The equation of the given line is $\frac{\mathit{x}-2}{2}$ = $\frac{\mathit{y}+1}{4}$ = $\frac{\mathit{z}-2}{2}$ ... (1)
Any point on the given line is (3λ + 2, 4λ - 1 , 2λ + 2)
If this point lies on the given plane x – y + z – 5 = 0, then
3λ + 2 -(4λ - 1) + 2λ + 2 - 5 = 0
⇒ λ = 0
Putting λ = 0 in (3λ + 2, 4λ - 1 , 2λ + 2) , we get the point of intersection of the given line and the plane is (2, -1, 2).
Let θ be the angle between the given line and the plane
∴ sin θ = $\frac{\overrightarrow{\mathit{a}}.\overrightarrow{\mathit{b}}}{\left|\overrightarrow{\mathit{a}}\right||\overrightarrow{\mathit{b}}|}$ = $\frac{\left(3\hat{\mathit{i}}+4\hat{\mathit{j}}+2\hat{\mathit{k}}\right).\left(\hat{\mathit{i}}-\hat{\mathit{j}}+\hat{\mathit{k}}\right)}{\sqrt{3^2+4^2+2^2}\sqrt{1^2+1^2+1^2}}$ = $\frac{3-4+2}{\sqrt{29}\sqrt{3}}$ = $\frac{1}{\sqrt{87}}$
⇒ θ = $\mathit{s}\mathit{i}{\mathit{n}}^{-1}\left(\frac{1}{\sqrt{87}}\right)$
Thus, the angle between the given line and the given plane is $\mathit{s}\mathit{i}{\mathit{n}}^{-1}\left(\frac{1}{\sqrt{87}}\right)$
OR
The equation of the given planes are
$\overrightarrow{\mathit{r}}.\hat{\mathit{i}}+2\hat{\mathit{j}}+3\hat{\mathit{k}}-4$ = 0 ... (1)
$\overrightarrow{\mathit{r}}.2\hat{\mathit{i}}+\hat{\mathit{j}}-\hat{\mathit{k}}+5$ = 0 ... (2)
The equation of the plane passing through the intersection of the planes (1) and (2) is
$\left|\overrightarrow{\mathit{r}}.\hat{\mathit{i}}+2\hat{\mathit{j}}+3\hat{\mathit{k}}-4\right|$ + λ $\left|\overrightarrow{\mathit{r}}.2\hat{\mathit{i}}+\hat{\mathit{j}}-\hat{\mathit{k}}+5\right|$ = 4 - 5λ ... (3)
Given that plane (3) is perpendicular to the plane $\overrightarrow{\mathit{r}}.5\hat{\mathit{i}}+3\hat{\mathit{j}}-6\hat{\mathit{k}}+8$ = 0
1 + 2λ × 5 + 2 + λ × 3 + 3 - λ × - 6 = 0
⇒ 19λ - 7 = 0
⇒ λ = $\frac{7}{19}$
Putting λ = $\frac{7}{19}$ in (3), we get
= 4 - $\frac{35}{19}$
⇒ $\overrightarrow{\mathit{r}}.\left(\frac{33}{19}\hat{\mathit{i}}+\frac{45}{19}\hat{\mathit{j}}+\frac{50}{19}\hat{\mathit{k}}\right)$ = $\frac{41}{19}$
⇒ $\overrightarrow{\mathit{r}}.33\hat{\mathit{i}}+45\hat{\mathit{j}}+50\hat{\mathit{k}}$ = 41. This is the equation of the required plane.