1. What is the Inner Product?

The inner product (or dot product) of two vectors is a scalar value that measures their similarity and angle between them. It's calculated by multiplying corresponding components and summing the results.

2. How Does the Calculator Work?

The calculator uses the standard inner product formula:

Where:

• \( u_x, u_y, u_z \) — Components of vector u
• \( v_x, v_y, v_z \) — Components of vector v

Explanation: The inner product combines both the magnitude of the vectors and the cosine of the angle between them.

3. Importance of Inner Product

Details: The inner product is fundamental in vector analysis, physics (work calculations), computer graphics, and machine learning (similarity measures).

4. Using the Calculator

Tips: Enter all six components (x,y,z for both vectors). The calculator works for 2D vectors too (just set z-components to 0).

5. Frequently Asked Questions (FAQ)

Q1: What does the inner product tell us?
A: It indicates whether vectors are pointing in similar directions (positive), opposite directions (negative), or are perpendicular (zero).

Q2: How is this related to vector length?
A: The length (norm) of a vector is the square root of its inner product with itself: \( \|u\| = \sqrt{\langle u,u \rangle} \).

Q3: Can I calculate angles between vectors?
A: Yes, the angle θ satisfies \( \cosθ = \frac{\langle u,v \rangle}{\|u\|\|v\|} \).

Q4: What's the difference between inner and cross product?
A: Inner product gives a scalar, cross product gives a vector perpendicular to both input vectors.

Q5: How is this used in Desmos?
A: Desmos can graph vectors and compute their inner product using similar formulas in its calculator interface.