The direction ratios of the line passing through (1,2,3) and perpendicular to the lines \( \displaystyle\frac{x-1}{1}=\displaystyle\frac{y-2}{2}=\displaystyle\frac{z-3}{3} \) and \( \displaystyle\frac{x}{-3}=\displaystyle\frac{y}{2}=\displaystyle\frac{z}{5} \) are:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 2, −7, 4
A line perpendicular to two lines is parallel to the cross product of their direction vectors. Here the direction ratios are 1, 2, 3 and −3, 2, 5.
\( \displaystyle \vec{b}_1\times\vec{b}_2=\begin{vmatrix}\hat{\imath} & \hat{\jmath} & \hat{k}\\1 & 2 & 3\\-3 & 2 & 5\end{vmatrix} \)
\( \displaystyle =\hat{\imath}(10-6)-\hat{\jmath}(5+9)+\hat{k}(2+6) \)
\( \displaystyle =4\hat{\imath}-14\hat{\jmath}+8\hat{k} \)
Dividing by 2, the direction ratios are 2, −7, 4. (The point (1, 2, 3) fixes the position of the line, not its direction.)
Shortcut: Test each option for perpendicularity with both direction vectors (dot product = 0):
\( \displaystyle (2,-7,4)\cdot(1,2,3)=2-14+12=0 \)
\( \displaystyle (2,-7,4)\cdot(-3,2,5)=-6-14+20=0 \)
For the others: \( (1,2,3)\cdot(1,2,3)=14 \), \( (2,-7,-4)\cdot(1,2,3)=-24 \), \( (2,7,4)\cdot(1,2,3)=28 \); none is 0.