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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 .
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.
In Cartesian coordinates, the divergence of a continuously differentiable vector field = + + is the scalar-valued function: = = (, , ) (, , ) = + +.. As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.
Vector algebra relations — regarding operations on individual vectors such as dot product, cross product, etc. Vector calculus identities — regarding operations on vector fields such as divergence, gradient, curl, etc.
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 ...
A cross-product function is not available, nor any function to extract individual elements from a vector, but these can be readily calculated by the user. Vectors can also be used to simply store up to three real numbers together, thereby increasing the calculator's storage capacity, though with more complexity and reduced speed.
The cross product occurs frequently in the study of rotation, where it is used to calculate torque and angular momentum. It can also be used to calculate the Lorentz force exerted on a charged particle moving in a magnetic field. The dot product is used to determine the work done by a constant force.
In three dimensions the cross product is invariant under the action of the rotation group, SO(3), so the cross product of x and y after they are rotated is the image of x × y under the rotation. But this invariance is not true in seven dimensions; that is, the cross product is not invariant under the group of rotations in seven dimensions, SO(7).