1. What is the Dot Product?

The dot product (or 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)

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

3. Applications of Dot Product

Details: Dot products are 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 which can be used to determine the angle between vectors (when combined with their magnitudes).

5. Frequently Asked Questions (FAQ)

Q1: What does the dot product tell us?
A: The dot product indicates how much one vector extends in the direction of another. A zero dot product means the vectors are perpendicular.

Q2: How is dot product different from cross product?
A: Dot product gives a scalar quantity while cross product gives a vector quantity. Dot product measures alignment while cross product measures perpendicularity.

Q3: Can dot product be negative?
A: Yes, a negative dot product indicates the vectors point in generally opposite directions (angle > 90°).

Q4: What's the dot product of a vector with itself?
A: It equals the square of the vector's magnitude (length).

Q5: How is dot product used to find angles?
A: The angle θ between vectors can be found using: cosθ = (a·b)/(|a||b|)