Show that the points (2, 3, 4), (-1, -2, 1), (5, 8, 7) are collin… - Vidyadip

Question

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

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Answer

Suppose the points are A(2, 3, 4), B(-1, -2, 1) and C(5, 8, 7).We know that the direction ratios of the line joining the points $\left(\mathrm{x}_1, \mathrm{y}_1, \mathrm{z}_1\right)$ and $\left(\mathrm{x}_2, \mathrm{y}_2, \mathrm{z}_2\right)$ are $\mathrm{x}_2-\mathrm{x}_1, \mathrm{y}_2-\mathrm{y}_1, \mathrm{z}_2-\mathrm{z}_1$
The direction ratios of AB are (-1 - 2), (-2 - 3), (1 - 4),
i.e. -3, -5, -3.
The direction ratios of BC are (5 - (-1)), (8 - (-2)), (7 - 1),
i.e. 6, 10, 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 in both AB and BC, points A, B, and C are collinear.

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