A hall has a square floor of dimension 10 m $$\times$$ 10 m (see the figure) and vertical walls. If the angle GPH between the diagonals AG and BH is $${\cos ^{ - 1}}{1 \over 5}$$, then the height of the hall (in meters) is :

Let $$\overrightarrow a = \widehat i + \widehat j + \widehat k$$ and $$\overrightarrow b = \widehat j - \widehat k$$. If $$\overrightarrow c $$ is a vector such that $$\overrightarrow a \times \overrightarrow c = \overrightarrow b $$ and $$\overrightarrow a .\overrightarrow c = 3$$, then $$\overrightarrow a .(\overrightarrow b \times \overrightarrow c )$$ is equal to :

Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three vectors such that $$\overrightarrow a $$ = $$\overrightarrow b $$ $$\times$$ ($$\overrightarrow b $$ $$\times$$ $$\overrightarrow c $$). If magnitudes of the vectors $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ are $$\sqrt 2 $$, 1 and 2 respectively and the angle between $$\overrightarrow b $$ and $$\overrightarrow c $$ is $$\theta \left( {0 < \theta < {\pi \over 2}} \right)$$, then the value of 1 + tan$$\theta$$ is equal to :

Let $$\overrightarrow a = \widehat i + \widehat j + 2\widehat k$$ and $$\overrightarrow b = - \widehat i + 2\widehat j + 3\widehat k$$. Then the vector product $$\left( {\overrightarrow a + \overrightarrow b } \right) \times \left( {\left( {\overrightarrow a \times \left( {\left( {\overrightarrow a - \overrightarrow b } \right) \times \overrightarrow b } \right)} \right) \times \overrightarrow b } \right)$$ is equal to :