The equation of the line passes through the point (2, -1, 3) and perpendicular to the lines
\( \displaystyle\frac{x-1}{1}=\displaystyle\frac{y}{2}=\displaystyle\frac{z+1}{-1} \) and \( \displaystyle\frac{x}{-1}=\displaystyle\frac{y}{1}=\displaystyle\frac{z-1}{2} \) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{2-x}{-5}=\displaystyle\frac{y+1}{-1}=\displaystyle\frac{z-3}{3} \)
The required line is perpendicular to both given lines, so its direction is along the cross product of their directions \( (1,2,-1) \) and \( (-1,1,2) \):
\( \displaystyle \begin{vmatrix}\hat{\imath} & \hat{\jmath} & \hat{k}\\ 1 & 2 & -1\\ -1 & 1 & 2\end{vmatrix}=5\hat{\imath}-\hat{\jmath}+3\hat{k} \)
The line passes through \( (2,-1,3) \) with direction ratios \( 5,\,-1,\,3 \):
\( \displaystyle \frac{x-2}{5}=\frac{y+1}{-1}=\frac{z-3}{3} \)
Option (3) is the same line, because \( \frac{2-x}{-5}=\frac{x-2}{5} \).