1. What is the Inner Product of Vectors?

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:

• \( \mathbf{a} \) — First vector (a₁, a₂, ..., aₙ)
• \( \mathbf{b} \) — Second vector (b₁, b₂, ..., bₙ)
• \( n \) — Dimension of the vectors

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

3. Importance of Inner Product

Details: The inner product is fundamental in vector algebra, used in physics, engineering, and machine learning for calculating angles between vectors, projections, and similarity measures.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What's the geometric interpretation of inner product?
A: It equals the product of vector magnitudes and the cosine of the angle between them: ⟨a,b⟩ = |a||b|cosθ.

Q2: What's the difference between inner product and cross product?
A: Inner product produces a scalar, while cross product (in ℝ³) produces a vector perpendicular to both input vectors.

Q3: What does a zero inner product mean?
A: Vectors are orthogonal (perpendicular) to each other.

Q4: Can inner product be negative?
A: Yes, when the angle between vectors is greater than 90°.

Q5: How is inner product used in machine learning?
A: It's used in kernel methods, similarity measures, and neural network operations.