If a → · b → = 1 , b → · c → = 2 and c → · a → = 3 , then…

We know that

$\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right),\overrightarrow{ b}\times \left(\overrightarrow{c}\times \overrightarrow{a}\right)$ and $\overrightarrow{c}\times \left(\overrightarrow{a}\times \overrightarrow{b}\right)$ are all coplanar as $\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right)+\overrightarrow{ b}\times \left(\overrightarrow{c}\times \overrightarrow{a}\right)+\overrightarrow{c}\times \left(\overrightarrow{a}\times \overrightarrow{b}\right)=\overrightarrow{0}$.

Also, the scalar triple product of $3$ vectors which are coplanar is $0$.

Hence,  $\left[\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right) \overrightarrow{b}\times \left(\overrightarrow{c}\times \overrightarrow{a}\right) \overrightarrow{c}\times \left(\overrightarrow{a}\times \overrightarrow{b}\right)\right]=0$