Discussion
The angle between the lines \(\frac{x - 3}{1} = \frac{y + 1}{-1} = \frac{z - 2}{-1}\) and \(\frac{x + 1}{2} = \frac{y - 2}{2} = \frac{z + 3}{-2}\) is
Answer: D
The angle is \(\cos^{-1}\left(\frac{1}{3}\right)\).
The direction ratios are (1, -1, -1) and (2, 2, -2):
\[ \begin{aligned} \cos\theta &= \frac{|2 - 2 + 2|}{\sqrt{3}\,\sqrt{12}} \\ &= \frac{2}{6} = \frac{1}{3} \end{aligned} \]
The dot product is 2 and the product of the lengths is \(\sqrt{3} \times 2\sqrt{3} = 6\), so only \(\frac{1}{3}\) fits. The surd options come from slips in the lengths, such as taking \(|(2, 2, -2)|\) as \(\sqrt{8}\) or \(\sqrt{6}\).
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