Let \( O A B C \) is a tetrahedron with equal edges and \( \overrightarrow{O A}=\vec{a}, \overri...

Let \( O A B C \) is a tetrahedron with equal edges and \( \overrightarrow{O A}=\vec{a}, \overrightarrow{O B}=\vec{b} \) and \( \overrightarrow{O C}=\vec{c} \) where \( |\vec{a}|=|\vec{b}|=|\vec{c}|=2 \). If \( \hat{p}, \hat{q} \) and \( \hat{r} \) are unit vectors along bisectors of angle between pair of edges represented by \( \overrightarrow{O A}, \overrightarrow{O B} ; \overrightarrow{O B}, \overrightarrow{O C} \) and \( \overrightarrow{O C}, \overrightarrow{O A} \) respectively, then the value of \( \frac{[\vec{a} \vec{b} \vec{c}]}{[\hat{p}+\hat{q} \hat{q}+\hat{r} \hat{r}+\hat{p}]} \) is equal to
(a) \( \frac{3 \sqrt{3}}{32} \)
(b) \( 4 \sqrt{2} \)
(c) \( 6 \sqrt{3} \)
(d) \( \frac{3 \sqrt{3}}{8} \)