I am not able to simplify this vector problem

Question

$\vec { a } ,\vec { b } ,\vec { c } $ are three unit vectors such that $|\vec { a } +\vec { b } +\vec { c } |=\sqrt { 3 } $ then the maximum value of $\left( \vec { a } \times \vec { b } \right) .\left( \vec { b } \times \vec { c } \right) +\left( \vec { b } \times \vec { c } \right) .\left( \vec { c } \times \vec { a } \right) +\left( \vec { c } \times \vec { a } \right) .\left( \vec { a } \times \vec { b } \right) $

I tried it by opening the given product using $\left( \vec { a } \times \vec { b } \right) .\left( \vec { c } \times \vec { d } \right) =\begin{vmatrix} \vec { a } .\vec { c } & \vec { a } .\vec { d } \\ \vec { b } .\vec { c } & \vec { b } .\vec { d } \end{vmatrix}$ but i am not able to find the maximum value . Please help

Answer 1

Hint:

$|\vec {a} +\vec { b } +\vec { c } |=\sqrt { 3 }$

$(\vec { a } +\vec { b } +\vec { c }) \cdot (\vec { a } +\vec { b } +\vec { c }) = 3$

So $ \sum \vec { a } \cdot\vec { b } = 0 \,$ (as $\vec {a}, \vec {b}, \vec {c} \,$ are unit vectors)

$(\vec { a } \times \vec { b }) \cdot (\vec {b} \times \vec {c}) = (\vec {a} \cdot \vec {b}) (\vec{b} \cdot \vec {c}) - \vec {a} \cdot \vec {c} \,$ (since $\vec{b} \cdot \vec{b} = 1,$ being unit vector)

Expand others similarly and add. If you get stuck, let me know.