Find the direction cosines of the vector joining the points $A(1,2, - 3)$ and $B( - 1, - 2,1)$ directed from $A$ to $B$.

Vector joining the points $A$ and $B$,
$\overrightarrow {AB} =$Position vector of $B -$position vector of $A$
$A = ( - \hat i - 2\hat j + \hat k) - (\hat i + 2\hat j - 3\hat k)$

$= - 2\hat i - 4\hat j + 4\hat k$
$\therefore$

Direction cosines of $\overrightarrow {AB}$ are
$< \cfrac{{ - 2}}{{\sqrt {4 + 16 + 16} }},\cfrac{{ - 4}}{{\sqrt {4 + 16 + 16} }},\cfrac{4}{{\sqrt {4 + 16 + 16} }} >$

i.e., $< \cfrac{{ - 2}}{6},\cfrac{{ - 4}}{6},\cfrac{4}{6} >$ i.e., $< \cfrac{{ - 1}}{3},\cfrac{{ - 2}}{3},\cfrac{2}{3} >$.