1. What is Inner Product?

The inner product (or dot product) of two vectors is a scalar value that measures their similarity and angle between them. For vectors u and v, it's 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:

• \( \mathbf{u} \) — First vector (u₁, u₂, ..., uₙ)
• \( \mathbf{v} \) — Second vector (v₁, v₂, ..., vₙ)
• \( n \) — Dimension of the vectors (must be equal for both)

Example: For vectors (1,0) and (0,2), the calculation is 1×0 + 0×2 = 0.

3. Importance of Inner Product

Details: Inner products are fundamental in vector spaces, used in geometry (calculating angles), 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 number of dimensions. The calculator shows both the result and the step-by-step calculation.

5. Frequently Asked Questions (FAQ)

Q1: What does a zero inner product mean?
A: A zero inner product indicates the vectors are orthogonal (perpendicular to each other).

Q2: Can I calculate inner product for 3D vectors?
A: Yes, the calculator works for vectors of any dimension as long as both have the same dimension.

Q3: What's the difference between inner product and cross product?
A: Inner product gives a scalar, while cross product (only in 3D) gives a vector perpendicular to both input vectors.

Q4: 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.

Q5: Can I use this for complex vectors?
A: This calculator handles real numbers only. Complex vectors require conjugate in the calculation.