How to find unit vectors that are the bisectors of a given tetrahedron?

Question

Let OABC is a tetrahedron with equal edges and OA=$\vec a$, OB=$\vec b$, OC=$\vec c$ where |$\vec a$|=|$\vec b$|=|$\vec c$|=2. If $\vec p$,$\vec q$,$\vec r$ are unit vectors along bisectors of angle between pair of edges represented by OA,OB; OB,OC and OC, OA respectively then what is the value of $$\frac {[\vec a\ \vec b\ \vec c]}{[\vec p+q\ \vec q+r\ \vec r+p]}$$?