#### Question

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

#### Solution

\[\text{ Suppose the points are A } \left( 2, 3, 4 \right), B \left( - 1 . - 2, 1 \right) \text { and } C \left( 5, 8, 7 \right) . \]

\[\text { We know that the direction ratios of the line joining the points } \left( x_1 , y_1 , z_1 \right) \text{ and } \left( x_2 , y_2 , z_2 \right) \text{ are } x_2 - x_1 , y_2 - y_1 , z_2 - z_1 . \]

\[\text{ The direction ratios of AB are } \left( - 1 - 2 \right), \left( - 2 - 3 \right), \left( 1 - 4 \right), \text{ i . e }. - 3, - 5, - 3 . \]

\[\text{ The direction ratios of BC are } \left( 5 - \left( - 1 \right) \right), \left( 8 - \left( - 2 \right) \right), \left( 7 - 1 \right), \text { i . e } . 6, 10, 6 . \]

\[ \text{ It can be seen that the direction ratios of BC are - 2 times that of AB, i . e . they are proportional . Therefore, AB is parallel to BC }. \]

\[\text { Since point B is common in both AB and BC, points A, B, and C are collinear } .\]