Discussion
If the lines \(\frac{2x - 1}{2} = \frac{3 - y}{1} = \frac{z - 1}{3}\) and \(\frac{x + 3}{2} = \frac{y + 2}{5} = \frac{z + 1}{a}\) are perpendicular to each other, then the value of \(a\) is equal to
Answer: E
\(a\) is 1.
Put the first line in standard form so that each numerator is just \(x\), \(y\) or \(z\) minus a constant:
\[ \begin{aligned} \frac{x - \frac{1}{2}}{1} &= \frac{y - 3}{-1} \\ &= \frac{z - 1}{3} \end{aligned} \]
Its direction ratios are \((1, -1, 3)\); those of the second line are \((2, 5, a)\). Perpendicular lines have zero dot product:
\[ \begin{aligned} 1(2) + (-1)(5) + 3a &= 0 \\ 3a &= 3, \quad a = 1 \end{aligned} \]
Reading the direction ratios directly from the printed denominators, (2, 1, 3), gives \(4 + 5 + 3a = 0\) and a negative value, which is not offered.
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