1. Conditions for an electromagnetic wave in vacuum

For an electromagnetic wave in vacuum:

• $\vec E \perp \vec B$
• Both $\vec E$ and $\vec B$ are perpendicular to the direction of propagation.
• The direction of propagation is along $\vec E \times \vec B$.
• Also, in vacuum, fields depend on the propagation coordinate and time, e.g. for propagation along $+x$, fields are functions of $(x,t)$.

2. Check option A

Given:

(i) Are $\vec E$ and $\vec B$ perpendicular?

Compute dot product of direction vectors:

So, $\vec E \perp \vec B$.

(ii) Direction of propagation

Expand:

Thus,

So the wave propagates in $+x$ direction, exactly as stated.

(iii) Functional dependence

Since propagation is along $+x$, dependence on $(x,t)$ is correct.

So, Option A is correct.

3. Check option B

Given:

Here $\vec E$ and $\vec B$ are along the same direction $(\hat y+\hat z)$, so they are not perpendicular.

But in an electromagnetic wave, $\vec E \perp \vec B$.

Also, for propagation along $+x$, fields should depend on $(x,t)$, not generally on $(y,z,t)$.

So, Option B is incorrect.

4. Check option C

Given for propagation in $+y$ direction:

For propagation along $+y$, both fields must be perpendicular to $\hat y$.

But here $\vec E$ is along $\hat y$, i.e. parallel to propagation direction, which is not allowed for a transverse electromagnetic wave.

So, Option C is incorrect.

5. Check option D

Given for propagation in $+y$ direction:

For propagation along $+y$, $\vec B$ must also be perpendicular to $\hat y$.

But here $\vec B$ is along $\hat y$, parallel to the propagation direction, which is impossible.

Also, direction of propagation is along $\vec E \times \vec B$:

which is not $+\hat y$.

So, Option D is incorrect.

6. Final conclusion

Only Option A satisfies all conditions for an electromagnetic wave in vacuum.