Show that the points (2, 3, 4), (–1, –2, 1), (5, 8, 7) are collinear.

Home » Show that the points (2, 3, 4), (–1, –2, 1), (5, 8, 7) are collinear.

Show that the points (2, 3, 4), (–1, –2, 1), (5, 8, 7) are collinear.

Solution:

If the direction ratios of two lines segments are proportional, then the lines are collinear.
It is given that
$\mathrm{A}(2,3,4), \mathrm{B}(-1,-2,1), \mathrm{C}(5,8,7)$
The direction ratio of line joining $\mathrm{A}(2,3,4)$ and $\mathrm{B}(-1,-2,1)$ , are
$(-1-2),(-2-3),(1-4)=(-3,-5,-3)$
Where, $a_{1}=-3, b_{1}=-5, c_{1}=-3$
The direction ratio of line joining $B(-1,-2,1)$ and $C(5,8,7)$ are
$(5-(-1)),(8-(-2)),(7-1)=(6,10,6)$
Where, $a_{2}=6, b_{2}=10$ and $c_{2}=6$
As a result, it is clear that the direction ratios of AB and BC are of same proportions.
By
$\begin{array}{l} \frac{a_{1}}{a_{2}}=\frac{-3}{6}=-2 \\ \frac{b_{1}}{b_{2}}=\frac{-5}{10}=-2 \end{array}$
And
$\frac{c_{1}}{c_{2}}=\frac{-3}{6}=-2$
As a result, $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are collinear.