I have the following proportion $\vec{JV} \times \vec{F_v} = \vec{JM} \times \vec{F_m}$ and all members are known except the magnitude of the vector $\vec F_m$, like described by another question here in greater details.
Can someone please suggest how to express $|\vec{F_m}|$ in terms of $\vec{JV}, \vec{JM}$, $\vec{F_v}$ and $\hat{F_m}$, where $\hat{F_m}$ is a direction of the $\vec{F_m}$?
Here is a graphical representation of what I'm talking about:
![2]](https://i.stack.imgur.com/YJB2R.png)
You are asking simply how to find $x$ such that $$x(\vec a\times\vec u) = \vec c$$ where $\vec a$, $\vec u$, $\vec c$ are given vectors (and $\vec u$ has unit length, although this is not relevant). Such $x$ exists if and only if $\vec a\times \vec u$ is a multiple of $\vec c$ and can be calculated simply as $$x = \frac{|\vec c|}{|\vec a\times\vec u|}.$$
In your case $\vec c=\vec{JV} \times \vec{F_v}$, $\vec a = \vec{JM}$, $\vec u = \hat{F_m}$ and you want to find $x=|\vec F_m|$.