Given a known Moment ($\vec{M}$), distance ($\vec{r}$), and unit vector $\hat{f}$ of force ($\vec{F}$), is it possible to find the resultant magnitude of force (using the formulation of the cross product)? Here the unit vector of force $\hat{f}$, $\vec{r}$, and $\vec{M}$ are all known.
I've been trying to think through using a Skew-Symmetric matrix but I'm not sure how to bound the solutions from:
$\vec{M} = \vec{r} \times \vec{F}\\$
$\vec{M} = [\vec{r}]_{x} \vec{F}\\$
$\vec{F} = [\vec{r}]^{-1}_{x} \vec{M}$
with a given unit vector $\hat{r}$
Is this even possible?
Note that you won't get an equation of the type $\vec F=k\vec M$ as that would imply $\vec F,\vec M$ are parallel while the original equation implies they are orthogonal.
Using $\vec F=|\vec F|\hat f$ we get $\vec M=|\vec F|(\vec r\times \hat f)$ so you can find $|\vec F|$ by taking the ratio of the $x$ components $[\vec M]_x$ and $[\vec r\times\hat f]_x$ (or the $y/z$ components) or the ratio of the magnitudes $|\vec M|$ and $|\vec r\times\hat f|$.
I am not sure what you want to do with a skew-symmetric matrix.