What are the properties of a dot product?

In mathematics, the dot product is an algebraic operation that takes two equal-length sequences of numbers (usually vectors) and returns a single number obtained by multiplying corresponding entries and adding up those products. The name is derived from the interpunct "●" that is often used to designate this operation; the alternative name scalar product emphasizes the scalar result, rather than a vector result.

The principal use of this product is the inner product in a Euclidean vector space: when two vectors are expressed in an Orthonormal basis, the dot product of their coordinate vectors gives their inner product. For this geometric interpretation, scalars must be taken to be Real. The dot product can be defined in a more general field, for instance the complex number field, but many properties would be different. In three dimensional space, the dot product contrasts with the cross product, which produces a vector as result.