Linear Algebra – Length of Vector Resulting from Cross Product Explained

Question

I have the following question. In physics forces are vectors. Now I may write a force as
\begin{equation}
\mathbf{F} = F \mathbf{e}_F
\end{equation}
with $F$ denoting the length and $\mathbf{e}_F$ denoting the direction vector. But some forces are the result of a cross product (pseudo vectors). The length is then
\begin{equation}
\vert\mathbf{a}\vert \vert\mathbf{b}\vert \sin(\theta)
\end{equation}
with $\mathbf{a}, \mathbf{b}$ some vectors (maybe position and velocity) and $\theta$ the angle between them.
However these vectors are also sometimes written in the first form. How can I check whether $F$ in the first form is the length of a cross product, or not?

Answer 1

You can always choose $|a|, |b|, \theta$, that will give you that vector so... ...$F$ is always a length of infinite cross products. What are the restrictions? You must pick $a, b$ in a plane orthogonal to $F$!