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Three Dimensional Geometry — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThree Dimensional Geometry3 Marks
Question
Show that the points (2, 3, 4), (-1, -2, 1), (5, 8, 7) are collinear.
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Answer

The given points are $A\left( {2,3,4} \right),B\left( { - 1, - 2,1} \right)$and $C\left( {5,8,7} \right)$
$\therefore$ Direction ratios of the line joining A and B are
$- 1 - 2, - 2 - 3,1 - 4\,\left[ {\because {x_2} - {x_1},{y_2} - {y_1},{z_2} - {z_1}} \right]$ …(i)
$ \Rightarrow - 3, - 5, - 3 = {a_1},{b_1},{c_1}$ (say)
Again Direction ratios of the line joining B and C are
$5 - \left( { - 1} \right),8 - \left( { - 2} \right),7 - 1 = 6,10,6 = {a_2},{b_2},{c_2}$   (say) ….(ii)
From eq. (i) and (ii),
$\frac{{ - 3}}{6} = \frac{{ - 1}}{2},\frac{{ - 5}}{{10}} = \frac{{ - 1}}{2},\frac{{ - 3}}{6} = \frac{{ - 1}}{2}$
$\Rightarrow \frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}} = \frac{{{c_1}}}{{{c_2}}}$
Therefore, AB is parallel to BC.
But point B is common to both AB and BC. Hence points A, B, C are collinear.

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