Generalised Scalar and Vector Products in $\mathbb{A}^3$

Anyone who wanders in here knows that if you have two vectors $u$ and $v$ in $\mathbb{V}^3$ (the associated vector space of $\mathbb{A}^3$) the Euclidean scalar product $u \cdot v$ and the Euclidean vector product $u \times v$ are defined as usual and satisfies some properties. A big problem with some of these properties is that taking "lengths" of vectors is highly problematic; square roots are very hard to compute, let alone non-existent in finite fields. Moreover, "sines" and "cosines" of "angles" (whatever those terms mean) are not well-defined and require the use of techniques from integral calculus & (infinite) power series (specifically those of the Maclaurin flavour).

Let us now strip everything back and generalise our definitions of scalar and vector products so that we can apply what we have for a general field (not of characteristic $2$) and for arbitrary geometries.

Starting with $\mathbb{V}^3$, the associated vector space of $\mathbb{A}^3$, we equip a symmetric bilinear form on it and represent it by the matrix $B$, so we can define such a symmetric bilinear form by

$u \cdot_B v \equiv uBv^T.$

This gives us a generalised scalar product, which we will call the $B$-scalar product. The Euclidean scalar product corresponds to the case when $B$ is the $3 \times 3$ identity matrix.

We can also define a generalised vector product on $\mathbb{V}^3$ by

$u \times_B v \equiv (u \times v) \text{adj}(B).$

Here, we use $u \times v$ to denote the Euclidean vector product. The generalised operation is called a $B$-vector product, and from here we can modify our usual results of vector products with respect to the matrix representation $B$.

To end this note, we see that by stripping back all geometrical meaning our definitions of $B$-scalar products and $B$-vector products easily generalise to arbitrary geometries, parameterised by the symmetric bilinear form with matrix representation $B$ (which gives different definitions of perpendicularity), as well as arbitrary fields not of characteristic $2$.

Want more? Keep a look out for my PhD thesis, to be submitted in 4 weeks time!