1. What is the Inner Product?

The inner product (also called dot product) is a fundamental operation in linear algebra that takes two equal-length sequences of numbers (vectors) and returns a single number. It measures the similarity between two vectors.

2. How Does the Calculator Work?

The calculator uses the inner product formula:

Where:

• \( x \) — First vector (x₁, x₂, ..., xₙ)
• \( y \) — Second vector (y₁, y₂, ..., yₙ)
• \( n \) — Length of the vectors
• \( x_i \) — i-th component of vector x
• \( y_i \) — i-th component of vector y

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 crucial in many areas including geometry (calculating angles between vectors), physics (work calculations), and machine learning (similarity measures).

4. Using the Calculator

Tips: Enter vectors as comma-separated values (e.g., "1, 2, 3"). Both vectors must have the same length. The calculator will automatically handle whitespace.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between inner product and dot product?
A: In most contexts, they are the same. Strictly speaking, dot product refers specifically to the Euclidean space version of inner product.

Q2: What does the inner product measure?
A: It measures the magnitude of projection of one vector onto another and their directional similarity.

Q3: What's the geometric interpretation?
A: For unit vectors, the inner product equals the cosine of the angle between them.

Q4: What are applications of inner product?
A: Used in physics, engineering, computer graphics, machine learning, and many other fields.

Q5: What's the inner product of orthogonal vectors?
A: The inner product of orthogonal (perpendicular) vectors is zero.