## Mathematical Analysis## Vector and Other Analyses |

## How is the product of two vectors determined? |

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**.