The direction ratios of the line perpendicular to the lines \(\displaystyle\frac{x-5}{2}=\frac{y+11}{-3}=\frac{z+3}{1}\) and \(\displaystyle\frac{x-7}{1}=\frac{y+2}{2}=\frac{z-4}{-2}\) are proportional to:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (4, 5, 7)
The direction ratios of the lines are the denominators: (2, −3, 1) and (1, 2, −2). A line perpendicular to both is along their cross product:
\( \displaystyle \begin{vmatrix}\hat{\imath}&\hat{\jmath}&\hat{k}\\2&-3&1\\1&2&-2\end{vmatrix} \)
\(= \hat{\imath}(6 - 2) - \hat{\jmath}(-4 - 1) + \hat{k}(4 + 3)\)
\(= 4\hat{\imath} + 5\hat{\jmath} + 7\hat{k}\)
So the direction ratios are proportional to (4, 5, 7).
Shortcut: The required direction must have zero dot product with both (2, −3, 1) and (1, 2, −2):
Option (1): \(8 - 15 - 7 = -14 \ne 0\)
Option (2): \(-8 - 15 + 7 = -16 \ne 0\)
Option (3): \(8 + 15 + 7 = 30 \ne 0\)
Option (4): \(8 - 15 + 7 = 0\) and \(4 + 10 - 14 = 0\)