I have written a paper which involves a "cross product" in $\mathbb{R}^n$ and I would like to have a reference to point to.

Let ${\bf e_1}, \dots, {\bf e_n}$ be the standard basis for $\mathbb{R}^n$ and let ${\bf w_1} = (w_{11},\dots,w_{1n}), \dots, {\bf w_{n-1}}=(w_{n-1\;1},\dots,w_{n-1\;n}) \in \mathbb{R}^n$. Then one can define

$${\bf w_1} \times {\bf w_2} \times \cdots \times {\bf w_{n-1}} = \begin{vmatrix} {\bf e_1} & {\bf e_2} & \cdots & {\bf e_n} \cr w_{11} & w_{12} & \cdots & w_{1n} \cr \vdots & \vdots & & \vdots \cr w_{n-1\;1} & w_{n-1\;2} & \cdots & w_{n-1\;n} \end{vmatrix}$$

where the right hand side is a "determinant".

Note: One can express this "cross product" in terms of exterior algebra operations. It is equivalent to $*({\bf w_1} \wedge {\bf w_2} \wedge \cdots \wedge {\bf w_{n-1}})$ where "$*$" is the Hodge dual operator.

Obviously if ${\bf a}, {\bf b} \in \mathbb{R}^3$, then ${\bf a} \times {\bf b}$ is the regular cross product and this $(n-1)$-airy product has the same properties as the regular cross product (it is a vector perpendicular to the vectors being multiplied and the length of this vector is given by the $(n-1)$-volume of the parallelotope spanned by the ${\bf w}$'s).

It was pointed out that this product appears in Susan Colley's "Vector Calculus" text [I have a second edition where this product is explored in problems 29-31 in section 1.6 on "$n$-dimensional geometry"]. Colley said she didn't have a good reference to point to (she couldn't recall where she'd seen it before).

I guess I could just refer to her book, but was wondering if anyone knows a better/historical reference? Or if not a paper does anyone know who I should attribute this to?

Alternatively, it would be nice to know if there is no "original" reference to point to and that this is just common knowledge/folklore.

Thank you!

binaryproduct (which only works in 1, 3, and 7 dimensions (a fact related to the existence of division algebras in 2, 4, and 8 dimensions). As for Paul's comment, what I have above really isn't the volume form (the standard volume form on $\mathbb{R}^n$isthe determinant) but instead this is somethingbuiltfrom the volume form. As far as I can tell, Eckmann's work is again about the binary products in 1, 3, and 7 dimensions. $\endgroup$2more comments