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

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Solution

Direction ratios of a line segment joining two points $(x_{1}, y_{1}, z_{1})$ and $(x_{2}, y_{2}, z_{2})$ are
$(x_{2} - x_{1}), (y_{2} - y_{1}), (z_{2} - z_{1})$

Direction ratios of line joining $A (2, 3, 4)$ and $B (- 1, - 2, 1)$ are
$- 1 - 2, - 2 - 3, 1 - 4$

i.e., $- 3, - 5, - 3$

Direction ratios of line joining $B (- 1, - 2, 1)$ and $C (5, 8, 7)$ are
$5 + 1, 8 + 2, 7 - 1$

i.e., $6, 10, 6$

Direction ratios of $A B$ and $B C$ are proportional, so $A B$ is parallel to $B C$ .
But $B$ is common to $A B$ and $B C$ , So, $A, B, C$ are collinear points.

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