Given, $$\overrightarrow a = \widehat i + 2\widehat j - \widehat k$$,

$$\overrightarrow b = \widehat i - \widehat j$$,

$$\overrightarrow c = \widehat i - \widehat j - \widehat k$$

$$\overrightarrow r \times \overrightarrow a = \overrightarrow c \times \overrightarrow a $$

$$ \Rightarrow \overrightarrow r \times \overrightarrow a - \overrightarrow c \times \overrightarrow a = 0$$

$$ \Rightarrow (\overrightarrow r - \overrightarrow c ) \times \overrightarrow a = 0$$

$$\therefore$$ $$\overrightarrow r - \overrightarrow c = \lambda \overrightarrow a $$

$$ \Rightarrow \overrightarrow r = \lambda \overrightarrow a + \overrightarrow c $$

$$ \Rightarrow \overrightarrow r \,.\,\overrightarrow b = \lambda \overrightarrow a \,.\,\overrightarrow b + \overrightarrow c \,.\,\overrightarrow b $$ (taking dot with $$\overrightarrow b $$)

$$ \Rightarrow 0 = \lambda \overrightarrow a \,.\,\overrightarrow b + \overrightarrow c \,.\,\overrightarrow b $$ [$$\because$$ $$\overrightarrow r \,.\,\overrightarrow b = 0$$]

$$ \Rightarrow \lambda (\widehat i + 2\widehat j - \widehat k)\,.\,(\widehat i - \widehat j) + (\widehat i - \widehat j - \widehat k)\,.\,(\widehat i - \widehat j) = 0$$

$$ \Rightarrow \lambda (1 - 2) + 2 = 0$$

$$ \Rightarrow \lambda = 2$$

$$\therefore$$ $$\overrightarrow r = 2\overrightarrow a + \overrightarrow c $$

$$ \Rightarrow \overrightarrow r \,.\,\overrightarrow a = 2\overrightarrow a \,.\,\overrightarrow a + \overrightarrow c \,.\,\overrightarrow a $$ [taking dot with $${\overrightarrow a }$$]

$$ = 2{\left| {\overrightarrow a } \right|^2} + \overrightarrow a \,.\,\overrightarrow c $$

$$ = 2(1 + 4 + 1) + (1 - 2 + 1)$$

$$ \Rightarrow \overrightarrow r \,.\,\overrightarrow a = 12$$