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Right-hand rule. In mathematics and physics, the right-hand rule is a convention and a mnemonic, utilized to define the orientation of axes in three-dimensional space and to determine the direction of the cross product of two vectors, as well as to establish the direction of the force on a current-carrying conductor in a magnetic field.
The cross product a × b is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a direction given by the right-hand rule [1] and a magnitude equal to the area of the parallelogram that the vectors span. [2] The cross product is defined by the formula [8] [9]
Specifically, the divergence of a vector is a scalar. The divergence of a higher-order tensor field may be found by decomposing the tensor field into a sum of outer products and using the identity, where is the directional derivative in the direction of multiplied by its magnitude. Specifically, for the outer product of two vectors,
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.
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 entire body of physics that deals with (right-handed) pseudovectors and the right hand rule could be replaced by using (left-handed) pseudovectors and the left hand rule without issue.
The Frenet–Serret formulas were generalized to higher-dimensional Euclidean spaces by Camille Jordan in 1874. Suppose that r(s) is a smooth curve in and that the first n derivatives of r are linearly independent. [2] The vectors in the Frenet–Serret frame are an orthonormal basis constructed by applying the Gram-Schmidt process to the ...
Right-hand rule for a current-carrying wire in a magnetic field B. When a wire carrying an electric current is placed in a magnetic field, each of the moving charges, which comprise the current, experiences the Lorentz force, and together they can create a macroscopic force on the wire (sometimes called the Laplace force).
Here F is the force on the particle, q is the particle's electric charge, v, is the particle's velocity, and × denotes the cross product. The direction of force on the charge can be determined by a mnemonic known as the right-hand rule (see the figure).