Trivector

An ordered family $[ \mathbf u , \mathbf v , \mathbf w ]$ of three vectors $\mathbf u , \mathbf v , \mathbf w$ in an affine space $A$, emanating from a common origin. A trivector is set equal to zero if the vectors defining it are coplanar (linearly dependent). A non-zero trivector determines the three-dimensional space supporting it. If $A$ has finite dimension $n$ and if in some basis $e = ( \mathbf e _ {1} \dots \mathbf e _ {n} )$ the vectors are

$$\mathbf u = \sum _ {i = 1 } ^ { n } u ^ {i} \mathbf e _ {i} ,\ \ \mathbf v = \sum _ {i = 1 } ^ { n } v ^ {i} \mathbf e _ {i} ,\ \ \mathbf w = \sum _ {i = 1 } ^ { n } w ^ {i} \mathbf e _ {i} ,$$

then the quantities

$$a ^ {ijk} = \ \left | \begin{array}{ccc} u ^ {i} &v ^ {i} &w ^ {i} \\ u ^ {j} &v ^ {j} &w ^ {j} \\ u ^ {k} &v ^ {k} &w ^ {k} \\ \end{array} \ \right | = 3! u ^ {[} i v ^ {j} w ^ {k]} ,\ \ 1 \leq i, j, k \leq n,$$

are called the coordinates of the trivector $[ \mathbf u , \mathbf v , \mathbf w ]$ with respect to the basis $e$. These coordinates are skew-symmetric with respect to any pair of indices; under a change of basis in $A$, they transform like coordinates of a triply-contravariant tensor. Of these coordinates, $n ( n - 1) ( n - 2)/6$ are essential. Two trivectors are said to be equal if their coordinates are equal in any basis of $A$. A class of equal trivectors is called a free trivector.

In the presence of a scalar product in $A$, several metric concepts of vector algebra can be applied to trivectors. The measure of a trivector $[ \mathbf u , \mathbf v , \mathbf w ]$ is the three-dimensional volume of the parallelopipedon formed by the set of ends of vectors of the form $\mathbf h = x \mathbf u + y \mathbf v + z \mathbf w$, where $0 \leq x, y, z \leq 1$, emanating from a common origin. In the case when $\mathop{\rm dim} A = 3$, the measure of the trivector is equal to the scalar triple product of $\mathbf u , \mathbf v , \mathbf w$. The scalar product of two trivectors is the number equal to the product of the measures of the factors by the cosine of the angle between the planes supporting them. The scalar product is a bilinear form of the coordinates of the factors. If $\mathop{\rm dim} A = 4$, then the trivector $[ \mathbf u , \mathbf v , \mathbf w ]$ can be identified with a vector of $A$, called the vector product of $\mathbf u , \mathbf v , \mathbf w$.

A trivector in tensor calculus is any contravariant skew-symmetric tensor of valency 3 (that is, a tensor of type $( 3, 0)$). Each such tensor can be represented as the sum of several tensors to which correspond trivectors with distinct supporting planes.