Explanation

(b)
We know that a vector perpendicular to the plane containing the points $\overrightarrow{\mathbf{A}},\overrightarrow{\mathbf{B}}\ ,\overrightarrow{\mathbf{C}}$ is given by $\overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}} + \overrightarrow{\mathbf{B}} \times \overrightarrow{\mathbf{C}} + \overrightarrow{\mathbf{C}} \times \overrightarrow{\mathbf{A}}$.
Given, $\overrightarrow{\mathbf{A}} = \widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}},\ \overrightarrow{\mathbf{B}} = 2\widehat{\dot{\mathbf{i}}} + 0\widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}}$
and $\overrightarrow{\mathbf{C}} = 0\widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}}$
$Now,\ \$
$\overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}} = \left| \begin{matrix}
\widehat{\dot{\mathbf{i}}} & \widehat{\dot{\mathbf{j}}} & \widehat{\mathbf{k}} \\
1 & - 1 & 2 \\
2 & 0 & - 1
\end{matrix} \right| = \widehat{\dot{\mathbf{i}}} + 5\widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}}\ \$
$\overrightarrow{\mathbf{B}} \times \overrightarrow{\mathbf{C}} = \left| \begin{matrix}
\widehat{\dot{\mathbf{i}}} & \widehat{\dot{\mathbf{j}}} & \widehat{\mathbf{k}} \\
2 & 0 & - 1 \\
0 & 2 & 1
\end{matrix} \right| = 2\widehat{\dot{\mathbf{i}}} - 2\widehat{\dot{\mathbf{j}}} + 4\widehat{\mathbf{k}}$
$\overrightarrow{\mathbf{C}} \times \overrightarrow{\mathbf{A}} = \left| \begin{matrix}
\widehat{\dot{\mathbf{i}}} & \widehat{\dot{\mathbf{j}}} & \widehat{\mathbf{k}} \\
0 & 2 & 1 \\
1 & - 1 & 2
\end{matrix} \right| = 5\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} - 2\widehat{\mathbf{k}}$
Thus,
$\overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}} + \overrightarrow{\mathbf{B}} \times \overrightarrow{\mathbf{C}} + \overrightarrow{\mathbf{C}} \times \overrightarrow{\mathbf{A}}$
$= \left( \widehat{\dot{\mathbf{i}}} + 5\widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}} \right) + \left( 2\widehat{\dot{\mathbf{i}}} - 2\widehat{\dot{\mathbf{j}}} + 4\widehat{\mathbf{k}} \right) + \left( 5\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} - 2\widehat{\mathbf{k}} \right)$
=$8\widehat{\dot{\mathbf{i}}} + 4\widehat{\dot{\mathbf{j}}} + 4\widehat{\mathbf{k}}\$