Let $$\overrightarrow a = - \widehat i - \widehat k,\overrightarrow b = - \widehat i + \widehat j$$ and $$\overrightarrow c = \widehat i + 2\widehat j + 3\widehat k$$ be three given vectors. If $$\overrightarrow r $$ is a vector such that $$\overrightarrow r \times \overrightarrow b = \overrightarrow c \times \overrightarrow b $$ and $$\overrightarrow r .\overrightarrow a = 0,$$ then the value of $$\overrightarrow r .\overrightarrow b $$ is

Let $$y'\left( x \right) + y\left( x \right)g'\left( x \right) = g\left( x \right),g'\left( x \right),y\left( 0 \right) = 0,x \in R,$$ where $$f'(x)$$ denotes $${{df\left( x \right)} \over {dx}}$$ and $$g(x)$$ is a given non-constant differentiable function on $$R$$ with $$g(0)=g(2)=0.$$ Then the value of $$y(2)$$ is