Multiplication of Vectors

Vector multiplication is a fundamental concept that allows us to combine two vectors to create either a scalar quantity or a vector quantity. Unlike adding or subtracting vectors (which always gives another vector), multiplying vectors can produce completely different types of results depending on the method used.​

Key Concept:

When we multiply vectors, we get new physical quantities that may be:

• Scalar (number only) - called Scalar Product or Dot Product
• Vector (magnitude + direction) - called Vector Product or Cross Product

The product of the magnitudes of two vectors and the cosine of the angle between them, giving a scalar quantity, is called the scalar or dot product.

The product of the magnitudes of two vectors and the sine of the angle between them, giving a vector quantity perpendicular to the plane of both vectors, is called the vector or cross product.

Scalar (Dot) Product:

• A ⋅ B = AB cos θ = A_x​B_x​ + A_y​B_y​ + A_z​B_z​
• Commutative:  A ⋅ B = B ⋅ A
• Distributive over addition: A ⋅ (B + C) = A ⋅ B + A ⋅ C
• Geometric interpretation: Product of the magnitude of one vector by the component of the other in the direction of the first
• A ⋅ A = A²
• If A ⊥ B, then A ⋅ B = 0

Vector (Cross) Product:

• A × B = (ABsinθ)\[\hat n\] = ​|\[\hat i\] \[\hat j\] \[\hat k\] A_x A_y A_z B_x B_y B_z|​​​
• Not commutative: A × B ≠ B × A
• Distributive over addition: A × (B + C) = A × B + A × C
• Geometric interpretation: Magnitude equals the area of the parallelogram whose adjacent sides are the two co-initial vectors
• A × A = 0
• If A ∥ B, then A × B = 0