Explanation

(a)
Let $\overrightarrow{\mathbf{a}} = \widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}},\ \overrightarrow{\mathbf{b}} = \widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}}$
and $\overrightarrow{\mathbf{c}} = \widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} + \lambda\widehat{\mathbf{k}}$
$Since,\ volume\ of\ tetrahedron = \frac{1}{6}\left\lbrack \overrightarrow{\mathbf{a}}\ \overrightarrow{\mathbf{b}}\ \overrightarrow{\mathbf{c}} \right\rbrack$
$\Longrightarrow \frac{2}{3} = \frac{1}{6}\left| \begin{matrix}
1 & 2 & - 1 \\
1 & 1 & 1 \\
1 & - 1 & \lambda
\end{matrix} \right|$
$\Longrightarrow \frac{2}{3} = \frac{1}{6}\left\lbrack 1(\lambda + 1) - 2(\lambda - 1) - 1( - 1 - 1) \right\rbrack$
$\Longrightarrow 4 = \lbrack - \lambda + 5\rbrack$
$\Longrightarrow \ \lambda = 1$