--- title: "array_negative_inner_product" slug: "dev-database-reference-sql-functions-vector-array-negative-inner-product" updated: 2026-10-30T18:18:51Z published: 2026-10-30T18:18:51Z canonical: "docs.vastdata.com/dev-database-reference-sql-functions-vector-array-negative-inner-product" --- > ## 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_negative_inner_product *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 minus the inner (dot) product of two vectors: -(a1 × b1 + a2 × b2 + …). The sign is flipped so that, as with the other distance functions, a smaller result means more similar and `ORDER BY ... LIMIT k` returns the best matches first. ## Syntax ``` array_negative_inner_product(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_negative_inner_product`. 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 3 has no embedding: ``` 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_negative_inner_product"' 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, "") 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, "")) 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_negative_inner_product(a, b)` multiplies the elements in the same positions, adds the products, and changes the sign: | Call | Result | | --- | --- | | `array_negative_inner_product([1.0, 2.0, 3.0]::FLOAT[3], [1.0, 2.0, 3.0]::FLOAT[3])` | `-14.0` | | `array_negative_inner_product([1.0, 2.0, 3.0]::FLOAT[3], [2.0, 4.0, 6.0]::FLOAT[3])` | `-28.0` | | `array_negative_inner_product([1.0, 0.0]::FLOAT[2], [0.0, 1.0]::FLOAT[2])` | `-0.0` (at right angles) | | `array_negative_inner_product([1.0, 2.0, 3.0]::FLOAT[3], [-1.0, -2.0, -3.0]::FLOAT[3])` | `14.0` | The result depends on the lengths of the vectors as well as on their directions: `[2.0, 4.0, 6.0]` scores better against `[1.0, 2.0, 3.0]` than `[1.0, 2.0, 3.0]` itself does. To compare directions only, use [array_cosine_distance](/develop/docs/dev-database-reference-sql-functions-vector-array-cosine-distance). A zero result is printed as `-0.0`: zero with its sign changed. 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`, and an integer array is accepted as long as the other argument is a floating-point array. ## NULLs and edge cases | Input | Result | | --- | --- | | A NULL vector (either argument) | `NULL` | | A zero vector, or vectors at right angles | `-0.0` | | Very large `FLOAT` values, such as `1e38` against `-1e38` | `inf`; with `DOUBLE` arrays, `3e+76` | | A NULL element, such as `[1.0, NULL, 3.0]::FLOAT[3]` | Error: `array_negative_inner_product: left argument can not contain NULL values` (or `right argument`) | | Arrays of different sizes | Error: `Binder Error: array_negative_inner_product: 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 'array_negative_inner_product(DECIMAL(2,1)[], DECIMAL(2,1)[])'` | | `array_negative_inner_product(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-negative-inner-product#set-up-the-examples): ``` run("SELECT array_negative_inner_product([1.0, 2.0, 3.0]::FLOAT[3], NULL), " "array_negative_inner_product([3.0, 4.0]::FLOAT[2], [0.0, 0.0]::FLOAT[2]), " "array_negative_inner_product([1.0, 0.0]::FLOAT[2], [0.0, 1.0]::FLOAT[2])") run("SELECT array_negative_inner_product([1e38, 1e38, 1e38]::FLOAT[3], [-1e38, -1e38, -1e38]::FLOAT[3]), " "array_negative_inner_product([1e38, 1e38, 1e38]::DOUBLE[3], [-1e38, -1e38, -1e38]::DOUBLE[3])") run("SELECT array_negative_inner_product([1.0, NULL, 3.0]::FLOAT[3], [1.0, 2.0, 3.0]::FLOAT[3])") run("SELECT array_negative_inner_product([1.0, 2.0, 3.0]::FLOAT[3], [1.0, 2.0, 3.0, 4.0]::FLOAT[4])") run("SELECT array_negative_inner_product([1.0, 2.0, 3.0], [1.0, 2.0, 3.0])") run("SELECT array_negative_inner_product(NULL, NULL)") ``` ``` [(None, -0.0, -0.0)] [(inf, 3e+76)] OSError: array_negative_inner_product: left argument can not contain NULL values InternalError: Binder Error: array_negative_inner_product: Array arguments must be of the same size InternalError: Binder Error: No function matches the given name and argument types 'array_negative_inner_product(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_negative_inner_product("NULL", "NULL")". In order to select one, please add explicit type casts. ``` ## Examples The negative inner product of each document's embedding with a query vector. Document 3 has no embedding, so its result is NULL: ``` run(f"SELECT id, title, array_negative_inner_product(e, [1.0, 1.0, 1.0]::FLOAT[3]) FROM {TABLE} ORDER BY id LIMIT 10") ``` ``` [(1, 'same', -6.0), (2, 'zero', -0.0), (3, 'none', None), (4, 'opposite', 6.0), (5, 'axis', -1.0), (6, 'three-four', -7.0)] ``` The three best matches for the query vector, best first. `three-four` comes before `same` because its dot product with `[1, 1, 1]` is larger (7 against 6). `ORDER BY` on the result needs a `LIMIT`: ``` run(f"SELECT id, title, array_negative_inner_product(e::FLOAT[3], [1.0, 1.0, 1.0]::FLOAT[3]) AS score " f"FROM {TABLE} WHERE e IS NOT NULL ORDER BY score LIMIT 3") ``` ``` [(6, 'three-four', -7.0), (1, 'same', -6.0), (5, 'axis', -1.0)] ``` The documents whose dot product with the query vector is more than 5, that is, whose result is below -5: ``` run(f"SELECT id, title FROM {TABLE} WHERE array_negative_inner_product(e, [1.0, 1.0, 1.0]::FLOAT[3]) < -5 ORDER BY id LIMIT 10") ``` ``` [(1, 'same'), (6, 'three-four')] ``` The same result from the list column, cast to a fixed-size array, as `FLOAT` and as `DOUBLE`: ``` run(f"SELECT id, array_negative_inner_product(raw::FLOAT[3], [1.0, 1.0, 1.0]::FLOAT[3]), " f"array_negative_inner_product(raw::DOUBLE[3], [1.0, 1.0, 1.0]::DOUBLE[3]) FROM {TABLE} ORDER BY id LIMIT 10") ``` ``` [(1, -6.0, -6.0), (2, -0.0, -0.0), (3, None, None), (4, 6.0, 6.0), (5, -1.0, -1.0), (6, -7.0, -7.0)] ``` ## 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_negative_inner_product()` is always computed by the Query Engine: - **In the `SELECT` list,** the scan reads the vector column and the Query Engine computes the result for each row it returns. - **In `WHERE`,** the scan sends the vector column for every row, and a `FILTER` step computes the result and keeps the rows that match. - **In `ORDER BY ... LIMIT k`,** a `TOP_N` step computes the result for the rows the scan sends and keeps the `k` smallest. The plan of the best-matches example shows the `TOP_N` step above the scan, which reads `id` and `e`: ``` plan(f"SELECT id, array_negative_inner_product(e::FLOAT[3], [1.0, 1.0, 1.0]::FLOAT[3]) AS score FROM {TABLE} ORDER BY score LIMIT 3") ``` ``` ┌───────────────────────────┐ │ TOP_N │ │ ──────────────────── │ │ Top: 3 │ │ │ │ Order By: │ │array_negative_inner_produc│ │ t(CAST(vastdb." │ │/docs_sql_fn_vector_array_n│ │ egative_inner_product" │ │ .docs.e AS FLOAT[3]), CAST│ │ (main.list_value(1.0, 1.0,│ │ 1.0) AS FLOAT[3])) ASC │ └─────────────┬─────────────┘ ┌─────────────┴─────────────┐ │ PROJECTION │ │ ──────────────────── │ │ id │ │ score │ │ │ └─────────────┬─────────────┘ ┌─────────────┴─────────────┐ │ VAST_SCAN │ │ ──────────────────── │ │ Function: VAST_SCAN │ │ │ │ Projections: │ │ id │ │ e │ │ │ └───────────────────────────┘ ``` For tables with a vector index: which index metric goes with `array_negative_inner_product`, 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 <> 'zero'`, can be evaluated in VAST DataBase ([pushdowns](/develop/docs/dev-database-reference-pushdowns)), so only the rows that pass it reach the computation. The plan has no `FILTER` step for it: ``` run(f"SELECT id, array_negative_inner_product(e::FLOAT[3], [1.0, 1.0, 1.0]::FLOAT[3]) AS score " f"FROM {TABLE} WHERE title <> 'zero' ORDER BY score LIMIT 3") plan(f"SELECT id, array_negative_inner_product(e::FLOAT[3], [1.0, 1.0, 1.0]::FLOAT[3]) AS score " f"FROM {TABLE} WHERE title <> 'zero' ORDER BY score LIMIT 3") ``` ``` [(6, -7.0), (1, -6.0), (5, -1.0)] ┌───────────────────────────┐ │ TOP_N │ │ ──────────────────── │ │ Top: 3 │ │ │ │ Order By: │ │array_negative_inner_produc│ │ t(CAST(vastdb." │ │/docs_sql_fn_vector_array_n│ │ egative_inner_product" │ │ .docs.e AS FLOAT[3]), CAST│ │ (main.list_value(1.0, 1.0,│ │ 1.0) AS FLOAT[3])) ASC │ └─────────────┬─────────────┘ ┌─────────────┴─────────────┐ │ PROJECTION │ │ ──────────────────── │ │ id │ │ score │ │ │ └─────────────┬─────────────┘ ┌─────────────┴─────────────┐ │ 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_negative_inner_product: Arguments must be arrays of FLOAT16, FLOAT or DOUBLE`. - **`FLOAT` can overflow.** Very large values give `inf`; cast both arguments to `DOUBLE[n]`. - **`ORDER BY` needs a `LIMIT`.** Without one it fails with `The operator 'ORDER_BY' type=1 is currently not supported for this usage.` - **Other SQL dialects' names aren't available.** `inner_product` and `dot_product` fail with `Scalar Function with name ... does not exist!`. ``` run(f"SELECT id, array_negative_inner_product(raw, [1.0, 1.0, 1.0]::FLOAT[3]) FROM {TABLE} ORDER BY id LIMIT 10") run("SELECT array_negative_inner_product([1, 2, 3]::INTEGER[3], [1, 2, 3]::INTEGER[3])") run(f"SELECT id FROM {TABLE} ORDER BY array_negative_inner_product(e::FLOAT[3], [1.0, 1.0, 1.0]::FLOAT[3])") run("SELECT inner_product([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_negative_inner_product(FLOAT[], FLOAT[3])'. You might need to add explicit type casts. InternalError: Binder Error: array_negative_inner_product: Arguments must be arrays of FLOAT16, FLOAT or DOUBLE InternalError: The operator 'ORDER_BY' type=1 is currently not supported for this usage. InternalError: Catalog Error: Scalar Function with name inner_product 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_negative_inner_product` | minus the dot product | `-28.0` | `14.0` | | `array_inner_product` | the dot product: larger means more similar, so sort it descending | `28.0` | `-14.0` | | [array_cosine_distance](/develop/docs/dev-database-reference-sql-functions-vector-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_inner_product` returns the same value with the opposite sign. For searches on a table with a vector index, see the note on `array_inner_product` in [distance functions](/develop/docs/dev-vector-search-reference-sql#distance-functions). ``` run("SELECT array_negative_inner_product([1.0, 2.0, 3.0]::FLOAT[3], [2.0, 4.0, 6.0]::FLOAT[3]), " "array_inner_product([1.0, 2.0, 3.0]::FLOAT[3], [2.0, 4.0, 6.0]::FLOAT[3]), " "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])") run("SELECT array_negative_inner_product([1.0, 2.0, 3.0]::FLOAT[3], [-1.0, -2.0, -3.0]::FLOAT[3]), " "array_inner_product([1.0, 2.0, 3.0]::FLOAT[3], [-1.0, -2.0, -3.0]::FLOAT[3]), " "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])") ``` ``` [(-28.0, 28.0, 0.0, 3.7416574954986572)] [(14.0, -14.0, 2.0, 7.4833149909973145)] ``` 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.