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Question
Find the value of λ so that the following lines are perpendicular to each other. \[\frac{x - 5}{5\lambda + 2} = \frac{2 - y}{5} = \frac{1 - z}{- 1}, \frac{x}{1} = \frac{2y + 1}{4\lambda} = \frac{1 - z}{- 3}\]
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Solution
The equations of the given line \[\frac{x - 5}{5\lambda + 2} = \frac{2 - y}{5} = \frac{1 - z}{- 1} \text{ and } \frac{x}{1} = \frac{2y + 1}{4\lambda} = \frac{1 - z}{- 3}\]s
can be re-written as
\[\frac{x - 5}{5\lambda + 2} = \frac{y - 2}{- 5} = \frac{z - 1}{1} \text{ and } \frac{x}{1} = \frac{y + \frac{1}{2}}{2\lambda} = \frac{z - 1}{3}\] Since the given lines are perpendicular to each other,
we have
\[\left( 5\lambda + 2 \right)1 - 5\left( 2\lambda \right) + 1\left( 3 \right) = 0\]
\[ \Rightarrow 5\lambda = 5\]
\[ \Rightarrow \lambda = 1\]
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