Direction ratios of the line perpendicular to the lines \( \displaystyle\frac{x-3}{2}=\frac{y+7}{-3}=\frac{z-2}{1} \) and \( \displaystyle\frac{x+2}{1}=\frac{y+3}{2}=\frac{z-5}{-2} \) are
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 4, 5, 7
The direction ratios of the lines are the denominators: \( \vec{b}_1=2\hat{\imath}-3\hat{\jmath}+\hat{k} \) and \( \vec{b}_2=\hat{\imath}+2\hat{\jmath}-2\hat{k} \). A line perpendicular to both is along their cross product:
\( \displaystyle \vec{b}_1\times\vec{b}_2=\begin{vmatrix}\hat{\imath} & \hat{\jmath} & \hat{k}\\2 & -3 & 1\\1 & 2 & -2\end{vmatrix} \)
\( \displaystyle =(6-2)\hat{\imath}-(-4-1)\hat{\jmath}+(4+3)\hat{k} \)
\( \displaystyle =4\hat{\imath}+5\hat{\jmath}+7\hat{k} \)
So the direction ratios are 4, 5, 7.
Shortcut: The dot product with (2, −3, 1) must be 0: Option (1) gives 8 + 15 + 7 = 30, Option (2) gives −8 − 15 + 7 = −16, Option (3) gives 8 − 15 − 7 = −14, and only Option (4) gives 8 − 15 + 7 = 0.
Tip: Keep the minus sign on the middle term of the cross product; dropping it gives 4, −5, 7.