The equations of the given planes are
$\overrightarrow{\mathit{r}}.\left(\hat{\mathit{i}}+2\hat{\mathit{j}}+3\hat{\mathit{k}}\right)$ - 4 = 0 ... (1)
$\overrightarrow{\mathit{r}}.\left(2\hat{\mathit{i}}+\hat{\mathit{j}}-\hat{\mathit{k}}\right)$ + 5 = 0 ... (2)
The equation of the plane passing through the line of intersection of the given planes is
$\left[\overrightarrow{\mathit{r}}.\left(\hat{\mathit{i}}+2\hat{\mathit{j}}+3\hat{\mathit{k}}\right)-4\right]$ + λ $\left[\overrightarrow{\mathit{r}}.\left(2\hat{\mathit{i}}+\hat{\mathit{j}}-\hat{\mathit{k}}\right)+5\right]$ = 0
$\overrightarrow{\mathit{r}}$ [(1 + 2λ) $\hat{\mathit{i}}$ + (2 + λ) $\hat{\mathit{j}}$ + (3 - λ) $\hat{\mathit{k}}$] + (- 4 + 5λ) = 0 ... (3)
The plane in equation (3) is perpendicular to the plane, $\overrightarrow{\mathit{r}}.\left(5\hat{\mathit{i}}+3\hat{\mathit{j}}-6\hat{\mathit{k}}\right)$ + 8 = 0
∴ 5 (1 + 2λ) + 3 (2 + λ) - 6 (3 - λ) = 0
⇒ 5 + 10λ + 6 + 3λ - 18 + 6λ = 0
⇒ 19λ - 7 = 0
⇒ λ = $\frac{7}{19}$
Substituting λ = $\frac{7}{19}$ in equation (3),
$\overrightarrow{\mathit{r}}.\left[\frac{33}{19}\hat{\mathit{i}}+\frac{45}{19}\hat{\mathit{j}}+\frac{50}{19}\hat{\mathit{k}}\right]$ - $\frac{41}{19}$ = 0
⇒ $\overrightarrow{\mathit{r}}.\left(33\hat{\mathit{i}}+45\hat{\mathit{j}}+50\hat{\mathit{k}}\right)$ - 41 = 0
This is the vector equation of the required plane.