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Question
The lines \[\frac{x}{1} = \frac{y}{2} = \frac{z}{3} \text { and } \frac{x - 1}{- 2} = \frac{y - 2}{- 4} = \frac{z - 3}{- 6}\]
Options
parallel
intersecting
skew
coincident
Solution
coincident
The equations of the given lines are
\[\frac{x}{1} = \frac{y}{2} = \frac{z}{3} . . . (1)\]
\[\frac{x - 1}{- 2} = \frac{y - 2}{- 4} = \frac{z - 3}{- 6}\]
\[ \Rightarrow \frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z - 3}{3} . . . \left( 2 \right)\]
Thus, the two lines are parallel to the vector
\[\overrightarrow{b} = \hat{i} + 2 \hat{j} + 3 \hat{k} \] and pass through the points (0, 0, 0) and (1, 2, 3).
Now,
\[\left( \overrightarrow{a_2} - \overrightarrow{a_1} \right) \times \overrightarrow{b} = \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) \times \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right)\]
\[ = \overrightarrow{0} \left[ \because \overrightarrow{a} \times \overrightarrow{a} = \overrightarrow{0} \right]\]
Since, the distance between the two parallel lines is 0, the given two lines are coincident lines.
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