Scalar product

The scalar product, also known as the dot product, of two vectors is a scalar value that represents the magnitude of one vector with respect to another. It can be calculated as the product of the magnitude of one vector and the cosine of the angle between the two vectors. Mathematically, it is expressed as:

A.B = |A| |B| cos(Θ)

where A and B are vectors, |A| and |B| are their magnitudes, and Θ is the angle between them. The scalar product has a number of useful properties, such as being distributive, associative, and commutative.

• • Commutativity: A.B = B.A
  • Distributivity: A.(B+C) = A.B + A.C
  • Associativity: (A.B).C = A.(B.C)
  • Positive definiteness: A.A >= 0, and A.A = 0 if and only if A = 0
  • Bilinearity: k(A.B) = (kA).B = A.(kB) where k is a scalar