Vector Inner Product Calculator

Inner Product Formula:

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

Inner Product:

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1. What is Vector Inner Product?

The inner product (or dot product) of two vectors is a scalar value obtained by summing the products of corresponding components. It measures the similarity between vectors and is fundamental in linear algebra and physics.

2. How Does the Calculator Work?

The calculator uses the inner product formula:

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

Where:

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

Explanation: The calculator multiplies corresponding components of both vectors and sums all these products.

3. Importance of Inner Product

Details: The inner product is used to determine angles between vectors, test for orthogonality, and in projections. It's essential in machine learning, physics, and computer graphics.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between inner product and cross product?
A: Inner product produces a scalar, while cross product (for 3D vectors) produces another vector perpendicular to both input vectors.

Q2: What does a zero inner product mean?
A: It means the vectors are orthogonal (perpendicular to each other).

Q3: Can I calculate inner product for complex vectors?
A: This calculator handles real numbers only. Complex vectors require conjugating one vector's components.

Q4: What's the geometric interpretation?
A: The inner product equals ||u||·||v||·cos(θ), where θ is the angle between vectors and ||·|| is the magnitude.

Q5: How is this used in machine learning?
A: Inner products are fundamental in kernel methods, neural networks, and similarity measurements between data points.