In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation (a transformation that can be expressed as an inversion followed by a proper rotation). Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In 3D Geometry, an improper rotation, also called rotoreflection or rotary reflection is depending on context a Linear transformation or The conceptual opposite of a pseudovector is a (true) vector or a polar vector.

A common way of constructing a pseudovector p is by taking the cross product of two vectors a and b:

p = a × b. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which

A simple example of an improper rotation in 3D (but not in 2D) is a coordinate inversion: x goes to −x, y to −y and z to −z. Under this transformation, a and b go to −a and −b (by the definition of a vector), but p clearly does not change. It follows that any improper rotation multiplies p by −1 compared to the rotation's effect on a true vector.

This concept can be further generalized to pseudoscalars and pseudotensors, both of which gain an extra sign flip under improper rotations compared to a true scalar or tensor. In Physics, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion such as Improper rotations In Physics and Mathematics, a pseudotensor is usually a quantity that transforms like a Tensor under a Proper rotation, but gains an additional In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually

Many occurrences of pseudovectors in mathematics and physics are more naturally analyzed as bivectors, following the calculus of differential forms; the double negation is natural for a bivector. In Differential geometry, a p -vector is the Tensor obtained by taking Linear combinations of the Wedge product of p In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is However, bivectors are "less intuitive" in some senses than ordinary vectors, and since in R³ every bivector a ʌ b has a unique dual vector a × b, it is this dual which is more often used. In Mathematics, the Hodge star operator or Hodge dual is a significant Linear map introduced in general by W

Physical examples

Physical examples of pseudovectors include the magnetic field, torque, vorticity, and the angular momentum. In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about Vorticity is a mathematical concept used in Fluid dynamics. It can be related to the amount of " circulation " or "rotation" (or more strictly the In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position

Often, the distinction between vectors and pseudovectors is overlooked, but it becomes important in understanding and exploiting the effect of symmetry on the solution to physical systems. Symmetry in physics refers to features of a Physical system that exhibit the property of Symmetry —that is under certain transformations, aspects of these For example, consider the case of an electrical current loop in the z=0 plane: this system is symmetric (invariant) under mirror reflections through the plane (an improper rotation), so the magnetic field should be unchanged by the reflection. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or But reflecting the actual magnetic field through that plane changes its sign—this contradiction is resolved by realizing that the mirror reflection of the field induces an extra sign flip because of its pseudovector nature.

As another example, consider the pseudovector angular momentum. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position Driving in a car, and looking forward, each of the wheels has an angular momentum vector pointing to the left. A wheel is a circular device that is capable of rotating on its axis facilitating movement or transportation whilst supporting a load ( Mass) or performing labour in machines If the world is reflected in a mirror which switches the left and right side of the car, the reflection of this angular momentum vector points to the right, but the actual angular momentum vector of the wheel still points to the left, corresponding to the minus sign.

To the extent that physical laws are the same for right-handed and left-handed coordinate systems (i. e. invariant under parity), the sum of a vector and a pseudovector is not meaningful. However, the weak force, which governs beta decay, does depend on the chirality of the universe, and in this case pseudovectors and vectors are added. The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four Fundamental interactions of nature In Nuclear physics, beta decay is a type of Radioactive decay in which a Beta particle (an Electron or a Positron) is emitted A phenomenon is said to be chiral if it is not identical to its Mirror image (see Chirality)

References

See also

• Grassmann algebra

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