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

(N/A) Let $AB$ be the line joining the points $(1, -1, 2)$ and $(3, 4, -2)$,and $CD$ be the line through the points $(0, 3, 2)$ and $(3, 5, 6)$.
The direction ratios $(a_1, b_1, c_1)$ of line $AB$ are given by $(x_2-x_1, y_2-y_1, z_2-z_1) = (3-1, 4-(-1), -2-2) = (2, 5, -4)$.
The direction ratios $(a_2, b_2, c_2)$ of line $CD$ are given by $(x_4-x_3, y_4-y_3, z_4-z_3) = (3-0, 5-3, 6-2) = (3, 2, 4)$.
Two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ are perpendicular if $a_1 a_2 + b_1 b_2 + c_1 c_2 = 0$.
Calculating the sum of products of direction ratios:
$a_1 a_2 + b_1 b_2 + c_1 c_2 = (2)(3) + (5)(2) + (-4)(4)$
$= 6 + 10 - 16$
$= 16 - 16$
$= 0$.
Since the sum is $0$,the line $AB$ is perpendicular to the line $CD$.