Calculate Inner Product Of Two Vectors

Inner Product Formula:

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

Inner Product Result:

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1. What Is The Inner Product?

The inner product (also called dot product) is an algebraic operation 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 standard inner product formula:

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

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 the vectors and sums all these products.

3. Importance Of Inner Product

Details: The inner product is fundamental in vector calculus, physics, and machine learning. It's used to determine angles between vectors, project one vector onto another, and in many 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 dimensions.

5. Frequently Asked Questions (FAQ)

Q1: What's the geometric interpretation of inner product?
A: The inner product relates to the angle θ between vectors: u·v = ||u|| ||v|| cosθ.

Q2: What's the difference between inner product and cross product?
A: Inner product gives a scalar, cross product gives a vector (in 3D). Inner product works in any dimension.

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

Q4: Can I calculate inner product of complex vectors?
A: This calculator handles real numbers only. Complex inner products require complex conjugation.

Q5: How is inner product used in machine learning?
A: It's fundamental in kernel methods, neural networks, and similarity measurements.