Dot Product of Two Vectors

The dot product is calculated by multiplying corresponding components of two vectors and adding the products. For vectors a = ⟨a₁, a₂, a₃⟩ and b = ⟨b₁, b₂, b₃⟩, the dot product equals a₁b₁ + a₂b₂ + a₃b₃. This algebraic method produces a scalar value from two vectors.

The geometric dot product formula is a·b = |a||b|cos θ, where θ is the angle between vectors. When the angle is small, cosine is large, maximizing the dot product. At 90 degrees, the dot product becomes zero, indicating orthogonal vectors with no alignment.

Solar panels use the dot product to maximize energy absorption. Sunlight is modeled as a direction vector, and the panel's normal vector represents its surface orientation. The dot product between these vectors measures how directly sunlight hits the panel, with larger values indicating better alignment and greater energy absorption.

Direction cosines measure angles between a vector and the coordinate axes. For vector a, they are cos α = a₁/|a|, cos β = a₂/|a|, and cos γ = a₃/|a|. These values satisfy cos²α + cos²β + cos²γ = 1 and are derived from dot product calculations with unit vectors along each axis.

Scalar projection (compₐ b) measures how much of vector b aligns with vector a as a single number: (a·b)/|a|. Vector projection (projₐ b) extends this by scaling the unit vector in a's direction: (a·b)/|a|² times a. Vector projection provides both magnitude and direction.

The dot product equals zero when vectors are orthogonal, meaning they meet at a 90-degree angle. This occurs because cos 90° = 0. Orthogonal vectors have no component in each other's direction, making them perpendicular in space. This property is useful for checking perpendicularity in vectors in space problem solving.

Work is calculated as the dot product of force and displacement vectors. The dot product captures how much force acts in the direction of motion. Only the force component aligned with displacement contributes to work, which the dot product efficiently quantifies through the cosine of the angle between them.