Let \(\mathbf{A}\) be vector parallel to line of intersection of planes \(P_1\) and \(P_2\) through origin. \(P_1\) is parallel to the vectors \(2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) and \(4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}\) and \(P_2\) is parallel to \(\hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\), then the angle between vector \(\mathbf{A}\) and \(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}\) is

Let vector \(\mathbf{A O}\) be parallel to line of intersection of planes \(P_1\) and \(P_2\) through, i.e.
\[
\begin{aligned}
& {[(2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \times(4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})] \times[(\hat{\mathbf{j}}-\hat{\mathbf{k}}) \times(3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}})]=54(\hat{\mathbf{j}}-\hat{\mathbf{k}}) .} \\
& \therefore \text { Angle between } 54(\hat{\mathbf{j}}-\hat{\mathbf{k}}) \text { and }(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}) \\
& \Rightarrow \cos \theta=\pm\left(\frac{54+108}{3.54 \cdot \sqrt{2}}\right)=\pm \frac{1}{\sqrt{2}} \\
& \therefore \quad \theta=\frac{\pi}{4}, \frac{3 \pi}{4}
\end{aligned}
\]