Explanation

(a)
Let $\overrightarrow{\mathbf{r}} = \overrightarrow{\mathbf{a}} + t\overrightarrow{\mathbf{b}}$
$\Longrightarrow \ \overrightarrow{\mathbf{r}} = \widehat{\dot{\mathbf{i}}}(1 + t) + \widehat{\dot{\mathbf{j}}}(2 - t) + \widehat{\mathbf{k}}(1 + t)$
Since, The projection of $\overrightarrow{\mathbf{r}}\ on\ \overrightarrow{\mathbf{c}},$
$\frac{\overrightarrow{\mathbf{r}} \bullet \overrightarrow{\mathbf{c}}}{\left| \overrightarrow{\mathbf{c}} \right|} = \frac{|1|}{\left| \sqrt{3} \right|}\ \ \ \ \ \ \ \lbrack given\rbrack$
$\Longrightarrow \ \frac{1 \bullet (1 + t) + 1 \bullet (2 - t) - 1 \bullet (1 + t)}{\sqrt{3}} = \pm \frac{1}{\sqrt{3}}$
$\Longrightarrow 2 - t = \pm 1$
$\Longrightarrow t = 1\ or\ 3$
When, $t = 1,\ \overrightarrow{\mathbf{r}} = 2\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}}$
When, $t = 3,\ \overrightarrow{\mathbf{r}} = 4\widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} + 4\widehat{\mathbf{k}}$