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. 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 ab 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 is antisymmetric, so ba is the negation of the bivector ab, producing the opposite orientation, and aa 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 ab 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. It does not have a definite location or position.

Read more about Bivector:  History, Formal Definition, Two Dimensions, Three Dimensions, Four Dimensions, Higher Dimensions, Projective Geometry

Other articles related to "bivector, bivectors":

Seven-dimensional Cross Product - Coordinate Expressions - Using Geometric Algebra
... The product starts with the exterior product, a bivector valued product of two vectors This is bilinear, alternate, has the desired magnitude, but is not vector valued ... vector, and so the cross product, comes from the product of this bivector with a trivector ... factor there is only one trivector, the pseudoscalar of the space, and a product of the above bivector and one of the two unit trivectors gives the vector result, the dual of the bivector ...
Geometric Algebra - Examples and Applications - Electrodynamics and Special Relativity
... In spacetime algebra the electromagnetic field tensor has a bivector representation where the imaginary unit is the volume element, and where Maxwell's ... rotation in Euclidean space, where is the bivector generated by the time and the space directions involved, whereas in the Euclidean case it is the bivector generated by the two space ...
Bivector - Projective Geometry - Tensors and Matrices
... As noted above a bivector can be written as a skew-symmetric matrix, which through the exponential map generates a rotation matrix that describes the same rotation as the ... But it is also used with other bivectors such as the angular velocity tensor and the electromagnetic tensor, respectively a 3×3 and 4×4 skew-symmetric matrix or tensor ... Real bivectors in Λ2ℝn are isomorphic to n×n skew-symmetric matrices, or alternately to antisymmetric tensors of order 2 on ℝn ...
Cross Product As An Exterior Product
... In exterior algebra the exterior product (or wedge product) of two vectors is a bivector ... A bivector is an oriented plane element, in much the same way that a vector is an oriented line element ... Given two vectors a and b, one can view the bivector a ∧ b as the oriented parallelogram spanned by a and b ...
Multivector - Examples - Bivectors
... A bivector is therefore an element of the antisymmetric tensor product of a tangent space with itself ... In geometric algebra, also, a bivector is a grade 2 element (a 2-vector) resulting from the wedge product of two vectors, and so it is geometrically an ... If a and b are two vectors, the bivector a ∧ b has a norm which is its area, given by a direction the plane where that area lies on, i.e ...