What is "bivector"

Bivector \Bi*vec"tor\, n. [Pref. bi- + vector.] (Math.) A term made up of the two parts ? + ?1 ?-1, where ? and ?1 are vectors.

n. (context mathematics English) An antisymmetric tensor of second rank

In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered an order zero quantity, and a vector is an order one quantity, then a bivector can be thought of as being of order two. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and quaternions in three dimensions. They can be used to generate rotations in any dimension, and are a useful tool for classifying such rotations. They also are used in physics, tying together a number of otherwise unrelated quantities.

Bivectors are generated by the exterior product on vectors: given two vectors a and b, their exterior product is a bivector, as is the sum of any bivectors. Not all bivectors can be generated as a single exterior product. More precisely, a bivector that can be expressed as an exterior product is called simple; in up to three dimensions all bivectors are simple, but in higher dimensions this is not the case. The exterior product of two vectors is antisymmetric, so is the negation of the bivector , producing the opposite orientation, and is the zero bivector.

Geometrically, a simple bivector can be interpreted as an oriented plane segment, much as vectors can be thought of as directed line segments. The bivector has a magnitude equal to the area of the parallelogram with edges a and b, has the attitude of the plane spanned by a and b, and has orientation being the sense of the rotation that would align a with b.

In mathematics, a bivector is the vector part of a biquaternion. For biquaternion q = w + x i + y j + zk, w is called the biscalar and xi + yj + zk is its bivector part. The coordinates w, x, y, z are complex numbers with imaginary unit h:

x = x + hx,  y = y + hy,  z = z + hz,  h =  − 1 = i = j = k.
A bivector may be written as the sum of real and imaginary parts:

(xi + yj + zk) + h(xi + yj + zk)
where r = xi + yj + zk,  r = xi + yj + zk) are vectors. Thus the bivector q = xi + yj + zk = r + hr.

The Lie algebra of the Lorentz group is expressed by bivectors. In particular, if r and r are right versors so that r =  − 1 = r, then the biquaternion curve {exp θ r : θ ∈ R} traces over and over the unit circle in the plane {x + y r : x,y ∈ R}. Such a circle corresponds to the space rotation parameters of the Lorentz group.

Now (h r) = (−1)(−1) = +1, and the biquaternion curve {exp(θ(hr)) : θ ∈ R} is a unit hyperbola in the plane {x + y r : x,y ∈ R}. The spacetime transformations in the Lorentz group that lead to FitzGerald contractions and time dilation depend on a hyperbolic angle parameter. In the words of Ronald Shaw, "Bivectors are logarithms of Lorentz transformations"

The commutator product of this Lie algebra is just twice the cross product on R, for instance, [i,j] = ij −ji = 2k which is twice i × j. As Shaw wrote in 1970:

Now it is well known that the Lie algebra of the homogeneous Lorentz group can be considered to be that of bivectors under commutation. ... The Lie algebra of bivectors is essentially that of complex 3-vectors, with the Lie product being defined to be the familiar cross product in (complex) 3-dimensional space.

William Rowan Hamilton coined both the terms vector and bivector. The first term was named with quaternions, and the second about a decade later, as in Lectures on Quaternions (1853). The popular text Vector Analysis (1901) used the term.

Given a bivector r = r + h r, the ellipse for which r and r are a pair of conjugate semi-diameters is called the directional ellipse of the bivector r.

In the standard linear representation of biquaternions as 2 x 2 complex matrices acting on the complex plane with basis {1, h},

$$\begin{pmatrix}hv & w+hx\\-w+hx & -hv\end{pmatrix}$$
represents bivector q = v i + w j + x k. The conjugate transpose of this matrix corresponds to −q, so the representation of bivector q is a skew-Hermitian matrix.

Ludwik Silberstein studied a complexified electromagnetic field E + h B, where there are three components, each a complex number, known as the Riemann-Silberstein vector.

"Bivectors ... help describe elliptically polarized homogeneous and inhomogeneous plane waves — one vector for direction of propagation, one for amplitude."