If \[\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c \]   are noncoplanar nonzero vectors then \[\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow a \times \overrightarrow c } \right) + \left( {\overrightarrow b \times \overrightarrow c } \right) \times \left( {\overrightarrow b \times \overrightarrow a } \right) + \left( {\overrightarrow c \times \overrightarrow a } \right) \times \left( {\overrightarrow c \times \overrightarrow b } \right)\]               is equal to :

Solution :
\[\begin{array}{l} \left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow a \times \overrightarrow c } \right) = \left( {\overrightarrow a \times \overrightarrow b .\overrightarrow c } \right)\overrightarrow a - \left( {\overrightarrow a \times \overrightarrow b .\overrightarrow a } \right)\overrightarrow c \\ = \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]\overrightarrow a - \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]\overrightarrow c \\ = \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]\overrightarrow a \\ {\rm{Similarly,}}\,\left( {\overrightarrow b \times \overrightarrow c } \right) \times \left( {\overrightarrow b \times \overrightarrow a } \right) = \left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow a } \right]\overrightarrow b \\ \left( {\overrightarrow c \times \overrightarrow a } \right) \times \left( {\overrightarrow c \times \overrightarrow b } \right) = \left[ {\overrightarrow c \,\,\overrightarrow a \,\,\overrightarrow b } \right]\overrightarrow c \\ \therefore {\rm{ the\, expression}} = \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]\left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right) \end{array}\]