Discussion
The angle between the lines \(\frac{x - 3}{-4} = \frac{y + 2}{3} = \frac{z - 1}{5}\) and \(\frac{x - 2}{2} = \frac{y - 4}{1} = \frac{z + 3}{3}\) is
Answer: E
The angle is \(\cos^{-1}\left(\frac{1}{\sqrt{7}}\right)\).
Use the direction ratios \((-4, 3, 5)\) and \((2, 1, 3)\):
\[ \begin{aligned} \cos\theta &= \frac{-8 + 3 + 15}{\sqrt{50}\,\sqrt{14}} = \frac{10}{\sqrt{700}} \\ &= \frac{10}{10\sqrt{7}} = \frac{1}{\sqrt{7}} \end{aligned} \]
The options with \(3\sqrt{7}\) or \(2\sqrt{7}\) come from errors in the magnitudes, \(\frac{2}{7}\) squares part of the result, and \(\sqrt{2}\) exceeds 1, so it cannot be a cosine.
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