Mathematical Methods — Important Questions

Commutative: A⃗ + B⃗ = B⃗ + A⃗, Associative: (A⃗ + B⃗) + C⃗ = A⃗ + (B⃗ + C⃗), Closure: Sum of two vectors is also a vector

Vector addition follows commutative and associative laws, and the sum of vectors is also a vector (closure property). The distributive property mentioned is incorrect for vector operations.

26

A⃗ · B⃗ = (2)(4) + (3)(6) = 8 + 18 = 26. For rectangular components, dot product is calculated as AxBx + AyBy + AzBz.

Along +z direction

î × ĵ = k̂, which is along the +z direction. This follows the right-hand rule: curl fingers from î to ĵ, thumb points in +z direction.

A⃗ × B⃗ = -B⃗ × A⃗, A⃗ × A⃗ = 0, |A⃗ × B⃗| = AB sin θ

Cross product is anti-commutative (not commutative), any vector crossed with itself gives zero, and the magnitude equals AB sin θ where θ is the angle between vectors.