Explanation

(b)
$The\ given\ line\ is\ parallel\ to\ the\ vector\ \overrightarrow{\mathbf{n}} = \widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}}.The\ required\ plane\ passing\$
$through\ the\ point\ (2,\ 3,\ 1)ie,\ 2\widehat{\dot{\mathbf{i}}} + 3\widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}}\ \ and\ is\ perpendicular\ to\ the\ vector$
$\ \overrightarrow{\mathbf{n}} = \widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}}$
$\therefore Its\ equation\ is$
$\lbrack\left( \overrightarrow{\mathbf{r}} - \left( 2\widehat{\dot{\mathbf{i}}} + 3\widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}} \right) \right\rbrack \bullet \left( \widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}} \right) = 0$
$\Longrightarrow \overrightarrow{\mathbf{r}} \bullet \left( \widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}} \right) = 1$