Overview
This lecture proves foundational properties of the dot product in vectors, specifically commutativity, distributivity, and associativity, by writing out components and using properties of real numbers.
Commutativity of the Dot Product
- The dot product of vectors v and w is v · w = w · v.
- Component-wise: v · w = v₁w₁ + v₂w₂ + ... + vₙwₙ.
- Since scalar multiplication is commutative, each term vᵢwᵢ = wᵢvᵢ, proving the property.
Distributivity of the Dot Product
- The dot product distributes over vector addition: (v + w) · x = v · x + w · x.
- v + w results in (v₁ + w₁, v₂ + w₂, ..., vₙ + wₙ).
- Component-wise, (v + w) · x = (v₁ + w₁)x₁ + ... + (vₙ + wₙ)xₙ.
- This matches v · x + w · x, so distributivity holds.
Associativity with Scalars
- For scalar c, c(v) · w = c(v · w).
- c(v) = (c v₁, c v₂, ..., c vₙ).
- The dot product: c v₁w₁ + c v₂w₂ + ... + c vₙwₙ.
- This equals c(v₁w₁ + v₂w₂ + ... + vₙwₙ) = c(v · w).
Key Terms & Definitions
- Vector — An ordered list of numbers (components), e.g., v = (v₁, v₂, ..., vₙ).
- Dot Product — For vectors v, w: v · w = v₁w₁ + v₂w₂ + ... + vₙwₙ.
- Commutative Property — Order of operation does not affect the result, e.g., a · b = b · a.
- Distributive Property — Operation distributes over addition, e.g., a(b + c) = ab + ac.
- Associative Property — Grouping of operations does not affect the result, e.g., (ab)c = a(bc).
Action Items / Next Steps
- Practice writing similar proofs for other vector operations.
- Prepare to use these properties to prove more advanced vector properties in upcoming lessons.