The ratio in which the plane $\vec{r}. (\hat{i}- 2\hat{j} + 3\hat{k}) = 17 $ divides the line joining points $-2\hat{i} + 4\hat{j} + 7 \hat{k} $ and $ 3\hat{i} - 5\hat{j} + 8 \hat{k}$ is

3 : 10

Let the required ratio be λ : 1. Then, the position vector of the point of division is

$\frac{(-2\hat{i} + 4\hat{j} + 7 \hat{k})+λ(3\hat{i} -5\hat{j} + 8 \hat{k})}{λ+1}$

$=\left(\frac{3λ-2}{λ+1}\right)\hat{i}+\left(\frac{4-5λ}{λ+1}\right)\hat{j}+\left(\frac{8λ+7}{λ+1}\right)\hat{k}$

The point of division lies on the plane.

$\vec{r}. (\hat{i} -2\hat{j} + 3 \hat{k})=17$

$⇒ \left(\frac{3λ-2}{λ+1}\right)\hat{i} -2\left(\frac{4-5λ}{λ+1}\right)\hat{j}+3\left(\frac{8λ+7}{λ+1}\right)\hat{k}=17$

$⇒ 3λ - 2 - 8 + 10 λ + 24 λ + 21 = 17(λ + 1) $

\(⇒ 3λ + 10λ + 24λ - 17λ = 17 + 2 + 8 - 21\)

\(⇒ 20λ =  6\)

\(⇒ λ = \frac{6}{20} = \frac{3}{10}\)

Hence, the required ratio is 3 : 10.