Show that the points $(2, 3, 4), (-1, -2, 1), (5, 8, 7)$ are collinear.
✓
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 $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are $x_2- x_{1,}y_2- y_{1,}z_2- z_{1.}$
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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