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Question
Using direction ratios show that the points A (2, 3, −4), B (1, −2, 3) and C (3, 8, −11) are collinear.
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Solution
\[\text{The given points are} \text{ A }\left( 2, 3, - 4 \right), B\left( 1, - 2, 3 \right) \text{and}\ C \left( 3, 8, - 11 \right) . \]
\[\text{We know that the direction ratios of the line joining the points, } \left( x_1 , y_1 , z_1 \right) \text{and}\ \left( x_2 , y_2 , z_2 \right) \text{are } \ x_2 - x_1 , y_2 - y_1 , z_2 - z_1 . \]
\[\text{The direction ratios of the line joining A and B are } 1 - 2, - 2 - 3, 3 + 4,\text{ i . e }. - 1, - 5, 7 . \]
\[\text{The direction ratios of the line joining B and C are } 3 - 1, 8 + 2, - 11 - 3, \text{i . e }. 2, 10, - 14 . \]
\[\text {It is clear that the direction ratios of BC are - 2 times that of AB, i . e . they are proportional . }\]
\[\text{Therefore, AB is parallel to BC . }\]
\[\text{Also, point B is common in both AB and BC . }\]
\[\text{Therefore, points A, B and C are collinear .}\]
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