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 the given points be $A(1, - 1,2),B(3,4, - 2),C(0,3,2)$ and $D(3,5,6)$ .

The direction ratios of AB are $< 3 - 1,4 + 1, - 2 - 2 >$

or $< 2,5, - 4 >$ and the direction ratios of CD

are$< 3 - 0,5 - 3,6 - 2 >$ or $< 3,2,4 >$

Two lines with direction ratios ${a_1},{b_1},{c_1}$ and ${a_2},{b_2},{c_2}$

are perpendicular if ${a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2} = 0$

$\Rightarrow$ $2 \times 3 + 5 \times 2 + ( - 4) \times 4 = 0$

therefore AB and CD are perpendicular.