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Parallel plane segments with the same orientation and area corresponding to the same bivector a ∧ b. [1] 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 ...
Given a bivector r = r 1 + hr 2, the ellipse for which r 1 and r 2 are a pair of conjugate semi-diameters is called the directional ellipse of the bivector r. [4]: 436 In the standard linear representation of biquaternions as 2 × 2 complex matrices acting on the complex plane with basis {1, h},
the bivector has two distinct principal null directions; in this case, the bivector is called non-null. Furthermore, for any non-null bivector, the two eigenvalues associated with the two distinct principal null directions have the same magnitude but opposite sign, λ = ±ν, so we have three subclasses of non-null bivectors: spacelike: ν = 0
Since the vector term of the vector bivector product the name dot product is zero when the vector is perpendicular to the plane (bivector), and this vector, bivector "dot product" selects only the components that are in the plane, so in analogy to the vector-vector dot product this name itself is justified by more than the fact this is the non ...
In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime.
A bivector is 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 oriented area , in the same way a vector is an oriented line segment.
A two-vector or bivector [1] is a tensor of type () and it is the dual of a two-form, meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars). The tensor product of a pair of vectors is a two-vector. Then, any two-form can be expressed as a linear combination of tensor products of pairs of ...
Any bivector field on a 2-dimensional manifold is automatically a Poisson structure; indeed, [,] is a 3-vector field, which is always zero in dimension 2. Given any Poisson bivector field π {\displaystyle \pi } on a 3-dimensional manifold M {\displaystyle M} , the bivector field f π {\displaystyle f\pi } , for any f ∈ C ∞ ( M ...