Product of Two Vectors

Multiplication of two vectors is defined in two ways:

  • Scalar (or dot) product where the result is a scalar
  • Vector (or cross) product where the result is a vector

Scalar (or dot) Product: The dot product may be defined algebraically or geometrically. The geometric definition is based on the notions of angle and distance (magnitude of vectors). The equivalence of these two definitions relies on having a Cartesian coordinate system for Euclidean space

The scalar product of two non-zero vectors a and b is a. b = |a| |b| cos (θ).Scalar ProductThe dot product of two vectors is zero if and only if they are perpendicular to each other.


  • Commutative: a.b = b.a
    Which follows from the definition (θ is the angle between a and b): a.b = |a||b| cosθ = |b||a| cosθ = b.a
  • Distributive over vector addition: a.(b + c) = a.b + a.c
  • Bi-linear:(rb + c) = r (a.b) + (a.c)
  • Scalar multiplication: (ac₁) (c₂b) = c₁c₂ (a.b)
  • Orthogonal: Two non-zero vectors a and b are orthogonal if and only if a.b = 0

Vector (or cross) Product: The vector product of two non-zero vectors a and b is a × b = |a||b| sin(θ) n

  • |a| is the magnitude (length) of vector a
  • |b| is the magnitude (length) of vector b
  • θ is the angle between aand b
  • n is the unit vector at right angles to both a and bRight Hand Rule
  • So the lengthis: the length of a times the length of b times the sine of the angle between a and b
  • Then we multiply by the vector nto make sure it heads in the right direction (at right angles to both a and b).

The cross product of two vectors is zero if and only if they are parallel (or collinear) to each other.

The cross product could point in the completely opposite direction and still be at right angles to the two other vectors, so we have the: With your right-hand, point your index finger along vector a, and point your middle finger along vector b: the cross product goes in the direction of your thumb