Show that the points $(2,3,4), (-1,-2,1),$ and $(5,8,7)$ are collinear.
(N/A) Let the given points be $A(2,3,4), B(-1,-2,1),$ and $C(5,8,7).$
The direction ratios of the line segment $AB$ are given by $(x_2 - x_1, y_2 - y_1, z_2 - z_1).$
Direction ratios of $AB = (-1 - 2, -2 - 3, 1 - 4) = (-3, -5, -3).$
Direction ratios of $BC = (5 - (-1), 8 - (-2), 7 - 1) = (6, 10, 6).$
We observe that the direction ratios of $BC$ are $-2$ times the direction ratios of $AB$:
$(6, 10, 6) = -2 \times (-3, -5, -3).$
Since the direction ratios are proportional,the lines $AB$ and $BC$ are parallel.
Because point $B$ is common to both $AB$ and $BC,$ the points $A, B,$ and $C$ must lie on the same line.
Therefore,the points $(2,3,4), (-1,-2,1),$ and $(5,8,7)$ are collinear.
The direction ratios of the line segment $AB$ are given by $(x_2 - x_1, y_2 - y_1, z_2 - z_1).$
Direction ratios of $AB = (-1 - 2, -2 - 3, 1 - 4) = (-3, -5, -3).$
Direction ratios of $BC = (5 - (-1), 8 - (-2), 7 - 1) = (6, 10, 6).$
We observe that the direction ratios of $BC$ are $-2$ times the direction ratios of $AB$:
$(6, 10, 6) = -2 \times (-3, -5, -3).$
Since the direction ratios are proportional,the lines $AB$ and $BC$ are parallel.
Because point $B$ is common to both $AB$ and $BC,$ the points $A, B,$ and $C$ must lie on the same line.
Therefore,the points $(2,3,4), (-1,-2,1),$ and $(5,8,7)$ are collinear.
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