How to find a vector given the scalar triple productScalar triple product formula: a (...

Unfortunately, you can't. Your real question is even simpler: if you know x and $\mathbf{a}\cdot \mathbf{x}=y$, can you solve for a?
The issue is that the dot product is not an invertible operation, so there are lots of ways to choose x such that $\mathbf{a}\cdot \mathbf{x}=y$. Think about it for the two-dimensional cases: let $\mathbf{a}=[a_1,a_2]$ and $\mathbf{x}=[x_1,x_2]$. The hypothesis is that we know $x_1$, $x_2$, $y$, and that
$a_1{x}_1+a_2{x}_2=y,$
but this is a single equation with two unknowns ($a_1$ and $a_2$), which has infinitely many solutions. If that seems odd, verify that $\mathbf{a}=[0,\frac{y}{x_2}]$ and $\mathbf{a}=[\frac{y}{x_1},0]$ are both solutions