$$[\matrix{ {\overrightarrow c } & {\overrightarrow a } & {\overrightarrow b } \cr } ] = - 25$$

Let $$\overrightarrow c = l\widehat i + n\widehat j + n\widehat k$$

$$\left| {\matrix{ l & m & n \cr 4 & 3 & 0 \cr 3 & { - 4} & 5 \cr } } \right| = - 25$$

$$ \Rightarrow 3l - 4m - 5n = - 5$$ ..... (i)

$$\overrightarrow c \,.\,(\widehat i + \widehat j + \widehat k) = 4$$

$$ \Rightarrow l + m + n = 4$$ ..... (ii)

$${{\overrightarrow c \,.\,\overrightarrow a } \over {|\overrightarrow a |}} = 1 \Rightarrow \overrightarrow c \,.\,\overrightarrow a = 5$$

$$ \Rightarrow 4l + 3m = 5$$ ...... (iii)

Using (i), (ii) and (iii)

$$l = 2,m = - 1,n = 3$$

Now, $${{\overrightarrow c \,.\,\overrightarrow b } \over {|\overrightarrow b |}} = {{25} \over {5\sqrt 2 }} = {5 \over {\sqrt 2 }}$$

$$\therefore$$ Option (2) is correct.