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The cross product with respect to a right-handed coordinate system. In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol .
Crossover experiments allow for experimental study of a reaction mechanism. Mechanistic studies are of interest to theoretical and experimental chemists for a variety of reasons including prediction of stereochemical outcomes, optimization of reaction conditions for rate and selectivity, and design of improved catalysts for better turnover number, robustness, etc. [6] [7] Since a mechanism ...
where denotes the number of moles of the reactant or product and is the stoichiometric number [4] of the reactant or product. Although less common, we see from this expression that since the stoichiometric number can either be considered to be dimensionless or to have units of moles, conversely the extent of reaction can either be considered to ...
Also, the dot, cross, and dyadic products can all be expressed in matrix form. Dyadic expressions may closely resemble the matrix equivalents. The dot product of a dyadic with a vector gives another vector, and taking the dot product of this result gives a scalar derived from the dyadic.
Conversion and its related terms yield and selectivity are important terms in chemical reaction engineering.They are described as ratios of how much of a reactant has reacted (X — conversion, normally between zero and one), how much of a desired product was formed (Y — yield, normally also between zero and one) and how much desired product was formed in ratio to the undesired product(s) (S ...
When del operates on a scalar or vector, either a scalar or vector is returned. Because of the diversity of vector products (scalar, dot, cross) one application of del already gives rise to three major derivatives: the gradient (scalar product), divergence (dot product), and curl (cross product).
This also relates to the handedness of the cross product; the cross product transforms as a pseudovector under parity transformations and so is properly described as a pseudovector. The dot product of two vectors is a scalar but the dot product of a pseudovector and a vector is a pseudoscalar, so the scalar triple product (of vectors) must be ...
The cross product and the Lie bracket operation [,] both satisfy the Jacobi identity. In analytical mechanics , the Jacobi identity is satisfied by the Poisson brackets . In quantum mechanics , it is satisfied by operator commutators on a Hilbert space and equivalently in the phase space formulation of quantum mechanics by the Moyal bracket .