INNER_PRODUCT

Calculates the inner product (dot product) of two vectors, which measures the similarity and projection between vectors.

Syntax

INNER_PRODUCT(vector1, vector2)

Arguments

• vector1: First vector (VECTOR Data Type)
• vector2: Second vector (VECTOR Data Type)

Returns

Returns a FLOAT value representing the inner product of the two vectors.

Description

The inner product (also known as dot product) calculates the sum of the products of corresponding elements in two vectors. The function:

1. Verifies that both input vectors have the same length
2. Multiplies corresponding elements from each vector
3. Sums all the products to produce a single scalar value

The mathematical formula implemented is:

inner_product(v1, v2) = Σ(v1ᵢ * v2ᵢ)

Where v1ᵢ and v2ᵢ are the elements of the input vectors.

The inner product is fundamental in:

• Measuring vector similarity (higher values indicate more similar directions)
• Computing projections of one vector onto another
• Machine learning algorithms (neural networks, SVM, etc.)
• Physics calculations involving work and energy

Examples

Basic Usage

SELECT INNER_PRODUCT([1,2,3]::VECTOR(3), [4,5,6]::VECTOR(3)) AS inner_product;

Result:

┌───────────────┐
│ inner_product │
├───────────────┤
│          32.0 │
└───────────────┘

Working with Table Data

Create a table with vector data:

CREATE TABLE vector_examples (
    id INT,
    vector_a VECTOR(3),
    vector_b VECTOR(3)
);

INSERT INTO vector_examples VALUES
    (1, [1.0, 2.0, 3.0], [4.0, 5.0, 6.0]),
    (2, [1.0, 0.0, 0.0], [0.0, 1.0, 0.0]),
    (3, [2.0, 3.0, 1.0], [1.0, 2.0, 3.0]);

Calculate inner products:

SELECT
    id,
    vector_a,
    vector_b,
    INNER_PRODUCT(vector_a, vector_b) AS inner_product
FROM vector_examples;

Result:

┌────┬───────────────┬───────────────┬───────────────┐
│ id │   vector_a    │   vector_b    │ inner_product │
├────┼───────────────┼───────────────┼───────────────┤
│  1 │ [1.0,2.0,3.0] │ [4.0,5.0,6.0] │          32.0 │
│  2 │ [1.0,0.0,0.0] │ [0.0,1.0,0.0] │           0.0 │
│  3 │ [2.0,3.0,1.0] │ [1.0,2.0,3.0] │          11.0 │
└────┴───────────────┴───────────────┴───────────────┘

Vector Similarity Analysis

-- Calculate inner products to measure vector similarity
SELECT
    INNER_PRODUCT([1,0,0]::VECTOR(3), [1,0,0]::VECTOR(3)) AS same_direction,
    INNER_PRODUCT([1,0,0]::VECTOR(3), [0,1,0]::VECTOR(3)) AS orthogonal,
    INNER_PRODUCT([1,0,0]::VECTOR(3), [-1,0,0]::VECTOR(3)) AS opposite;

Result:

┌────────────────┬─────────────┬──────────┐
│ same_direction │ orthogonal  │ opposite │
├────────────────┼─────────────┼──────────┤
│           1.0  │         0.0 │     -1.0 │
└────────────────┴─────────────┴──────────┘