For any vector $$\vec a,$$  the value of $${\left( {\vec a \times \hat i} \right)^2} + {\left( {\vec a \times \hat j} \right)^2} + {\left( {\vec a \times \hat k} \right)^2}\,$$       is equal to :

Solution :
$$\eqalign{ & {\text{Let }}\vec a = x\vec i + y\vec j + z\vec k \cr & \vec a \times \vec i = z\vec j - y\vec k\,\,\,\,\, \Rightarrow {\left( {\vec a \times \hat i} \right)^2} = {y^2} + {z^2} \cr & {\text{Similarly, }}{\left( {\vec a \times \hat j} \right)^2} = {x^2} + {z^2}{\text{ and }}{\left( {\vec a \times \hat k} \right)^2} = {x^2} + {y^2} \cr & \Rightarrow {\left( {\vec a \times \hat i} \right)^2} + {\left( {\vec a \times \hat j} \right)^2} + {\left( {\vec a \times \hat k} \right)^2} = 2\left( {{x^2} + {y^2} + {z^2}} \right) \cr & \Rightarrow {\left( {\vec a \times \hat i} \right)^2} + {\left( {\vec a \times \hat j} \right)^2} + {\left( {\vec a \times \hat k} \right)^2} = 2{{\vec a}^2} \cr} $$