If $$\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c $$   are three noncoplanar nonzero vectors and $$\overrightarrow r $$ is any vector in space then $$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow r \times \overrightarrow c } \right) + \left( {\overrightarrow b \times \overrightarrow c } \right) \times \left( {\overrightarrow r \times \overrightarrow a } \right) + \left( {\overrightarrow c \times \overrightarrow a } \right) \times \left( {\overrightarrow r \times \overrightarrow b } \right)$$               is equal to :

Solution :
$$\eqalign{ & \left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow r \times \overrightarrow c } \right) = \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]\overrightarrow r - \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow r } \right]\overrightarrow c ......\left( 1 \right) \cr & \left( {\overrightarrow b \times \overrightarrow c } \right) \times \left( {\overrightarrow r \times \overrightarrow a } \right) = \left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow a } \right]\overrightarrow r - \left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow r } \right]\overrightarrow a \cr & \left( {\overrightarrow c \times \overrightarrow a } \right) \times \left( {\overrightarrow r \times \overrightarrow b } \right) = \left[ {\overrightarrow c \,\,\overrightarrow a \,\,\overrightarrow b } \right]\overrightarrow r - \left[ {\overrightarrow c \,\,\overrightarrow a \,\,\overrightarrow r } \right]\overrightarrow b \cr} $$
$$\therefore $$  the expression $$ = 3\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]\overrightarrow r - \left\{ {\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow r } \right]\overrightarrow c + \left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow r } \right]\overrightarrow a + \left[ {\overrightarrow c \,\,\overrightarrow a \,\,\overrightarrow r } \right]\overrightarrow b } \right\}......\left( 2 \right)$$
Again, $$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow r \times \overrightarrow c } \right) = \left[ {\overrightarrow a \,\,\overrightarrow r \,\,\overrightarrow c } \right]\overrightarrow b - \left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow r } \right]\overrightarrow a ......\left( 3 \right)$$
From (1) and (3), $$\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]\overrightarrow r - \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow r } \right]\overrightarrow c = \left[ {\overrightarrow a \,\,\overrightarrow r \,\,\overrightarrow c } \right]\overrightarrow b - \left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow r } \right]\overrightarrow a $$
$$\therefore \,\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]\overrightarrow r = \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow r } \right]\overrightarrow c + \left[ {\overrightarrow c \,\,\overrightarrow a \,\,\overrightarrow r } \right]\overrightarrow b + \left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow r } \right]\overrightarrow a $$
$$\therefore $$  from (2), the expression $$ = 3\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]\overrightarrow r - \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]\overrightarrow r = 2\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]\overrightarrow r .$$