aren't available.** `cosine_distance` fails with `Scalar Function with name cosine_distance does not exist!`.

```
run(f"SELECT id, array_cosine_distance(raw, [1.0, 1.0, 1.0]::FLOAT[3]) FROM {TABLE} ORDER BY id LIMIT 10")
run("SELECT array_cosine_distance([1, 2, 3]::INTEGER[3], [1, 2, 3]::INTEGER[3])")
run("SELECT array_cosine_distance([1e-30, 1e-30]::FLOAT[2], [2e-30, 2e-30]::FLOAT[2])")
run(f"SELECT id FROM {TABLE} ORDER BY array_cosine_distance(e::FLOAT[3], [1.0, 1.0, 1.0]::FLOAT[3])")
run("SELECT cosine_distance([1.0, 2.0]::FLOAT[2], [1.0, 2.0]::FLOAT[2])")
```

```
InternalError: Binder Error: No function matches the given name and argument types 'array_cosine_distance(FLOAT[], FLOAT[3])'. You might need to add explicit type casts.
InternalError: Binder Error: array_cosine_distance: Arguments must be arrays of FLOAT16, FLOAT or DOUBLE
[(2.0,)]
InternalError: The operator 'ORDER_BY' type=1 is currently not supported for this usage.
InternalError: Catalog Error: Scalar Function with name cosine_distance does not exist!
```

## Similar functions

| Function | Measures | `[1, 2, 3]` and `[2, 4, 6]` (same direction) | `[1, 2, 3]` and `[-1, -2, -3]` (opposite) |
| --- | --- | --- | --- |
| `array_cosine_distance` | the angle between the vectors, not their length | `0.0` | `2.0` |
| [array_distance](/develop/docs/dev-database-reference-sql-functions-vector-array-distance) | straight-line distance | `3.7416574954986572` | `7.4833149909973145` |
| [array_negative_inner_product](/develop/docs/dev-database-reference-sql-functions-vector-array-negative-inner-product) | minus the dot product | `-28.0` | `14.0` |

For all three, a smaller result means closer or more similar.

```
run("SELECT array_cosine_distance([1.0, 2.0, 3.0]::FLOAT[3], [2.0, 4.0, 6.0]::FLOAT[3]), "
    "array_distance([1.0, 2.0, 3.0]::FLOAT[3], [2.0, 4.0, 6.0]::FLOAT[3]), "
    "array_negative_inner_product([1.0, 2.0, 3.0]::FLOAT[3], [2.0, 4.0, 6.0]::FLOAT[3])")
run("SELECT array_cosine_distance([1.0, 2.0, 3.0]::FLOAT[3], [-1.0, -2.0, -3.0]::FLOAT[3]), "
    "array_distance([1.0, 2.0, 3.0]::FLOAT[3], [-1.0, -2.0, -3.0]::FLOAT[3]), "
    "array_negative_inner_product([1.0, 2.0, 3.0]::FLOAT[3], [-1.0, -2.0, -3.0]::FLOAT[3])")
```

```
[(0.0, 3.7416574954986572, -28.0)]
[(2.0, 7.4833149909973145, 14.0)]
```

Clean up:

```
run(f"DROP TABLE IF EXISTS {TABLE}")
run(f"DROP SCHEMA IF EXISTS {SCHEMA}")
conn.close()
```

```
[]
[]
```

## See also

- [Vector search SQL](/develop/docs/dev-database-reference-sql-vector-search) and the [SQL reference for VAST Vector Search](/develop/docs/dev-vector-search-reference-sql).
- [Where filters run](/develop/docs/dev-database-reference-sql-select#where-filters-run) and [pushdowns](/develop/docs/dev-database-reference-pushdowns).
- [EXPLAIN](/develop/docs/dev-database-reference-sql-explain): how to read a plan.