There are two distinct types of products of two vectors: scalar and vector products, sometimes called the inner and outer products (mostly in reference to tensor products; see below). The scalar (or dot) product of two vectors is not a vector because the product has a magnitude but not a direction. For example, if A and B are vectors (of magnitude A and B, respectively), their scalar product is: A • B = AB cos θ, in which θ is the angle between the two vectors. This scalar quantity is also called the dot product of the vectors. These equations obey the commutative and distributive laws of algebra (for more information, see “Algebra”). Thus, A • B = B • A; A • (B + C) = A • B + A • C. If A is perpendicular to B, then A • B = 0.

The vector (or cross or skew) product of A and B is the length C = AB sin θ; its direction is perpendicular to the plane determined by A and B. In this case, this kind of multiplication does not follow the commutative law, as A • B = -B • A.