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Vector algebra relations
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The relations below apply to vectors in a three-dimensional Euclidean space. Some, but not all of them, extend to vectors of higher dimensions. In particularQuaternionic polytope (238 words) [view diff] exact match in snippet view article find links to article

quaternions are non-commutative, a convention must be made for the multiplication of vectors by scalars, which is usually in favour of left-multiplicationSubalgebra (479 words) [view diff] exact match in snippet view article find links to article

ring or field is a vector subspace which is closed under the multiplication of vectors. The restriction of the algebra multiplication makes it an algebraPolar sine (1,036 words) [view diff] exact match in snippet view article find links to article

_{n}\end{bmatrix}}\\&=-\Omega \end{aligned}}} Invariance under scalar multiplication of vectors The polar sine does not change if all of the vectors v1, ..., vnComplex spacetime (929 words) [view diff] exact match in snippet view article find links to article

field). To "complexify" a space means extending ordinary scalar multiplication of vectors by real numbers to scalar multiplication by complex numbers. ForVectors in three-dimensional space (278 words) [view diff] exact match in snippet view article find links to article

displacements including these sections: Introduction, Scalar multiplication of vectors, Addition and subtraction of vectors, Displacements in EuclideanCommutative property (2,444 words) [view diff] exact match in snippet view article find links to article

addition and multiplication of complex numbers, addition and scalar multiplication of vectors, and intersection and union of sets. Concatenation, the act ofLinear algebra (8,005 words) [view diff] exact match in snippet view article find links to article

coordinate n-space Fn. Under this identification, addition and scalar multiplication of vectors in V correspond to addition and scalar multiplication of theirRing (mathematics) (13,594 words) [view diff] exact match in snippet view article

concept of a vector space (over a field) by generalizing from multiplication of vectors with elements of a field (scalar multiplication) to multiplicationClassical Hamiltonian quaternions (5,235 words) [view diff] exact match in snippet view article find links to article

the numerator times the reciprocal of the denominator. Since multiplication of vectors is not commutative, the order cannot be changed in the following