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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 .
In physics, pseudovectors are generally the result of taking the cross product of two polar vectors or the curl of a polar vector field. The cross product and curl are defined, by convention, according to the right hand rule, but could have been just as easily defined in terms of a left-hand rule.
The direction of the cross product may be found by application of the right-hand rule as follows: The index finger points in the direction of the velocity vector v. The middle finger points in the direction of the magnetic field vector B. The thumb points in the direction of the cross product F. For example, for a positively charged particle ...
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 .
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).
In Poynting's original paper and in most textbooks, the Poynting vector is defined as the cross product [4] [5] [6] =, where bold letters represent vectors and E is the electric field vector; H is the magnetic field's auxiliary field vector or magnetizing field.
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.
A common example of a screw is the wrench associated with a force acting on a rigid body. Let P be the point of application of the force F and let P be the vector locating this point in a fixed frame. The wrench W = (F, P × F) is a screw.