Find the vector equation of the line passing through $(1,2,3)$ and perpendicular to the plane $\vec r \cdot (\hat i + 2\hat j - 5\hat k) + 9 = 0$.

.: The given plane is $\vec r \cdot (\hat i + 2\hat j - 5\hat k) + 9 = 0$ ...(1)
$\therefore$ The direction ratios of the normal to the plane (1) are $< 1,2, - 5 >$

$\therefore$ The equation of the required line is

$\vec r = (\hat i + 2\hat j + 3\hat k) + \lambda (\hat i + 2\hat j - 5\hat k)$