Euclidean Inner Product Calculator

Euclidean Inner Product Formula:

\[ \langle \mathbf{u}, \mathbf{v} \rangle = \sum_{i=1}^{n} u_i v_i \]

Euclidean Inner Product:

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1. What is Euclidean Inner Product?

The Euclidean inner product (also known as dot product) is a fundamental operation in linear algebra that takes two equal-length vectors and returns a single scalar value. It measures the similarity between two vectors and is defined as the sum of the products of their corresponding components.

2. How Does the Calculator Work?

The calculator uses the Euclidean inner product formula:

\[ \langle \mathbf{u}, \mathbf{v} \rangle = \sum_{i=1}^{n} u_i v_i \]

Where:

\( \mathbf{u} = (u_1, u_2, ..., u_n) \) — First vector
\( \mathbf{v} = (v_1, v_2, ..., v_n) \) — Second vector
\( n \) — Dimension of the vectors
\( u_i v_i \) — Product of corresponding components

Explanation: The calculator sums the products of corresponding components from both vectors to compute the inner product.

3. Applications of Inner Product

Details: The Euclidean inner product is used in vector projection, angle calculation between vectors, determining orthogonality, machine learning algorithms, signal processing, and physics calculations involving work and energy.

4. Using the Calculator

Tips: Enter vectors as comma-separated values (e.g., "1,2,3,4"). Both vectors must have the same dimension. The calculator accepts integer and decimal values.

5. Frequently Asked Questions (FAQ)

Q1: What is the geometric interpretation of inner product?
A: The inner product relates to the cosine of the angle between vectors: \( \mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\|\|\mathbf{v}\|\cos\theta \).

Q2: What does a zero inner product indicate?
A: A zero inner product indicates that the vectors are orthogonal (perpendicular) to each other.

Q3: Can I calculate inner product for complex vectors?
A: This calculator is for real vectors only. Complex vectors require the conjugate of one vector in the calculation.

Q4: What are the units of the inner product?
A: The units are the square of the original vector units. If vectors represent physical quantities, the inner product has units of those quantities squared.

Q5: How is inner product related to vector length?
A: The length (norm) of a vector is the square root of its inner product with itself: \( \|\mathbf{u}\| = \sqrt{\mathbf{u} \cdot \mathbf{u}} \).