Which of the following cannot be the direction ratios of the straight…

The correct answer is Option (3) → 2, 3, −1

In the standard form \(\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\), every numerator must be (variable − number). Here the middle term has 2 − y, so write it as

    \( \displaystyle \frac{2-y}{3} = \frac{y-2}{-3} \)

The line is

    \( \displaystyle \frac{x-3}{2}=\frac{y-2}{-3}=\frac{z+4}{-1} \)

So its direction ratios are 2, −3, −1, or any set proportional to them. Checking the options:

    \(2, -3, -1\): the same set
    \(-2, 3, 1 = -1 \times (2, -3, -1)\)
    \(2, 3, -1\): ratios \(\frac{2}{2} = 1\) but \(\frac{3}{-3} = -1\), not proportional
    \(6, -9, -3 = 3 \times (2, -3, -1)\)

So 2, 3, −1 cannot be direction ratios of the line.