If $(\hat{i} + \lambda\hat{j}) \times (5\hat{i} + 3\hat{j} + \sigma\hat{k}) = 0$, what are the values of $\lambda$ and $\sigma$?

$\lambda = \frac{3}{5}, \sigma = 0$

The correct answer is Option (1) → $\lambda = \frac{3}{5}, \sigma = 0$ ##

Given that, $(\hat{i} + \lambda\hat{j}) \times (5\hat{i} + 3\hat{j} + \sigma\hat{k}) = 0$

$\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & \lambda & 0 \\  5 & 3 & \sigma \end{vmatrix} = 0$

$(\lambda\sigma - 0)\hat{i} - (\sigma - 0)\hat{j} + (3 - 5\lambda)\hat{k} = 0$

$\lambda\sigma\hat{i} - \sigma\hat{j} + (3 - 5\lambda)\hat{k} = 0\hat{i} + 0\hat{j} + 0\hat{k}$

Comparing both sides:

$\lambda\sigma = 0 \quad ...{i}$

$\sigma = 0 \quad ...{ii}$

$3 - 5\lambda = 0 \quad ...{iii}$

Solving above equations: $\sigma = 0, \lambda = \frac{3}{5}$.