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).

Home » 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).

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:

The points $(4,7,8),(2,3,4)$ and $(-1,-2,1),(1,2,5)$ .
Consider $A B$ be the line joining the points, $(4,7,8),(2,3,4)$ and $C D$ be the line through the points $(-1,-2$ , 1), $(1,2,5)$ .
So now,
The direction ratios, $a_{1}, b_{1}, c_{1}$ of $A B$ are
$(2-4),(3-7),(4-8)=-2,-4,-4$
Direction ratios, $\mathrm{a}_{2}, \mathrm{~b}_{2}, \mathrm{c}_{2}$ of $\mathrm{CD}$ are
$(1-(-1)),(2-(-2)),(5-1)=2,4,4$
Then $\mathrm{AB}$ will be parallel to $\mathrm{CD}$ , if
$\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$
Therefore, $a_{1} / a_{2}=-2 / 2=-1$
$\begin{array}{l} \mathrm{b}_{1} / \mathrm{b}_{2}=-4 / 4=-1 \\ \mathrm{c}_{1} / \mathrm{c}_{2}=-4 / 4=-1 \end{array}$
Therefore, we can say that,
$\begin{array}{c} \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}} \\ -1=-1=-1 \end{array}$
As a result, $\mathrm{AB}$ is parallel to $\mathrm{CD}$ where the line through the points $(4,7,8),(2,3,4)$ is parallel to the line through the points $(-1,-2,1),(1,2,5)$