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

If the direction ratios of two lines segments are proportional, then the lines are collinear.

Given A(2, 3, 4), B(−1, −2, 1), C(5, 8, 7)

Direction ratio of line joining A (2,3,4) and B (−1, −2, 1), are

(−1−2), (−2−3), (1−4)

= (−3, −5, −3)

So a_{1} = -3, b_{1} = -5, c_{1} = -3

Direction ratio of line joining B (−1, −2, 1) and C (5, 8, 7) are

(5− (−1)), (8−(−2)), (7−1)

= (6, 10, 6)

So a_{2} = 6, b_{2} = 10 and c_{2} =6

It is clear that the direction ratios of AB and BC are of same proportions

As

and

Therefore A, B, C are collinear.

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