There are two sets of basis vectors $\vec{a}_1$, $\vec{a}_2$ and $\vec{b}_1$, $\vec{b}_2$. I want to find the third basis vectors which norm are minimum: $$\vec{R}_1=a_{11}\vec{a}_1+a_{12}\vec{a}_2=b_{11}\vec{b}_1+b_{12}\vec{b}_2$$ $$\vec{R}_2=a_{21}\vec{a}_1+a_{22}\vec{a}_2=b_{21}\vec{b}_1+b_{22}\vec{b}_2$$
and $a_{11},a_{12},a_{21},a_{22},b_{11},b_{12},b_{21},b_{22}$ are all integers.
I don't have any idea