In a certain region of space, a plane electromagnetic wave is...

The direction of propagation of an electromagnetic wave is given by the direction of the Poynting vector, which is parallel to $\vec{E} \times \vec{B}$.

Since the wave is traveling along the negative $z$-axis, the direction of $\vec{E} \times \vec{B}$ must be $-\hat{k}$.

Also, the electric field $\vec{E}$ and magnetic field $\vec{B}$ must be mutually perpendicular, so $\vec{E} \cdot \vec{B} = 0$.

Checking the given options:

For $\vec{E} = \hat{i} + \hat{j}$ and $\vec{B} = \hat{i} - \hat{j}$:
$\vec{E} \cdot \vec{B} = (1)(1) + (1)(-1) = 0$
$\vec{E} \times \vec{B} = (\hat{i} + \hat{j}) \times (\hat{i} - \hat{j}) = -\hat{i} \times \hat{j} + \hat{j} \times \hat{i} = -\hat{k} - \hat{k} = -2\hat{k}$

The direction of $\vec{E} \times \vec{B}$ is along $-\hat{k}$, which corresponds to the negative $z$-axis.

Answer: $\hat{i} + \hat{j} , \ \hat{i} - \hat{j}$