Vector Dot Product Calculator

Understanding the Dot Product

The **Dot Product** (also known as the scalar product) is a fundamental algebraic operation that takes two equal-length sequences of numbers (vectors) and returns a single number (scalar). It is widely used in physics, engineering, and computer graphics.

Algebraic Definition

The dot product of two vectors A = [a₁, a₂, ..., aₙ] and B = [b₁, b₂, ..., bₙ] is defined as:

A · B = ∑(aᵢbᵢ) = a₁b₁ + a₂b₂ + ... + aₙbₙ

Geometric Definition

Geometrically, the dot product relates to the angle θ between the two vectors:

A · B = ||A|| ||B|| cos(θ)

Where ||A|| is the magnitude (length) of vector A.

Interpreting the Result

• Positive (> 0): The vectors point in the same general direction (angle < 90°).
• Zero (= 0): The vectors are **Orthogonal** (perpendicular) to each other (angle = 90°).
• Negative (< 0): The vectors point in opposite general directions (angle > 90°).

Applications

1. Work Done (Physics): Work is calculated as the dot product of the Force vector and the Displacement vector ($W = F \cdot d$).

2. Lighting (Computer Graphics): To calculate how bright a polygon is, engines calculate the dot product between the surface normal and the light direction. If they are perpendicular, the light doesn't hit the surface.

3. Projection: Finding how much of one vector goes in the direction of another.