Physical quantity that changes sign with improper rotation
A loop of wire (black), carrying a current I, creates a magnetic field B (blue). If the position and current of the wire are reflected across the plane indicated by the dashed line, the magnetic field it generates would not be reflected: Instead, it would be reflected and reversed. The position and current at any point in the wire are "true" vectors, but the magnetic field B is a pseudovector.[1]
In physics and mathematics, a pseudovector (or axial vector)[2] is a quantity that behaves like a vector in many situations, but its direction does not conform when the object is rigidly transformed by rotation, translation, reflection, etc. This can also happen when the orientation of the space is changed. For example, the angular momentum is a pseudovector because it is often described as a vector, but by just changing the position of reference (and changing the position vector), angular momentum can reverse direction, which is not supposed to happen with true vectors (also known as polar vectors).[3]
One example of a pseudovector is the normal to an oriented plane. An oriented plane can be defined by two non-parallel vectors, a and b,[4] that span the plane. The vector a × b is a normal to the plane (there are two normals, one on each side – the right-hand rule will determine which), and is a pseudovector. This has consequences in computer graphics, where it has to be considered when transforming surface normals.
In three dimensions, the curl of a polar vector field at a point and the cross product of two polar vectors are pseudovectors.[5]
A number of quantities in physics behave as pseudovectors rather than polar vectors, including magnetic field and angular velocity. In mathematics, in three dimensions, pseudovectors are equivalent to bivectors, from which the transformation rules of pseudovectors can be derived. More generally, in n-dimensional geometric algebra, pseudovectors are the elements of the algebra with dimension n − 1, written ⋀n−1Rn. The label "pseudo-" 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.
^Stephen A. Fulling; Michael N. Sinyakov; Sergei V. Tischchenko (2000). Linearity and the mathematics of several variables. World Scientific. p. 343. ISBN 981-02-4196-8.
^"Details for IEV number 102-03-33: "axial vector"". International Electrotechnical Vocabulary (in Japanese). Retrieved 2023-11-07.
^"Details for IEV number 102-03-34: "polar vector"". International Electrotechnical Vocabulary (in Japanese). Retrieved 2023-11-07.
^
RP Feynman: §52-5 Polar and axial vectors, Feynman Lectures in Physics, Vol. 1
^
Aleksandr Ivanovich Borisenko; Ivan Evgenʹevich Tarapov (1979). Vector and tensor analysis with applications (Reprint of 1968 Prentice-Hall ed.). Courier Dover. p. 125. ISBN 0-486-63833-2.
In physics and mathematics, a pseudovector (or axial vector) is a quantity that behaves like a vector in many situations, but its direction does not conform...
In high energy physics, a pseudovector meson or axial vector meson is a meson with total spin 1 and even parity (+) (usually noted as J P = 1+ ). Compare...
A pseudovector boson is a vector boson that has even parity, whereas "regular" vector bosons have odd parity. There are no fundamental pseudovector bosons...
lowercase Greek letter omega), also known as angular frequency vector, is a pseudovector representation of how the angular position or orientation of an object...
decreases counterclockwise. In three dimensions, angular acceleration is a pseudovector. For rigid bodies, angular acceleration must be caused by a net external...
not. A pseudoscalar, when multiplied by an ordinary vector, becomes a pseudovector (or axial vector); a similar construction creates the pseudotensor. A...
mesons contrast with the pseudovector mesons, which also have a total spin 1 but instead have even parity. The vector and pseudovector mesons are also dissimilar...
over a line. In more advanced treatments, one further distinguishes pseudovector fields and pseudoscalar fields, which are identical to vector fields...
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase...
In continuum mechanics, vorticity is a pseudovector (or axial vector) field that describes the local spinning motion of a continuum near some point (the...
quantity of magnetic moment per unit volume. It is represented by a pseudovector M. Magnetization can be compared to electric polarization, which is the...
cross product transforms as a pseudovector under parity transformations and so is properly described as a pseudovector. The dot product of two vectors...
angular momentum for a point particle is classically represented as a pseudovector r × p, the cross product of the particle's position vector r (relative...
transform in a similar way under changes of the coordinate system include pseudovectors and tensors. The vector concept, as it is known today, is the result...
around which it is being determined. In three dimensions, the torque is a pseudovector; for point particles, it is given by the cross product of the displacement...
momentum is the cross product of position x with momentum p to obtain a pseudovector x × p, or alternatively as the exterior product to obtain a second order...
vector to each point of space, called a vector field (more precisely, a pseudovector field). In electromagnetics, the term magnetic field is used for two...
waves). Angular frequency (or angular speed) is the magnitude of the pseudovector quantity angular velocity. Angular frequency can be obtained multiplying...
Due to the dependence on handedness, the cross product is said to be a pseudovector. In connection with the cross product, the exterior product of vectors...
Measure for the strength of the magnetic field tesla (T = Wb/m2) M T−2 I−1 pseudovector field Magnetic moment (or magnetic dipole moment) m The component of...
another. Note the cross product of two vectors is a pseudovector, while the cross product of a pseudovector with a vector is another vector. Other identities...
symmetry plane), it is important to distinguish between vectors and pseudovectors (as well as scalars and pseudoscalars, and in general between tensors...
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