Cross products and determinants in $\mathbb{R}^3$

Question

I know that the absolute value of determinant of three vectors in $\mathbb{R}^3$ is the volume of the parallelepiped determined by the three vectors. The volume can be computed by basic calculation involving cross products, etc. Cross products are definitely related to determinants. I was just wondering if the origin of cross products was from determinants or vies versa or that they were invented independently but coincidentally happened to have a relation. I would appreciate any info!

Answer 1

The more analytical definition of the cross product that I know is: given ${\bf u},{\bf v}\in \Bbb R^3$, the cross product of ${\bf u}$ and $\bf v$, written ${\bf u}\times {\bf v}$, is the (only) vector satisfying: $$\det({\bf u},{\bf v},{\bf w}) = \langle {\bf u}\times {\bf v},{\bf w}\rangle, \quad \forall\,{\bf w}\in \Bbb R^3,$$ where $\det({\bf u},{\bf v},{\bf w})$ is the determinant of the matrix obtained by putting the components of the vectors in columns. The relation must hold for all $\bf w$, so if we write ${\bf u}\times {\bf v} = (x_1,x_2,x_3)$ and make ${\bf w} = {\bf e}_i$, $i = 1,2,3$, we get: $$x_i = \langle {\bf u}\times{\bf v},{\bf e}_i\rangle = \det({\bf u},{\bf v},{\bf e}_i),$$ and with this you can deduce the usual expression that you know, with the formal determinant. This definition is made possible by the Riesz Representation Theorem: the vector ${\bf u}\times {\bf v}$ corresponds to the linear functional $\det({\bf u},{\bf v}, \cdot)$. This definition also makes obvious the fact that $({\bf u}\times {\bf v}) \perp {\bf u}$ and $({\bf u}\times{\bf v})\perp {\bf v}$. Also, note that if we take another inner product and use this definition, we will get another cross-product. We can have several cross products, and all of them come of the deteminant.

Answer 2

The idea of the determinant definitely came first. In fact, the concept of the determinant predates the application of actually using matrices themselves. You can check out the origins of the word determinant and you will see in fact it stems from the idea that it was used originally to find if a linear system had solutions. The first recorded instance goes all the way back to the third century, BC, in the Chinese manuscript titled "The Nine Chapters on the Mathematical Art" .

Ofcourse the years following there was much more development and new advances, most notably, Lebinez used determinants of smaller dimensions (such as 2x2) for solving general problems (17th century).

Lot of work on minors and theory by Gauss and Lagrange, followed by Jacobi and later by Cayley and wronskian.

I believe the cross product was first used in the late nineteenth century, as well as the dot product. Both were introduced by Josia Gibbs and and Oliver Heavside.

Answer 3

They were invented independently.

The determinant was first made to analyze how many solutions a system of equations could have as well as for Cramer's Rule (see http://www-history.mcs.st-and.ac.uk/HistTopics/Matrices_and_determinants.html , for example).

The cross product arose from Hamilton's extension of the complex numbers, the "quaternions" (see http://www.math.mcgill.ca/labute/courses/133f03/VectorHistory.html , for example).