1. What is Vector Inner Product?

The inner product (or dot product) of two vectors is a scalar value that measures their similarity and is calculated as the sum of the products of their corresponding components.

2. How Does the Calculator Work?

The calculator uses the inner product formula:

Where:

• \( u \) — First vector (u₁, u₂, ..., uₙ)
• \( v \) — Second vector (v₁, v₂, ..., vₙ)
• \( n \) — Dimension of the vectors

Explanation: The inner product is calculated by multiplying corresponding components of the vectors and summing all these products.

3. Importance of Inner Product

Details: The inner product is fundamental in vector algebra, used in projections, determining angles between vectors, and in many machine learning algorithms.

4. Using the Calculator

Tips: Enter vectors as comma-separated values (e.g., "1,2,3"). Both vectors must have the same number of components.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between inner product and dot product?
A: In most contexts, they refer to the same operation, though "inner product" is more general while "dot product" specifically refers to Euclidean space.

Q2: What does the inner product measure?
A: It measures the similarity between vectors, with larger values indicating more similar direction and magnitude.

Q3: What's the geometric interpretation?
A: The inner product equals the product of the vectors' magnitudes and the cosine of the angle between them.

Q4: What's the inner product of orthogonal vectors?
A: Zero, since the cosine of 90° is zero.

Q5: Can I calculate inner products for complex vectors?
A: This calculator handles real vectors only. Complex vectors require conjugate of one vector in the product.