Unit 1: Vectors and Differential Equations

Vector Algebra

Dot and Cross product and their properties

Scalar (Dot) Product: A product of two vectors that results in a scalar.

A · B = |A| |B| cos(θ)

• Properties:
  • Commutative: A · B = B · A
  • Distributive: A · (B + C) = A · B + A · C
  • If A ⊥ B (perpendicular), then θ = 90°, and A · B = 0.
  • If A || B (parallel), then θ = 0°, and A · B = |A||B|.
  • In components: A · B = A_xB_x + A_yB_y + A_zB_z
• Application: Work done, W = F · d.

Vector (Cross) Product: A product of two vectors that results in a new vector.

A × B = |A| |B| sin(θ) n̂

Where n̂ is a unit vector perpendicular to the plane of A and B, given by the Right-Hand Rule.

• Properties:
  • Anti-commutative: A × B = - (B × A)
  • Distributive: A × (B + C) = (A × B) + (A × C)
  • If A || B (parallel), then θ = 0°, and A × B = 0 (the null vector).
  • In determinant form:

     | î ĵ k̂ | A × B = | Ax Ay Az | | Bx By Bz |