Bivector

See also: Exterior algebra

Various identically oriented and equal area plane segments corresponding to the same bivector a$\wedge$b

A bivector is a directed number that characterizes a directed plane segment, analogous to the way a vector is related to a directed line segment. The bivector has magnitude, direction and orientation, as does a vector, but the notion of direction is somewhat different, corresponding to the direction of an oriented flat plane. The bivector does not describe a unique set of points in that plane, nor where in the plane the points are located. As described by Dorst et al.: “The algebraic bivector is not specific on shape; geometrically it is an amount of oriented area in a specific plane, that's all.” Such a plane segment can be generated by by sweeping a point along one vector a and then sweeping this vector along another b to form a parallelogram. The orientation changes sign depending upon whether it is formed by first sweeping along a and then b, or the reverse sequence. Mathematically the corresponding bivector B is denoted by:

${\boldsymbol {B}}={\boldsymbol {a\wedge b}}\ ,$

and is called the wedge product, outer product or Grassman product.

Properties

See also: Wedge product § Formal definitions and algebraic properties

The bivector satisfies the relations:

${\boldsymbol {B}}={\boldsymbol {b\wedge a=a\wedge (-b)=(-b)\wedge (-a)=(-a)\wedge b}}.\,$

The magnitude of the bivector is the area of the corresponding parallelogram:

$|{\boldsymbol {B}}|=|{\boldsymbol {a}}||{\boldsymbol {b}}|\ \sin \theta ,\,$

where θ is the angle between the vectors. This result shows that for non-zero vectors B is zero if and only if a and b are collinear.

Bivectors can be multiplied by scalars:

$(\alpha {\boldsymbol {b}})\wedge {\boldsymbol {a}}={\boldsymbol {b}}\wedge (\alpha {\boldsymbol {a}})=\alpha {\boldsymbol {B}}$

So if

${\boldsymbol {C}}=\alpha {\boldsymbol {B}},\,$

then

$|{\boldsymbol {C}}|=|\alpha ||{\boldsymbol {B}}|,\,$

and if α is negative, the orientation of C is reversed from that of B.

The wedge product is left and right distributive over addition:

${\boldsymbol {a\wedge (b+c)=a\wedge b+a\wedge c}}.\,$

${\boldsymbol {(b+c)\wedge a=b\wedge a+c\wedge a}}.\,$

Vector cross product

In three dimensions the vector cross product of a and b can be written in terms of the the bivector a $\wedge$ b as:

${\boldsymbol {a\times b}}=-i\ {\boldsymbol {a\wedge b}},\,$

where i is the three-dimensional, unit pseudoscalar. The unit pseudoscalar commutes with all vectors in the space and, like the imaginary i = √−1, satisfies:

$i^{2}=-1.\,$

The cross product a × b is dual to the bivector a $\wedge$ b, and the vectors a, b, a × b form a right-handed set. So, in the sense of a dual, in three dimensions the cross product is a vector representation of the bivector related to it in the above way. The cross-product vector form is usually preferred over the bivector form because it is more familiar.

Geometric product

In three dimensional Euclidean vector space, an orthonormal basis can be introduced, e₁, e₂, e₃ satisfying:

${\boldsymbol {e}}_{i}\cdot {\boldsymbol {e}}_{j}=\delta _{ij},\,$

where '•' denotes the standard vector dot product and δ_ij is the Kronecker delta. A new element denoted by the juxtaposition of two vectors is introduced, the geometric product of the basis vectors:

${\begin{aligned}{\boldsymbol {e}}_{i}{\boldsymbol {e}}_{j}&={\boldsymbol {e}}_{i}\cdot {\boldsymbol {e}}_{i}{\text{ when }}i=j,\\&={\boldsymbol {e}}_{ij},{\text{ when }}i\neq j.\end{aligned}}$

Multiple products can be reduced by pairing:

${\boldsymbol {(e_{i}e_{j}e_{k})e_{k}=(e_{i}e_{j})(e_{k}e_{k})=e_{i}e_{j}}}.\,$

In addition:

${\boldsymbol {e_{i}e_{j}}}=-{\boldsymbol {e_{j}e_{i}}},{\text{ if }}i\neq j.$

These formalities allow the bivector and the cross product to be compared. One finds:

${\boldsymbol {a\wedge b}}=(a^{2}b^{3}-a^{3}b^{2}){\boldsymbol {e_{23}}}+(a^{3}b^{1}-a^{1}b^{3}){\boldsymbol {e_{31}}}+(a^{1}b^{2}-a^{2}b^{1}){\boldsymbol {e_{12}}},\,$

where superscripts indicate the components of the vectors, while:

${\boldsymbol {a\times b}}=(a^{2}b^{3}-a^{3}b^{2}){\boldsymbol {e}}_{1}+(a^{3}b^{1}-a^{1}b^{3}){\boldsymbol {e}}_{2}+(a^{1}b^{2}-a^{2}b^{1}){\boldsymbol {e}}_{3},\,$

which are related by introducing the notion of the dual of e₁ as e₂₃ = e₂e₃, and so forth. That is, the dual of e₁ is the subspace perpendicular to e₁, namely the subspace spanned by e₂ and e₃. The cross product of a and b is thus the dual to the wedge product of these two vectors. An intuitive view of this connection is provided by Cook.

The advantage of the wedge product is that it generalizes to dimensions other than three, while the cross product does not.

Bivectors and rotations

The geometric product of two orthonormal basis vectors e₁ and e₂, say, denoted by e₁e₂ = e₁₂ is a bivector because the dot-product of two orthogonal vectors is zero. The fact is used to show that rotations in a vector space can be represented by bivectors. In three dimensions, applying the unit bivector e₁e₂ to the basis vectors e₁ and e₂ by multiplying from the right:

${\boldsymbol {e_{1}\ (e_{1}e_{2})=e_{2}}}\ ,$

${\boldsymbol {e_{2}\ (e_{1}e_{2})=-e_{1}}}\ ,$

indicating a rotation through a right angle in the e₁e₂ plane. Likewise, any linear combination of e₁ and e₂ is rotated the same way.

Rotations in n-dimensions also are discussed by Baylis.

Tensors

The field of geometric algebra is closely related to tensor analysis, being a subalgebra of general tensor algebra. In tensor analysis the bivector is discussed as a particular second rank tensor.

Complex numbers

If the square of the bivector is evaluated, for example:

$({\boldsymbol {e_{12})^{2}=-({\boldsymbol {e_{1}e_{2}}})({\boldsymbol {e_{2}e_{1}}})=-{\boldsymbol {e_{1}}}({\boldsymbol {e_{2}e_{2}}}){\boldsymbol {e_{1}}}=-({\boldsymbol {e}}_{1}e_{1}}})=-1,\,$

the connection to the imaginary i = √−1 is established. Using this result, which applies to numerous such combinations, it can be shown that an entity like w= a + b e₁₂ behaves in the same way as complex numbers w = a + ib.

Historical remarks

J.W. Gibbs regarded what are now called bivectors as vectors, describing them, for example, as "areas when looked upon as vectors". He also introduced the notions of direct (now dot) products and skew (now cross) products. Confusingly, Gibbs used the name "bivector" to refer to "imaginary vectors", that is, vectors with real and imaginary parts, such as a + i b, which he associated with the axes of ellipses. This usage of the term "bivector" persists in some fields, and this usage sometimes is qualified as the "Gibbs bivector".

In the late 19th century a debate arose over the utility of vector algebra versus the algebra of quaternions. Today these two are merged in geometric algebra using the distinction between vectors and bivectors. In this connection, the common distinction between polar vectors (normal vectors) and axial vectors like a × b is nothing more than the distinction between vectors and bivectors.

General references

• Leo Dorst, Daniel Fontijne, Stephen Mann (2009). "§ 2.3.3 Visualizing bivectors". Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry (2nd ed.). Morgan Kaufmann. p. 31 ff. ISBN 0123749425.{{cite book}}: CS1 maint: multiple names: authors list (link)

• Lounesto, Pertti (2001). "§ 1.8 The Clifford product of vectors; the bivector". Clifford algebras and spinors. Cambridge: Cambridge University Press. p. 8 ff. ISBN 978-0-521-00551-7. {{cite book}}: More than one of |pages= and |page= specified (help)

• Chris Doran and Anthony Lasenby (2003). "§ 1.6 The outer product". Geometric Algebra for Physicists. Cambridge: Cambridge University Press. p. 11 et seq. ISBN 978-0-521-71595-9.

See also

• p-vector
• Multivector
• Clifford algebra
• Exterior algebra
• Vector algebra