The acute angle between the line \(\bar{r}=(\hat{\imath}+2 \hat{\jmath}+\hat{k})+\lambda(\hat{\imath}+\hat{\jmath}+\hat{k})\) and the plane
\(\bar{r} \cdot(2 \hat{\imath}-\hat{\jmath}+\hat{k})=5\) is

Given \(\overline{\mathrm{r}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})\) and the plane \(\overline{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}})=5\)
The angle between the line \(\overline{\mathrm{r}}=\overline{\mathrm{a}}+\lambda \overline{\mathrm{b}}\) and the plane \(\overline{\mathrm{r}} \cdot \overline{\mathrm{n}}=p\) is given by \(\sin \theta=\frac{\overline{\mathrm{b}} \cdot \overline{\mathrm{n}}}{|\overline{\mathrm{b}}| \overline{\mathrm{n}} \mid}\)
Here \(\bar{b}=\hat{i}+\hat{j}+\hat{k} \Rightarrow|\bar{b}|=\sqrt{3}\) and \(\bar{n}=2 \hat{i}-\hat{j}+\hat{k} \Rightarrow|\bar{n}|=\sqrt{6}\)
Here \(\bar{b} \cdot \bar{n}=(\hat{i}+\hat{j}+\hat{k}) \cdot(2 \hat{i}-\hat{j}+\hat{k})=2-1+1=2\)
\(\therefore \sin \theta=\frac{2}{\sqrt{3} \sqrt{6}}=\frac{2}{\sqrt{3} \times \sqrt{3} \times \sqrt{2}} \Rightarrow \sin \theta=\frac{\sqrt{2}}{3} \Rightarrow \theta=\) \(\sin ^{-1} \frac{\sqrt{2}}{3}\)