Let $$\overrightarrow c $$ be a vector perpendicular to the vectors, $$\overrightarrow a $$ = $$\widehat i$$ + $$\widehat j$$ $$-$$ $$\widehat k$$ and
$$\overrightarrow b $$ = $$\widehat i$$ + 2$$\widehat j$$ + $$\widehat k$$. If $$\overrightarrow c \,.\,\left( {\widehat i + \widehat j + 3\widehat k} \right)$$ = 8 then the value of
$$\overrightarrow c $$ . $$\left( {\overrightarrow a \times \overrightarrow b } \right)$$ is equal to __________.

Let $$\overrightarrow a = \widehat i + \alpha \widehat j + 3\widehat k$$ and $$\overrightarrow b = 3\widehat i - \alpha \widehat j + \widehat k$$. If the area of the parallelogram whose adjacent sides are represented by the vectors $$\overrightarrow a $$ and $$\overrightarrow b $$ is $$8\sqrt 3 $$ square units, then $$\overrightarrow a $$ . $$\overrightarrow b $$ is equal to __________.

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

Let three vectors $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ be such that $$\overrightarrow c $$ is coplanar
with $$\overrightarrow a $$ and $$\overrightarrow b $$, $$\overrightarrow a .\overrightarrow c $$ = 7 and $$\overrightarrow b $$ is perpendicular to $$\overrightarrow c $$, where
$$\overrightarrow a = - \widehat i + \widehat j + \widehat k$$ and $$\overrightarrow b = 2\widehat i + \widehat k$$ , then the
value of $$2{\left| {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right|^2}$$ is _____.