When is the $B$ defined through $\forall i:a_i=v_i\times B$ existent and unique?

Question

Of course the motivation is to understand what data is needed to define the (electric and) magnetic field through the lorentz force. So suppose we are given $a_1,\ldots,a_n,v_1,\ldots,v_n\in\mathbb R^3$ and we want to verify the existence and uniqueness of a $B\in\mathbb R^3$ such that $\forall i:a_i=v_i\times B$. If I had to make a guess, then I would say that $B$ exists and is unique if $n=3$ and $v_1,v_2,v_3$ is a basis, but I don't know how to prove this. Any help is much appreciated :)

Answer 1

Uniqueness holds for $n>2$, provided the $v_i$ are distinct and non-zero. Existence is not guaranteed for any $n$ without conditions on the $F_i$ and $v_i$. Namely, if there is such a $B$, then we need $v_i\times F_j+v_j\times F_i=0$*. In fact, this condition will give existence.

*I have just realised I made a mistake. The conditions should be $v_i\cdot F_j+v_j\cdot F_i=0$.

The span of the $v_i$ doesn't need to be 3-dimensional: in fact for $n=2$, there is a unique solution provided the conditions hold. We can choose a basis in which $v_1=(1,0,0)^t$, $v_2=(0,1,0)^t$. Then the conditions $v_i\cdot F_j+v_j\cdot F_i=0$ mean that the two forces take the form $F_1=(0,a,b)^t$, $F_2=(-a,0,c)^t$. Then the equations $F_i=v_i\times B$ read

$$ \begin{pmatrix} 0\\ a\\ b \end{pmatrix} = \begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix} \times \begin{pmatrix} B_1\\ B_2\\ B_3 \end{pmatrix} = \begin{pmatrix} 0\\ -B_3\\ B_2 \end{pmatrix} $$ and $$ \begin{pmatrix} -a\\ 0\\ c \end{pmatrix} = \begin{pmatrix} 0\\ 1\\ 0 \end{pmatrix} \times \begin{pmatrix} B_1\\ B_2\\ B_3 \end{pmatrix} = \begin{pmatrix} B_3\\ 0\\ -B_1 \end{pmatrix} $$

So, we get the unique solution $B=(-c,b,-a)^t$.