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