![]() The straightforward algorithm for calculating a floating-point dot product of vectors can suffer from catastrophic cancellation. In modern presentations of Euclidean geometry, the points of space are defined in terms of their Cartesian coordinates, and Euclidean space itself is commonly identified with the real coordinate space R n, see Tensor contraction for details. ![]() The equivalence of these two definitions relies on having a Cartesian coordinate system for Euclidean space. The geometric definition is based on the notions of angle and distance (magnitude) of vectors. The dot product may be defined algebraically or geometrically. " 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 with the vector product in three-dimensional space).The name "dot product" is derived from the centered dot " 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 between two vectors is the quotient of their dot product by the product of their lengths). In modern geometry, Euclidean spaces are often defined by using vector spaces. These definitions are equivalent when using Cartesian coordinates. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. It is often called the inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more).Īlgebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. 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. For the product of a vector and a scalar, see Scalar multiplication. For the abstract scalar product, see Inner product space.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |