P-vector
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A p-vector in differential geometry is the tensor obtained by taking linear combinations of the wedge product of p tangent vectors, for some integer p ≥ 1. It is the dual concept to a p-form.
For p = 2 and 3, these are often called respectively bivectors and trivectors; they are dual to 2-forms and 3-forms.
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 (see exterior algebra) of two vectors, and so it is geometrically an oriented area, in the same way a vector is an oriented line segment. If a and b are two vectors, the bivector <math>\mathbf a \wedge \mathbf b<math> has
- a norm which is its area, given by
- <math>\Vert \mathbf a \wedge \mathbf b \Vert = \Vert \mathbf{a} \Vert \,
\Vert \mathbf{b} \Vert \, \sin(\phi_{a,b})<math>
- a direction: the plane where that area lies on, i.e., the plane determined by a and b, as long as they are linearly independent;
- an orientation (out of two), determined by the order in which the originating vectors are multiplied.
Bivectors are connected to polar vectors, and are used to represent rotations in geometric algebra.
(Alternatively, four-vector is used in relativity to mean a quantity related to the four-dimensional spacetime. In analogy, the term three-vector is sometimes used as a synonym for a spatial vector in three dimensions. These meanings are different from p-vectors for p equal to 3 or 4.)