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Three Dimensional Geometry — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThree Dimensional Geometry5 Marks
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 $(x_1, y_1, z_1)$ 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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