6vector

A 6vector, or six-vector, is a mathematical object consisting of six components. As an element of a six‑dimensional vector space such as R^6 or C^6, it is the direct generalization of ordinary vectors to six dimensions. In practice, a 6vector may represent two three‑component triples that are treated together under a symmetry or transformation, but at its core it is simply a six‑component column or row of numbers.

In linear algebra, a 6vector is simply a vector with six entries. Arithmetic operations such as addition and scalar multiplication are defined componentwise, and a dot product can be defined using a metric to yield notions of length and angle. A standard Euclidean 6‑space uses the usual dot product, summing the squares of the six components.

In physics, the term is used in a few contexts. A common application is the six‑component representation of the electromagnetic field, where the electric field E and magnetic field B are arranged into a six‑component object, for example as (E1, E2, E3, B1, B2, B3)^T. This six‑vector can simplify certain formal manipulations, though the full Lorentz symmetry is more precisely encoded in the antisymmetric field strength tensor Fμν, which has six independent components.

In robotics and rigid‑body mechanics, a 6vector frequently denotes a twist, an element of the Lie algebra se(3) representing instantaneous motion. This twist combines angular velocity ω and linear velocity v into a six‑component vector, often written as (v, ω) or (ω, v). The 6vector is central to screw theory and the mathematical description of rigid‑body motions via exponential coordinates.

The precise definition and ordering of the six components vary by domain, but the core idea is a vector with six components used to encode two coupled three‑component quantities or a six‑dimensional state.