The vector equation of line passing through (2, -1, 3) and perpendicular to the lines $\frac{x-2}{3}=\frac{y-1}{1}=\frac{z+2}{2}$ and $\frac{x+3}{-4}=\frac{y-5}{-3}=\frac{z+1}{2}$ is

(Here $λ$ is a parameter)

$\vec r= (2\hat i-\hat j+3\hat k) +λ (8\hat i-14\hat j-5\hat k)$

The correct answer is Option (1) → $\vec r= (2\hat i-\hat j+3\hat k) +λ (8\hat i-14\hat j-5\hat k)$

Given lines in symmetric form:

$\frac{x-2}{3}=\frac{y-1}{1}=\frac{z+2}{2}\;\Rightarrow\;$ direction $\vec{v}_{1}=\langle 3,1,2\rangle$

$\frac{x+3}{-4}=\frac{y-5}{-3}=\frac{z+1}{2}\;\Rightarrow\;$ direction $\vec{v}_{2}=\langle -4,-3,2\rangle$

Required line ⟂ to both ⇒ direction $\vec{v}=\vec{v}_{1}\times\vec{v}_{2}$

$\vec{v}=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\3&1&2\\-4&-3&2\end{vmatrix} =8\hat{i}-14\hat{j}-5\hat{k}$

Passing through $(2,-1,3)$:

$\vec{r}=(2\hat{i}-\hat{j}+3\hat{k})+\lambda\,(8\hat{i}-14\hat{j}-5\hat{k})$