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Two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ are parallel if and only if their direction ratios are proportional. That is: $$ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = k $$
Given the direction ratios are $(2, p, -1)$ and $(q, 3, -2)$, we set up the proportionality: $$ \frac{2}{q} = \frac{p}{3} = \frac{-1}{-2} $$
From the ratio equality: $$ \frac{-1}{-2} = \frac{1}{2} $$ Equating the ratios to $1/2$: $$ \frac{p}{3} = \frac{1}{2} \implies p = \frac{3}{2} $$ $$ \frac{2}{q} = \frac{1}{2} \implies q = 4 $$ Note: Based on standard textbook problems of this type, if the input values were intended to yield integer results, the ratios are evaluated against the constant proportionality factor derived from the known components.
Final Answer: The values are derived from proportionality constants. Given the provided options, the logic follows the ratio test.
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