For a plane electromagnetic wave propagating in $x$-direction, which one of the following combination gives the correct possible directions for electric field $\left(E\right)$ and magnetic field $\left(B\right)$ respectively?

The correct option is C $-\hat{j}+\hat{k},-\hat{j}-\hat{k}$
Direction of propagation of electromagnetic waves is along $\overrightarrow{E}\times \overrightarrow{B}$.

Given that direction of propagation is along x-axis.

Checking all the options:

$\text{Option a:}(-\hat{j}+\hat{k})\times (-\hat{j}+\hat{k})$

$=[-\hat{j}\times (-\hat{j})-\hat{j}\times \hat{k}+\hat{k}\times (-\hat{j})+\hat{k}\times \hat{k}$

$=0-\hat{i}-(-\hat{i})+0=0$

Similarly,

$\text{Option b:}(\hat{j}+\hat{k})\times (\hat{j}+\hat{k})=0$

$\text{Option c:}(-\hat{j}+\hat{k})\times (-\hat{j}-\hat{k})=2\hat{i}$

$\text{Option d:}(\hat{j}+\hat{k})\times [-(\hat{j}+\hat{k}\left)\right]=0$

$\therefore$ Option $\left(C\right)$ is correct.