Let AB be the line through the points, (4, 7, 8) and (2, 3, 4), and CD be the line through the points,

Question:
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).

Solution:

Let AB be the line through the points, (4, 7, 8) and (2, 3, 4), and CD be the line through the points, (−1, −2, 1) and (1, 2, 5).

The directions ratios, $a_{1}, b_{1}, c_{1}$, of $A B$ are $(2-4),(3-7)$, and $(4-8)$ i.e., $-2,-4$, and $-4$

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

AB will be parallel to $\mathrm{CD}$, if $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$

$\frac{a_{1}}{a_{2}}=\frac{-2}{2}=-1$

$\frac{b_{1}}{b_{2}}=\frac{-4}{4}=-1$

$\frac{c_{1}}{c_{2}}=\frac{-4}{4}=-1$

$\therefore \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$

Thus, AB is parallel to CD.