Prove that for any given $c_1,c_2,c_3\in \mathbb{Z}$,the equations set has integral solution. $$ \left\{
\begin{aligned}
c_1 & = a_2b_3-b_2a_3 \\
c_2 & = a_3b_1-b_3a_1 \\
c_3 & = a_1b_2-b_1a_2
\end{aligned}
\right.
$$
$c_1,c_2,c_3\in  \mathbb{Z}$ is given,prove that $\exists a_1,a_2,a_3,b_1,b_2,b_3\in\mathbb{Z}$ meet the equations set.
Apparently,the question equal to how to decompose the integer vector into the cross product (vector product) of two integer vector.And the real question I want to ask just is it.

The given integral vector v has at least one rational vector w such that  w ,v are perpendicular. We take the cross product u of v and w and this is rational. Now we find a suitable common multiple of u and w to eliminate the denominators. Thus we get new integral vectors u',w' perpendicular to v. 
