Find the direction cosines of the line passing through two points $(2, 1, 0)$ and $(1, -2, 3)$.

$\frac{-1}{\sqrt{19}}, \frac{-3}{\sqrt{19}}, \frac{3}{\sqrt{19}}$

The correct answer is Option (2) → $\frac{-1}{\sqrt{19}}, \frac{-3}{\sqrt{19}}, \frac{3}{\sqrt{19}}$ ##

Here $P(2, 1, 0)$ and $Q(1, -2, 3)$.

So, $PQ = \sqrt{(1 - 2)^2 + (-2 - 1)^2 + (3 - 0)^2}$

$PQ = \sqrt{(-1)^2 + (-3)^2 + (3)^2} = \sqrt{1 + 9 + 9} = \sqrt{19}$

Thus, the direction cosines of the line joining the two points are

$< \frac{1 - 2}{\sqrt{19}}, \frac{-2 - 1}{\sqrt{19}}, \frac{3 - 0}{\sqrt{19}} >\text{i.e., } < \frac{-1}{\sqrt{19}}, \frac{-3}{\sqrt{19}}, \frac{3}{\sqrt{19}} >$