Show that the line through the points (1, −1, 2) (3, 4, −2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Let AB be the line joining the points, (1, −1, 2) and (3, 4, − 2), and CD be the line joining the points, (0, 3, 2) and (3, 5, 6).

The direction ratios, $a_{1}, b_{1}, c_{1}$, of AB are $(3-1),(4-(-1))$, and $(-2-2)$ i.e., 2,5, and $-4$.

The direction ratios, $a_{2}, b_{2}, c_{2}$, of CD are $(3-0),(5-3)$, and $(6-2)$ i.e., 3,2 , and 4 .

$A B$ and $C D$ will be perpendicular to each other, if $a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0$

$a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=2 \times 3+5 \times 2+(-4) \times 4$

$=6+10-16$

$=0$

Therefore, AB and CD are perpendicular to each other.