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Question
If the lines \[\frac{x - 1}{- 3} = \frac{y - 2}{2 \lambda} = \frac{z - 3}{2} \text{ and } \frac{x - 1}{3\lambda} = \frac{y - 1}{1} = \frac{z - 6}{- 5}\] are perpendicular, find the value of λ.
Solution
The equations of the given lines are
\[\frac{x - 1}{- 3} = \frac{y - 2}{2 \lambda} = \frac{z - 3}{2}\]
\[\frac{x - 1}{3\lambda} = \frac{y - 1}{1} = \frac{z - 6}{- 5}\]
Since the given lines are perpendicular to each other, we have
\[- 3\left( 3\lambda \right) + 2\lambda\left( 1 \right) + 2\left( - 5 \right) = 0\]
\[ \Rightarrow - 9\lambda + 2\lambda - 10 = 0\]
\[ \Rightarrow \lambda = - \frac{10}{7}\]
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