Explanation

(b)
Let $P$ be the point and it divides the line segment in the ratio $\lambda:1$. Then,
$\overrightarrow{OP} = \overrightarrow{r} = \frac{- 3\lambda + 2}{\lambda + 1}\widehat{i} + \frac{5\lambda - 4}{\lambda + 1}\widehat{j} + \frac{- 8\lambda - 7}{\lambda + 1}\widehat{k}$
It satisfies $\overrightarrow{r}.\left( \widehat{i} - 2\widehat{j} + 3\widehat{k} \right) = 13$. So,
$\frac{- 3\lambda + 2}{\lambda + 1} - 2\frac{5\lambda - 4}{\lambda + 1} + 3\frac{- 8\lambda - 7}{\lambda + 1} = 13$
or $- 3\lambda + 2 - 2(5\lambda - 4) + 3( - 8\lambda - 7) = 13(\lambda + 1)$
or $- 37\lambda - 11 = 13\lambda + 13$ or $50\lambda = - 24$ or $\lambda = - \frac{12}{25}$