Explanation

(c)
Given equation can be rewritten as
$\overrightarrow{\mathbf{r}} = 3\widehat{\dot{\mathbf{j}}} + \left( \widehat{\dot{\mathbf{i}}} + 2\widehat{\mathbf{k}} \right)s + \left( - 2\widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}} \right)t$
which is a plane passing through$\ \overrightarrow{\mathbf{a}} = 3\widehat{\dot{\mathbf{j}}}\$and parallel to the vectors $\overrightarrow{\mathbf{b}} = \widehat{\dot{\mathbf{i}}} + 2\widehat{\mathbf{k}}\$and$\ \$
$\overrightarrow{\mathbf{c}} = - 2\widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}}.$
Therefore, it is perpendicular to the vector
$\overrightarrow{\mathbf{n}} = \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} = 2\widehat{\dot{\mathbf{i}}} - 5\widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}}$
$Hence,\ its\ vector\ equation\ is\ \left( \overrightarrow{\mathbf{r}} - \overrightarrow{\mathbf{a}} \right) \bullet \overrightarrow{\mathbf{n}} = 0$
$\Longrightarrow \ \overrightarrow{\mathbf{r}} \bullet \overrightarrow{\mathbf{n}} = \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{n}}$
$\Longrightarrow \left( x\widehat{\dot{\mathbf{i}}} + y\widehat{\dot{\mathbf{j}}} + z\widehat{\mathbf{k}} \right) \bullet \left( 2\widehat{\dot{\mathbf{i}}} - 5\widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}} \right) = 3\widehat{\dot{\mathbf{j}}} \bullet \left( 2\widehat{\dot{\mathbf{i}}} - 5\widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}} \right)$
$\Longrightarrow 2x - 5y - z + 15 = 0$