If $$\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c $$   be three vectors such that $$\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] = 4$$    then $$\left[ {\overrightarrow a \times \overrightarrow b \,\,\overrightarrow b \times \overrightarrow c \,\,\overrightarrow c \times \overrightarrow a } \right]$$     is equal to :

Solution :
$$\eqalign{ & \left[ {\overrightarrow a \times \overrightarrow b \,\,\overrightarrow b \times \overrightarrow c \,\,\overrightarrow c \times \overrightarrow a } \right] = \left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow b \times \overrightarrow c } \right).\overrightarrow c \times \overrightarrow a \cr & = \left\{ {\left( {\overrightarrow a \times \overrightarrow b .\overrightarrow c } \right)\overrightarrow b - \left( {\overrightarrow a \times \overrightarrow b .\overrightarrow b } \right)\overrightarrow c } \right\}.\overrightarrow c \times \overrightarrow a \cr & = \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]\,\left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow a } \right] \cr & = {\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]^2} \cr & = {4^2} \cr & = 16 \cr} $$