I was experimenting with the triple scalar product and forces in equilibrium when I came to this result:
Consider 4 forces $ \pmb{F_i}$ for $i=1,2,3,4$. $\pmb{F_i}=F_i\hat{e_i}$ where $\hat{e_i}$ is the unit vector in the direction of the corresponding force and $F_i$ is the magnitude.
$$F_1\hat{e_1}+F_2\hat{e_2}+F_3\hat{e_3}+F_4\hat{e_4}=0$$
Take the scalar product of the system with $e_3 \times e_4$ (the vector product of $e_3,e_4$).
$$F_1\hat{e_1}\cdot (e_3 \times e_4)+F_2\hat{e_2}\cdot (e_3 \times e_4)+F_3\hat{e_3}\cdot (e_3 \times e_4)+F_4\hat{e_4}\cdot (e_3 \times e_4)=0$$
$$\Rightarrow F_1\hat{e_1}\cdot (e_3 \times e_4)+F_2\hat{e_2}\cdot (e_3 \times e_4)+0+0=0$$
$$\Rightarrow \frac{F_1}{\hat{e_2}\cdot (e_3 \times e_4)}=\frac{F_2}{-\hat{e_1}\cdot (e_3 \times e_4)}$$
$$\Rightarrow \frac{F_1}{V_1}=\frac{F_2}{V_2}$$
Where $V_1$ is the volume of the parallelepiped with edges $\hat{e_{2}},\hat{e_{3}},\hat{e_{4}}$.
This can be extended to any other pair of forces.
If one expresses Lami's theorem as
If one has 3 forces in equilibrium acting at a single point in 2D then the magnitude of each force is proportional to the area of the parallelogram made from the unit vectors of the other 2 forces.
Then the similarities between my result and this are pretty clear. This also leads me to conjecture for that for two forces in 1 dimension, the forces are proportional to the magnitude of the direction of the other force.
Firstly, is my result correct and valid? Does this theorem have a name, or are there any resources on it I could read? Can this formulation and result be extended into other (n?) dimensions?