Explanation

(a)
Let $\overrightarrow{\mathbf{a}} =$ $\widehat{\dot{\mathbf{i}}} - 2\widehat{\dot{\mathbf{j}}} + 3\widehat{\mathbf{k}}$, $\overrightarrow{\mathbf{b}} = - 2\widehat{\dot{\mathbf{i}}} + 3\widehat{\dot{\mathbf{j}}} - 4\widehat{\mathbf{k}}$
$and\ \overrightarrow{\mathbf{c}} = \lambda\widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}}$
$\therefore\ \ \left\lbrack \overrightarrow{\mathbf{a}},\overrightarrow{\mathbf{b}},\overrightarrow{\mathbf{c}} \right\rbrack = 0$
$\Longrightarrow \ \left| \begin{matrix}
1 & - 2 & 3 \\
- 2 & 3 & - 4 \\
\lambda & - 1 & 2
\end{matrix} \right| = 0$
$\Longrightarrow 1(6 - 4) + 2( - 4 + 4\lambda) + 3(2 - 3\lambda) = 0$
$\Longrightarrow \ \lambda = 0$