If $$\overrightarrow p = \lambda \left( {\overrightarrow u \times \overrightarrow v } \right) + \mu \left( {\overrightarrow v \times \overrightarrow w } \right) + \nu \left( {\overrightarrow w \times \overrightarrow u } \right)$$          and $$\left[ {\overrightarrow u \overrightarrow v \overrightarrow w } \right] = \frac{1}{5},$$    then $$\lambda + \mu + \nu $$   is equal to :

Solution :
$$\eqalign{ & \overrightarrow p = \lambda \left( {\overrightarrow u \times \overrightarrow v } \right) + \mu \left( {\overrightarrow v \times \overrightarrow w } \right) + \nu \left( {\overrightarrow w \times \overrightarrow u } \right) \cr & \Rightarrow \overrightarrow p .\overrightarrow w = \lambda \left( {\overrightarrow u \times \overrightarrow v } \right).\overrightarrow w + \mu \left( {\overrightarrow v \times \overrightarrow w } \right).\overrightarrow w + \nu \left( {\overrightarrow w \times \overrightarrow u } \right).\overrightarrow w \cr & \Rightarrow \lambda \left[ {\overrightarrow u \overrightarrow v \overrightarrow w } \right] + 0 + 0 = \frac{\lambda }{5} \cr & \Rightarrow \lambda = 5\left( {\overrightarrow p .\overrightarrow w } \right) \cr & {\text{Similarly, }}\mu = 5\left( {\overrightarrow p .\overrightarrow u } \right){\text{ and }}\nu = 5\left( {\overrightarrow p .\overrightarrow v } \right) \cr & \therefore \,\lambda + \mu + \nu \cr & = 5\left( {\overrightarrow p .\overrightarrow w } \right) + 5\left( {\overrightarrow p .\overrightarrow u } \right) + 5\left( {\overrightarrow p .\overrightarrow v } \right) \cr & = 5\overrightarrow p \left( {\overrightarrow u + \overrightarrow v + \overrightarrow w } \right) \cr & {\text{Hence, }}\lambda + \mu + \nu {\text{ depends on the vectors}}{\text{.}} \cr} $$