Explanation

(d) If the three vectors are coplanar then their scalar triple product is zero. So ($\overrightarrow{A} \times \overrightarrow{C}$).$\overrightarrow{B} = 0$
or $\left\lbrack \left( 2\widehat{i} + 3\widehat{j} - 2\widehat{k} \right) \times \left( - \widehat{i} + 2\widehat{j} + 3\widehat{k} \right) \right\rbrack.\left\lbrack 5\widehat{i} + a\widehat{j} + \widehat{k} \right\rbrack = 0$
or $\left\lbrack 13\widehat{i} - 4\widehat{j} + 7\widehat{k} \right\rbrack.\left\lbrack 5\widehat{i} + a\widehat{j} + 5\widehat{k} \right\rbrack = 0$
or $65 - 4a + 7 = 0$ or $a = 18$