1. What is the Dot Product?

The dot product (also called scalar product) is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It measures the magnitude of one vector in the direction of another.

2. How Does the Calculator Work?

The calculator uses the dot product formula:

Where:

• \( \vec{a} \) — First vector with components (a₁, a₂, a₃)
• \( \vec{b} \) — Second vector with components (b₁, b₂, b₃)
• The result is a scalar value

Explanation: The dot product is calculated by multiplying corresponding components and summing the results.

3. Applications of Dot Product

Details: The dot product is used in physics to calculate work done, in computer graphics for lighting calculations, in machine learning for similarity measures, and in engineering for projection calculations.

4. Using the Calculator

Tips: Enter the x, y, and z components for both vectors. The calculator will compute the dot product. All fields are required.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between dot product and cross product?
A: Dot product gives a scalar result while cross product gives a vector result. Dot product measures similarity while cross product measures perpendicularity.

Q2: What does a dot product of zero mean?
A: A zero dot product indicates the vectors are perpendicular (90° angle between them).

Q3: Can I calculate dot product for 2D vectors?
A: Yes, just set the z-components to 0 or leave them blank (they'll be treated as 0).

Q4: What's the geometric interpretation of dot product?
A: It equals the product of the vectors' magnitudes and the cosine of the angle between them.

Q5: How is dot product used in machine learning?
A: It's used in cosine similarity, neural networks, and various other algorithms to measure vector similarity.