Discussion
The straight line passing through the points (3,2,3) and (5,-1,-2) is perpendicular to the straight line passing through the points (1,3,1) and \((\alpha, \alpha, \alpha)\). Then the value of \(\alpha\) is equal to
Answer: B
\(\alpha = 2\).
Two lines are perpendicular when the dot product of their direction ratios is zero.
\[ \begin{aligned} \vec{d_{1}} &= (5 - 3, -1 - 2, -2 - 3) = (2, -3, -5) \\ \vec{d_{2}} &= (\alpha - 1, \alpha - 3, \alpha - 1) \end{aligned} \]
\[ \begin{aligned} 2(\alpha - 1) - 3(\alpha - 3) - 5(\alpha - 1) &= 0 \\ -6\alpha + 12 &= 0 \\ \alpha &= 2 \end{aligned} \]
Check: with \(\alpha = 2\), \(\vec{d_{2}} = (1, -1, 1)\) and \(2 + 3 - 5 = 0\).
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