# Pseudovector

**axial vectorpolar vector(axial) vectoraxialpolarpseudo-vector**

In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but in three dimensions gains an additional sign flip under an improper rotation such as a reflection.wikipedia

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### Euclidean vector

**vectorvectorsvector addition**

In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but in three dimensions gains an additional sign flip under an improper rotation such as a reflection.

Other vector-like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors.

### Bivector

In mathematics, pseudovectors are equivalent to three-dimensional bivectors, from which the transformation rules of pseudovectors can be derived. On the other hand, the plane of the two vectors is represented by the exterior product or wedge product, denoted by a ∧ b. In this context of geometric algebra, this bivector is called a pseudovector, and is the Hodge dual of the cross product.

They are related to complex numbers in two dimensions and to both pseudovectors and quaternions in three dimensions.

### Angular velocity

**angular speedangular velocitiesOrders of magnitude (angular velocity)**

A number of quantities in physics behave as pseudovectors rather than polar vectors, including magnetic field and angular velocity. Physical examples of pseudovectors include torque, angular velocity, angular momentum, magnetic field, and magnetic dipole moment.

In three dimensions, angular velocity is a pseudovector, with its magnitude measuring the rate at which an object rotates or revolves, and its direction pointing perpendicular to the instantaneous plane of rotation or angular displacement.

### Rotation (mathematics)

**rotationrotationsrotate**

In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but in three dimensions gains an additional sign flip under an improper rotation such as a reflection.

### Pseudoscalar

**pseudo-scalar**

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.

Any scalar product between a pseudovector and an ordinary vector is a pseudoscalar.

### Angular momentum

**conservation of angular momentumangular momentamomentum**

Physical examples of pseudovectors include torque, angular velocity, angular momentum, magnetic field, and magnetic dipole moment.

In three dimensions, the angular momentum for a point particle is a pseudovector r × p, the cross product of the particle's position vector r (relative to some origin) and its momentum vector; the latter is p = mv in Newtonian mechanics.

### Torque

**moment armmomentlever arm**

Physical examples of pseudovectors include torque, angular velocity, angular momentum, magnetic field, and magnetic dipole moment.

In three dimensions, the torque is a pseudovector; for point particles, it is given by the cross product of the position vector (distance vector) and the force vector.

### Pseudotensor

**pseudo differential formpseudo-formpseudo-tensor**

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.

This is a generalization of a pseudovector.

### Normal (geometry)

**normalnormal vectorsurface normal**

This has consequences in computer graphics where it has to be considered when transforming surface normals.

If the normal is constructed as the cross product of tangent vectors (as described in the text above), it is a pseudovector.

### Improper rotation

**rotoreflectionproper rotationrotoinversion**

In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but in three dimensions gains an additional sign flip under an improper rotation such as a reflection.

When studying the symmetry of a physical system under an improper rotation (e.g., if a system has a mirror symmetry plane), it is important to distinguish between vectors and pseudovectors (as well as scalars and pseudoscalars, and in general between tensors and pseudotensors), since the latter transform differently under proper and improper rotations (in 3 dimensions, pseudovectors are invariant under inversion).

### Parity (physics)

**parityparity violationP-symmetry**

(See parity violation.)

However, angular momentum \vec{L} is an axial vector,

### Right-hand rule

**right hand ruleright-handedright hand grip rule**

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.

### Cross product

**vector cross productvector productcross-product**

In three dimensions, the pseudovector p is associated with the curl of a polar vector or with the cross product of two polar vectors a and b:

The product can be generalized in various ways; it can be made independent of orientation by changing the result to pseudovector, or in arbitrary dimensions the exterior product of vectors can be used with a bivector or two-form result.

### Weak interaction

**weak forceweakweak nuclear force**

In fact, this is exactly what happens in the weak interaction: Certain radioactive decays treat "left" and "right" differently, a phenomenon which can be traced to the summation of a polar vector with a pseudovector in the underlying theory.

In 1957, Robert Marshak and George Sudarshan and, somewhat later, Richard Feynman and Murray Gell-Mann proposed a V−A (vector minus axial vector or left-handed) Lagrangian for weak interactions.

### Blade (geometry)

**blade2-blades2-blade**

A k-fold wedge product also is referred to as a k-blade.

is called a pseudovector or an antivector.

### Hodge star operator

**Hodge dualHodge starcodifferential**

On the other hand, the plane of the two vectors is represented by the exterior product or wedge product, denoted by a ∧ b. In this context of geometric algebra, this bivector is called a pseudovector, and is the Hodge dual of the cross product.

Applied to three dimensions, the Hodge star provides an isomorphism between axial vectors and bivectors, so each axial vector a is associated with a bivector

### Multivector

**trivectorp-vectork''-vector**

For example, a multivector is a summation of k-fold wedge products of various k-values.

### Orientation (vector space)

**orientationorientedorientation-preserving**

### Physics

**physicistphysicalphysicists**

### Mathematics

**mathematicalmathmathematician**

### Reflection (mathematics)

**reflectionreflectionsmirror plane**

### Mirror image

**reflectionmirror imagesmirror-image**

Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image.

### Curl (mathematics)

**curlcurl operatorcross product of a curl**

In three dimensions, the pseudovector p is associated with the curl of a polar vector or with the cross product of two polar vectors a and b:

### Plane (geometry)

**planeplanarplanes**

One example is the normal to an oriented plane.

### Magnetic field

**magnetic fieldsmagneticmagnetic flux density**

A number of quantities in physics behave as pseudovectors rather than polar vectors, including magnetic field and angular velocity. Physical examples of pseudovectors include torque, angular velocity, angular momentum, magnetic field, and magnetic dipole moment.