<P> In mathematics, the dot product or scalar product is an algebraic operation that takes two equal - length sequences of numbers (usually coordinate vectors) and returns a single number . In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used and often called inner product (or rarely projection product); see also inner product space . </P> <P> Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers . Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them . These definitions are equivalent when using Cartesian coordinates . In modern geometry, Euclidean spaces are often defined by using vector spaces . In this case, the dot product is used for defining lengths (the length of a vector is the square root of the dot product of the vector by itself) and angles (the cosine of the angle of two vectors is the quotient of their dot product by the product of their lengths). </P> <P> The name "dot product" is derived from the centered dot "" that is often used to designate this operation; the alternative name "scalar product" emphasizes that the result is a scalar, rather than a vector, as is the case for the vector product in three - dimensional space . </P> <P> The dot product may be defined algebraically or geometrically . The geometric definition is based on the notions of angle and distance (magnitude of vectors). The equivalence of these two definitions relies on having a Cartesian coordinate system for Euclidean space . </P>

The dot product of two vectors results in a