Explanation

(d)
Here, ${\overrightarrow{\mathbf{a}}}_{1} = 3\widehat{\dot{\mathbf{i}}} + 6\widehat{\dot{\mathbf{j}}}$, ${\overrightarrow{\mathbf{a}}}_{2} = - 2\widehat{\dot{\mathbf{i}}} + 7\widehat{\mathbf{k}}$
${\overrightarrow{\mathbf{b}}}_{1} = - 4\widehat{\dot{\mathbf{i}}} + 3\widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}}$ and ${\overrightarrow{\mathbf{b}}}_{2} = - 4\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}}$
Now, ${\overrightarrow{\mathbf{a}}}_{2} - {\overrightarrow{\mathbf{a}}}_{1} = \widehat{\dot{\mathbf{i}}} - 6\widehat{\dot{\mathbf{j}}} + 7\widehat{\mathbf{k}}$
and
$\ {\overrightarrow{\mathbf{b}}}_{1} \times \ {\overrightarrow{\mathbf{b}}}_{2} = \left| \begin{matrix}
\widehat{\dot{\mathbf{i}}} & \widehat{\dot{\mathbf{j}}} & \widehat{\mathbf{k}} \\
- 4 & 3 & 2 \\
- 4 & 1 & 1
\end{matrix} \right| = \widehat{\dot{\mathbf{i}}} - 4\widehat{\dot{\mathbf{j}}} + 8\widehat{\mathbf{k}}$
$\Longrightarrow \left| {\overrightarrow{\mathbf{b}}}_{1} \times \ {\overrightarrow{\mathbf{b}}}_{2} \right| = \sqrt{1 + 16 + 64} = 9$
Now,
$\left( {\overrightarrow{\mathbf{a}}}_{2} - {\overrightarrow{\mathbf{a}}}_{1} \right) \bullet \left( {\overrightarrow{\mathbf{b}}}_{1} \times \ {\overrightarrow{\mathbf{b}}}_{2} \right) = \left( \widehat{\dot{\mathbf{i}}} - 6\widehat{\dot{\mathbf{j}}} + 7\widehat{\mathbf{k}} \right) \bullet \left( \widehat{\dot{\mathbf{i}}} - 4\widehat{\dot{\mathbf{j}}} + 8\widehat{\mathbf{k}} \right)$
$= 1 + 24 + 56 = 81$
∴ Shortest distance,
$d = \left| \frac{\left( {\overrightarrow{\mathbf{a}}}_{2} - {\overrightarrow{\mathbf{a}}}_{1} \right) \bullet \left( {\overrightarrow{\mathbf{b}}}_{1} \times \ {\overrightarrow{\mathbf{b}}}_{2} \right)}{\left| {\overrightarrow{\mathbf{b}}}_{1} \times \ {\overrightarrow{\mathbf{b}}}_{2} \right|} \right|$
$= \frac{81}{9} = 9\ unit$