Question #41290

Let $\vec{a}, \vec{b}, \vec{c}$ be three mutually orthogonal vectors forming a left-handed system. Consider three lines $L_1, L_2, L_3$ defined as: $L_1: \vec{r} = \vec{a} \times \vec{b} + \lambda (\vec{b} \times \vec{c})$ $L_2: \vec{r} = \vec{b} \times \vec{c} + \mu (\vec{c} \times \vec{a})$ $L_3: \vec{r} = \vec{c} \times \vec{a} + \gamma (\vec{a} \times \vec{b})$ If the shortest distances between the pairs of lines $(L_1, L_2)$, $(L_2, L_3)$, and $(L_3, L_1)$ are $2$, $3$, and $6$ respectively, find the value of $[\vec{a}' + \vec{b}', \vec{b}' + \vec{c}', \vec{c}' + \vec{a}']$, where $\vec{a}', \vec{b}', \vec{c}'$ represent the reciprocal system of vectors for $\vec{a}, \vec{b}, \vec{c}$.