If $$\overrightarrow a = \left( {\widehat i + \widehat j + \widehat k} \right),\overrightarrow a .\overrightarrow b = 1$$ and $$\overrightarrow a \times \overrightarrow b = \widehat j - \widehat k,$$ then $$\overrightarrow b $$ is

If the lines $${{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 1} \over 4}$$ and $$\,{{x - 3} \over 1} = {{y - k} \over 2} = {z \over 1}$$ intersect, then the value of $$k$$ is

The unit vector which is orthogonal to the vector $$3\overrightarrow i + 2\overrightarrow j + 6\overrightarrow k $$ and is coplanar with the vectors $$\,2\widehat i + \widehat j + \widehat k$$ and $$\,\widehat i - \widehat j + \widehat k$$$$\,\,\,$$ is

The value of the integral $$\int\limits_0^1 {\sqrt {{{1 - x} \over {1 + x}}} dx} $$ is