Let $$\overrightarrow a $$ and $$\overrightarrow b $$ be two noncollinear unit vectors. If $$\overrightarrow u = \overrightarrow a - \left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow b $$     and $$\overrightarrow v = \overrightarrow a \times \overrightarrow b $$    then $$\left| {\overrightarrow v } \right|$$  is :

Solution :
$$\eqalign{ & \left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow b = \left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow b - \left( {\overrightarrow b .\overrightarrow b } \right)\overrightarrow a \cr & \therefore \,\left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow b = \left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow b + \overrightarrow a \cr & \therefore \,\overrightarrow u = \overrightarrow a - \left\{ {\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow b + \overrightarrow a } \right\} = \overrightarrow b \times \left( {\overrightarrow a \times \overrightarrow b } \right) \cr & \therefore \,\overrightarrow u = \overrightarrow b \times \overrightarrow v \cr & \Rightarrow \left| {\overrightarrow u } \right| = \left| {\overrightarrow b } \right|\left| {\overrightarrow v } \right|{\text{ because }}\overrightarrow b .\overrightarrow v = 0,{\text{ i}}{\text{.e}}{\text{., }}\overrightarrow b \bot \overrightarrow v \cr & \Rightarrow \left| {\overrightarrow u } \right| = \left| {\overrightarrow v } \right|{\text{ because }}\left| {\overrightarrow b } \right| = 1. \cr} $$