Let $$\overrightarrow r $$ be a vector perpendicular to $$\overrightarrow a + \overrightarrow b + \overrightarrow c ,$$   where $$\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] = 2.$$    If $$\overrightarrow r = l\left( {\overrightarrow b \times \overrightarrow c } \right) + m\left( {\overrightarrow c \times \overrightarrow a } \right) + n\left( {\overrightarrow a \times \overrightarrow b } \right)$$          then $$l + m + n$$   is :

Solution :
$$\eqalign{ & {\text{Here }}\overrightarrow r .\left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right) = 0 \cr & \therefore \,\overrightarrow r .\overrightarrow a + \overrightarrow r .\overrightarrow b + \overrightarrow r .\overrightarrow c = 0 \cr & {\text{Now }}\overrightarrow r .\overrightarrow a = l\left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow a } \right]\,\,\overrightarrow r .\overrightarrow b = m\left[ {\overrightarrow c \,\,\overrightarrow a \,\,\overrightarrow b } \right]\,\,\overrightarrow r .\overrightarrow c = n\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] \cr & \therefore l\left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow a } \right] + m\left[ {\overrightarrow c \,\,\overrightarrow a \,\,\overrightarrow b } \right] + n\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] = 0 \cr & \Rightarrow l + m + n = 0\,\,\,\,\,\,\,\left( {\because \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] = 2 \ne 0} \right) \cr} $$