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The commutative property can also be easily proven with the algebraic ... This property is sometimes called the "associative law for scalar and dot product", [8] ...
The dyadic product is also associative with the dot and cross products with other vectors, which allows the dot, cross, and dyadic products to be combined to obtain other scalars, vectors, or dyadics.
Frobenius inner product, the dot product of matrices considered as vectors, or, equivalently the sum of the entries of the Hadamard product; Hadamard product of two matrices of the same size, resulting in a matrix of the same size, which is the product entry-by-entry; Kronecker product or tensor product, the generalization to any size of the ...
The generalization of the dot product formula to Riemannian manifolds is a defining property of a Riemannian connection, which differentiates a vector field to give a vector-valued 1-form. Cross product rule
The scalar triple product (also called the mixed product, box product, or triple scalar product) is defined as the dot product of one of the vectors with the cross product of the other two. Geometric interpretation
The associative property is closely related to the commutative property. The associative property of an expression containing two or more occurrences of the same operator states that the order operations are performed in does not affect the final result, as long as the order of terms does not change.
The space together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket. Like the dot product, it depends on the metric of Euclidean space , but unlike the dot product, it also depends on a choice of orientation (or ...
Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or scalar product of Cartesian coordinates. Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces.