A vector perpendicular to the line $\vec{r} = \hat{i} + \hat{j} - \hat{k} + \lambda(3\hat{i} - \hat{j})$ is

$\hat{i} + 3\hat{j} + 5\hat{k}$

The correct answer is Option (2) → $\hat{i} + 3\hat{j} + 5\hat{k}$ ##

$\vec{r} = \hat{i} + \hat{j} - \hat{k} + \lambda(3\hat{i} - \hat{j} + 0\hat{k})$

$a_1 = 3, b_1 = -1, c_1 = 0$

On checking options one by one taking option (B) as:

$a_2 = 1, b_2 = 3, c_2 = 5$

We know that, vector perpendicular to line, only when

$a_1a_2 + b_1b_2 + c_1c_2 = 0$

$= (3)(1) + (-1)(3) + (0)(5)$

$= 3 - 3 + 0 = 0$