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In mathematics, the dot product or scalar product [note 1] 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.
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
Specifically, for the outer product of two vectors, ... The divergence of a vector field A is a scalar, and the divergence of a scalar quantity is undefined.
Tensor product – for two vectors and , where and are vector spaces, their tensor product belongs to the tensor product of the vector spaces. Geometric product or Clifford product – for two vectors, the geometric product = + is a mixed quantity consisting of a scalar plus a bivector.
The dot product takes in two vectors and returns a scalar, while the cross product [a] returns a pseudovector. Both of these have various significant geometric interpretations and are widely used in mathematics, physics, and engineering. The dyadic product takes in two vectors and returns a second order tensor called a dyadic in this context. A ...
The scalar and vector part of this Hamilton product corresponds to the negative of dot product and cross product of the two vectors. In 1881, Josiah Willard Gibbs , [ 10 ] and independently Oliver Heaviside , introduced the notation for both the dot product and the cross product using a period ( a ⋅ b ) and an "×" ( a × b ), respectively ...
In common geometrical contexts, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction. Scalar multiplication is the multiplication of a vector by a scalar (where the product is a vector), and is to be distinguished from inner product of two vectors (where ...
A scalar beside a vector (either or both of which may be in parentheses) implies scalar multiplication. The two common operators, a dot and a rotated cross, are also acceptable (although the rotated cross is almost never used), but they risk confusion with dot products and cross products, which operate on two vectors. The product of a scalar k ...