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

Solution :
As $$\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c $$   are noncoplanar, $$\overrightarrow b \times \overrightarrow c ,\,\overrightarrow c \times \overrightarrow a ,\,\overrightarrow a \times \overrightarrow b $$      are also noncoplanar.
So, any vector can be expressed as a linear combination of these vectors.
$$\eqalign{ & {\text{Let }}\overrightarrow a = \lambda \overrightarrow b \times \overrightarrow c + \mu \overrightarrow c \times \overrightarrow a + \nu \overrightarrow a \times \overrightarrow b \cr & \therefore \overrightarrow a .\overrightarrow a = \lambda \left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow a } \right],\,\,\overrightarrow a .\overrightarrow b = \mu \left[ {\overrightarrow c \,\,\overrightarrow a \,\,\overrightarrow b } \right],\,\,\overrightarrow a .\overrightarrow c = \nu \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] \cr & \therefore \,\overrightarrow a = \frac{{\left( {\overrightarrow a .\overrightarrow a } \right)\overrightarrow b \times \overrightarrow c }}{{\left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow a } \right]}} + \frac{{\left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow c \times \overrightarrow a }}{{\left[ {\overrightarrow c \,\,\overrightarrow a \,\,\overrightarrow b } \right]}} + \frac{{\left( {\overrightarrow a .\overrightarrow c } \right)\overrightarrow a \times \overrightarrow b }}{{\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]}} \cr} $$