#### Question

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

#### Solution

The given points are A (2, 3, 4), B (− 1, − 2, 1), and C (5, 8, 7).

It is known that the direction ratios of line joining the points, (*x*_{1}, *y*_{1}, *z*_{1}) and (*x*_{2}, *y*_{2}, *z*_{2}), are given by, *x*_{2} − *x*_{1}, *y*_{2} − *y*_{1}, and *z*_{2} − *z*_{1}.

The direction ratios of AB are (−1 − 2), (−2 − 3), and (1 − 4) i.e., −3, −5, and −3.

The direction ratios of BC are (5 − (− 1)), (8 − (− 2)), and (7 − 1) i.e., 6, 10, and 6.

It can be seen that the direction ratios of BC are −2 times that of AB i.e., they are proportional.

Therefore, AB is parallel to BC. Since point B is common to both AB and BC, points A, B, and C are collinear.

Is there an error in this question or solution?

Solution Show that the Points (2, 3, 4), (−1, −2, 1), (5, 8, 7) Are Collinear. Concept: Direction Cosines and Direction Ratios of a Line.