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

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Solution

Three points A,B,C are collinear if direction ratios of AB and BC are proportional.
A( $2, 3, 4$ ) and B( $- 1, - 2, 1$ )
Direction ratios = $- 1 - 2, - 2 - 3, 1 - 4$
$= - 3, - 5, - 3$
So, $a_{1} = - 3, b_{1} = - 5, c_{1} = - 3$
B( $- 1, - 2, 1$ ) and C( $5, 8, 7$ )
Direction ratios = $5 - (- 1), 8 - (- 2), 7 - 1$
$= 6, 10, 6$
so, $a_{2} = 6, b_{2} = 10, c_{2} = 6$
Now,
$\begin{matrix} \frac{a_{2}}{a_{1}} = \frac{6}{- 3} = - 2 \frac{b_{2}}{b_{1}} = \frac{10}{- 5} = - 2 \frac{c_{2}}{c_{1}} = \frac{6}{- 3} = - 2 S i n c e \frac{a_{2}}{a_{1}} = \frac{b_{2}}{b_{1}} = \frac{c_{2}}{c_{1}} = - 2 \end{matrix}$
Therefore, A,B,C are collinear.

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