How to rewrite a vector cross product? Can this vector cross product $ \vec F = q \vec v \times \vec B $ be rewritten to $  \vec B = \vec F \times q \vec v $ ?
Edit: Take the special case, that all three vectors are orthogonal to each other.
 A: In fact the vector triple product identity says
$$ (\vec {qv} \times \vec{B}) \times \vec{qv} = \|\vec{qv}\|^2 \vec B - (\vec{qv} \cdot \vec B) \vec{qv} $$
Assuming $\vec{qv}$ and $\vec{B}$ are linearly independent, this is $\vec B$ if and only if $\vec{qv}$ is a unit vector orthogonal to $\vec B$.
EDIT: In particular, if $\vec{qv}$ is orthogonal to $\vec B$, and is nonzero (but not necessarily a unit vector), you can write
$$ \vec B = \dfrac{\vec F \times \vec{qv}}{\|\vec{qv}\|^2}$$ 
A: You cannot perform this task.
Notice that you are taking the cross products of vectors. If you are simply finding the magnitude of the values present, then you can simply rewrite to get:
$$F = qvB,\ B = \frac{F}{qv}$$
However, you are taking the cross products of vectors, which isn't a scalar quantity, it is a vector quantity. The direction of $F$ is determined using the right-hand rule. You cannot determine the direction of $B$ given the vectors $F$ and $v$. The only thing you can do is find the scalar value of $B$ by the formula above.
