If $$\overrightarrow {{r_1}} = \lambda \hat i + 2\hat j + \hat k,\,\overrightarrow {{r_2}} = \hat i + \left( {2 - \lambda } \right)\hat j + 2\hat k$$         are such that $$\left| {\overrightarrow {{r_1}} } \right| > \left| {\overrightarrow {{r_2}} } \right|,$$   then $$\lambda $$ satisfies which one of the following ?

Solution :
$$\eqalign{ & {\text{Given, }}\overrightarrow {{r_1}} = \lambda \hat i + 2\hat j + \hat k{\text{ and }}\,\overrightarrow {{r_2}} = \hat i + \left( {2 - \lambda } \right)\hat j + 2\hat k \cr & \therefore \,\left| {\overrightarrow {{r_1}} } \right| > \left| {\overrightarrow {{r_2}} } \right| \cr & \Rightarrow \sqrt {{\lambda ^2} + {{\left( 2 \right)}^2} + {{\left( 1 \right)}^2}} > \sqrt {{{\left( 1 \right)}^2} + {{\left( {2 - \lambda } \right)}^2} + {{\left( 2 \right)}^2}} \cr & \Rightarrow {\lambda ^2} + 4 + 1 > 1 + 4 + {\lambda ^2} - 4\lambda + 4 \cr & \Rightarrow 5 > 9 - 4\lambda \cr & \Rightarrow 4\lambda > 4 \cr & \Rightarrow \lambda > 1 \cr} $$