Electromagnetic waves are transverse in nature. The electric field ($\vec{E}$), magnetic field ($\vec{B}$), and the direction of propagation (velocity $\vec{v}$) are mutually perpendicular. The direction of wave propagation is given by the direction of the cross product $\vec{E} \times \vec{B}$ .

Given: Direction of propagation ($\vec{v}$): $+x$ axis ($\hat{i}$) Direction of Electric Field ($\vec{E}$): $+y$ axis ($\hat{j}$)

We need to find the direction of $\vec{B}$ (let's call it unit vector $\hat{n}$) such that: $\text{Direction of } (\vec{E} \times \vec{B}) = \text{Direction of } \vec{v}$ $\hat{j} \times \hat{n} = \hat{i}$

Using the cyclic property of cross products for unit vectors ($\hat{i} \times \hat{j} = \hat{k}, \hat{j} \times \hat{k} = \hat{i}, \hat{k} \times \hat{i} = \hat{j}$): Since $\hat{j} \times \hat{k} = \hat{i}$, the direction of the magnetic field $\hat{n}$ must be $\hat{k}$, which corresponds to the $+z$ direction.