$\text{ Let }\overline{a}\text{ be vector parallel to line intersection of planes }{P}_1\text{ and }{P}_2\text{ through origin. If }{P}_1\text{ is }$$\text{ parallel to the vectors }2\overline{j}+3\overline{k}\text{ and }4\overline{j}-3\overline{k}\text{ and }{P}_2\text{ is parallel to }\overline{j}-\overline{k}\text{ and }\overline{3i}+3\overline{j}\text{ , the }$$\text{ angle between }\overline{a}\text{ and }2\overline{i}+\overline{j}-2\overline{k}$

$\text{ Let }{\overline{n}}_1,{\overline{n}}_2\text{ be normal to planes }{P}_1\&P_2\text{ respectively. Then }$

$\overline{n_1}=\left(2\overline{j}+3\overline{k}\right)\times \left(4\overline{j}-3\overline{k}\right)=-6\overline{i}-12\overline{i}=-18\overline{i}$

$$\begin{gathered} and \overline{n_2}=(\overline{j}-\overline{k})\times (3\overline{i}+3\overline{j})=-3\overline{k}-3\overline{j}+3\overline{i} \\ \therefore \overline{a}=\overline{n_1}\times {\overline{n}}_2=\left|\begin{array}{ccc}\overline{i}& \overline{j}& \overline{k}\\ -18& 0& 0\\ 3& -3& -3\end{array}\right| \end{gathered}$$

$=54\left|\begin{array}{ccc}\overline{i}& \overline{j}& \overline{k}\\ 1& 0& 0\\ -1& 1& 1\end{array}\right|$

$=54(0.\overline{i}-\overline{j}+\overline{k})$

$\text{ Let }\theta \text{ be the angle between }-\overline{j}+\overline{k}\text{ and }2\overline{i}+\overline{j}-2\overline{k}\text{ then }$

$\cos \text{θ}=\frac{\left|0-1-2\right|}{\sqrt{2}.3}=\frac{1}{\sqrt{2}}$

$\text{ So, }\theta =\frac{\pi }{4}$

$\therefore \text{ Angle }=\frac{\pi }{4}$