Vector Operations and Calculus

Vector Projections

• Projection of a vector onto another vector:
  • Given vectors a and b, the projection of a onto b is defined as:
    • proj_b(a) = (a · b / ||b||^2) * b
  • This gives the component of a in the direction of b
• Scalar projection:
  • The scalar projection of a onto b is the magnitude of the projection vector
  • Calculated as: a · b / ||b||

Vector Operations

• Vector addition:
  • Commutative: a + b = b + a
  • Associative: (a + b) + c = a + (b + c)
• Scalar multiplication:
  • Distributive: k(a + b) = ka + kb
  • Associative: (km)a = k(ma)
• Dot product (inner product):
  • a · b = b · a (commutative)
  • a · (b + c) = a · b + a · c (distributive)

Vector Calculus

• Gradient:
  • A vector pointing in the direction of the maximum rate of increase of a function
  • Calculated as: ∇f(x, y, z) = (∂f/∂x, ∂f/∂y, ∂f/∂z)
• Divergence:
  • Measures how much a vector field spreads out or converges at a point
  • Calculated as: ∇ · F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z
• Curl:
  • Measures the rotational tendency of a vector field around a point
  • Calculated as: ∇ × F = (∂F_z/∂y - ∂F_y/∂z, ∂F_x/∂z - ∂F_z/∂x, ∂F_y/∂x - ∂F_x/∂y)

Unit Vectors

• Definition: A vector with a magnitude of 1
• Notation: Typically denoted with a hat symbol, e.g., î, ĵ, ķ
• Properties:
  • ||u|| = 1
  • u · u = 1
  • u × u = 0
• Examples:
  • î = <1, 0, 0> (standard basis vector in the x-direction)
  • ĵ = <0, 1, 0> (standard basis vector in the y-direction)
  • ķ = <0, 0, 1> (standard basis vector in the z-direction)