Show that the line through the points $(4,7,8),(2,3,4)$ is parallel to the line through the points $( - 1, - 2,1),(1,2,5).$

Let the given points be $A(4,7,8),B(2,3,4),C( - 1, - 2,1)$ and $D(1,2,5)$

Direction ratios of AB are$< 2 - 4,3 - 7,4 - 8 >$or $< - 2, - 4, - 4 >$and

direction ratios of CD are $< 1 + 1,2 + 2,5 - 1 >$ or $< 2,4,4 >$

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

are parallel,
if $\cfrac{{{a_1}}}{{{a_2}}} = \cfrac{{{b_1}}}{{{b_2}}}$

$= \cfrac{{{c_1}}}{{{c_2}}} \Rightarrow \cfrac{{ - 2}}{2} = \cfrac{{ - 4}}{4} = \cfrac{{ - 4}}{4}$ $\Rightarrow$ $AB||CD$