Bivector

In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. Considering a scalar as a degree-zero quantity and a vector as a degree-one quantity, a bivector is of degree 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 vector quaternions in three dimensions. They can be used to generate rotations in a space of any number of dimensions, and are a useful tool for classifying such rotations.

Geometrically, a simple bivector can be interpreted as characterizing an directed plane segment, much as vectors can be thought of as characterizing directed line segments.^[2] The bivector $a \land b$ has an attitude (direction) of the plane spanned by $a$ and $b$ , has an area that is a scalar multiple of any reference plane segment with the same attitude (and in geometric algebra, it has a magnitude equal to the area of the parallelogram with edges $a$ and $b$ ), and has an orientation being the side of $a$ on which $b$ lies within the plane spanned by $a$ and $b$ .^[2]^[3] In layman terms, any surface defines the same bivector if it is parallel to the same plane (same attitude), has the same area, and same orientation (see figure).

Bivectors are generated by the exterior product on vectors: given two vectors $a$ and $b$ , their exterior product $a \land b$ is a bivector, as is any sum of bivectors. Not all bivectors can be expressed as an exterior product without such summation. 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.^[4] The exterior product of two vectors is alternating, so $a \land a$ is the zero bivector, and $b \land a$ is the negative of the bivector $a \land b$ , producing the opposite orientation. Concepts directly related to bivector are rank-2 antisymmetric tensor and skew-symmetric matrix.

^ Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2009). Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry (2nd ed.). Morgan Kaufmann. p. 32. ISBN 978-0-12-374942-0. The algebraic bivector is not specific on shape; geometrically it is an amount of directed area in a specific plane, that's all.
^ ^a ^b Hestenes, David (1999). New foundations for classical mechanics: Fundamental Theories of Physics (2nd ed.). Springer. p. 21. ISBN 978-0-7923-5302-7.
^ Lounesto 2001, p. 33
^ Cite error: The named reference Lounesto 2001 p. 87 was invoked but never defined (see the help page).

[Dorst-1] Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2009). Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry (2nd ed.). Morgan Kaufmann. p. 32. ISBN 978-0-12-374942-0. The algebraic bivector is not specific on shape; geometrically it is an amount of directed area in a specific plane, that's all.

[Hestenes-2] Hestenes, David (1999). New foundations for classical mechanics: Fundamental Theories of Physics (2nd ed.). Springer. p. 21. ISBN 978-0-7923-5302-7.

[3] Lounesto 2001, p. 33

[Lounesto_2001_p._87-4] Cite error: The named reference Lounesto 2001 p. 87 was invoked but never defined (see the help page).

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