An electromagnetic wave is propagating along the positive z -axis....

The direction of propagation of an electromagnetic wave is given by the direction of $\vec{E} \times \vec{B}$.

Given that the wave propagates along the positive $z$-axis ($+\hat{k}$) and the electric field $\vec{E}$ is parallel to $(\hat{i} + \hat{j})$.

Let us check the cross product for the given options:

For option 1: $(\hat{i} + \hat{j}) \times (-\hat{i} + \hat{j}) = (\hat{i} \times -\hat{i}) + (\hat{i} \times \hat{j}) + (\hat{j} \times -\hat{i}) + (\hat{j} \times \hat{j}) = 0 + \hat{k} - (-\hat{k}) + 0 = 2\hat{k}$. This points in the $+z$ direction, which matches the given propagation direction.

For option 2: $(\hat{i} + \hat{j}) \times (\hat{i} - \hat{j}) = -\hat{k} - \hat{k} = -2\hat{k}$ (points in $-z$ direction).

For options 3 and 4: The cross product with $(\hat{i} + \hat{j})$ yields zero, meaning they are parallel or anti-parallel, which violates the condition that $\vec{E}$ and $\vec{B}$ are perpendicular.

Thus, the magnetic field must be parallel to $(-\hat{i} + \hat{j})$.

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