An electromagnetic wave in vacuum has the electric and magnetic field `vecE` and `vecB`, which are always perpendicular to each other. The direction of polarization is given by `vecX` and that of wave propagation by `vecK`. Then

Options

A. `vec(X)||vec(B) and vec(k)||vec(B)xxvec(E)`
B. `vec(X)||vec(E) and vec(k)||vec(E)xxvec(B)`
C. `vec(X)||vec(B) and vec(k)||vec(E)xxvec(B)`
D. `vec(X)||vec(E) and vec(k)||vec(B)xxvec(E)`

Answer

To solve the problem, we need to analyze the relationships between the electric field \(\vec{E}\), the magnetic field \(\vec{B}\), the direction of polarization \(\vec{X}\), and the direction of wave propagation \(\vec{K}\) in an electromagnetic wave.

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- In an electromagnetic wave, the electric field \(\vec{E}\) and magnetic field \(\vec{B}\) are always perpendicular to each other and also to the direction of wave propagation \(\vec{K}\).
- Given that the direction of polarization is along the x-axis, we can denote \(\vec{E}\) as being in the x-direction.

\[ \vec{E} \parallel \hat{i} \quad (\text{where } \hat{i} \text{ is the unit vector in the x-direction}) \]

2. **Determining the Direction of \(\vec{B}\)**:
- Since \(\vec{E}\) and \(\vec{B}\) are perpendicular, we can use the right-hand rule to determine the direction of \(\vec{B}\).
- If \(\vec{E}\) is in the x-direction, then \(\vec{B}\) can be in the y-direction or z-direction. For simplicity, let’s assume \(\vec{B}\) is in the y-direction.

\[ \vec{B} \parallel \hat{j} \quad (\text{where } \hat{j} \text{ is the unit vector in the y-direction}) \]

3. **Finding the Direction of Wave Propagation \(\vec{K}\)**:
- The direction of wave propagation \(\vec{K}\) is given by the cross product of \(\vec{E}\) and \(\vec{B}\):

\[ \vec{K} \parallel \vec{E} \times \vec{B} \]

- Using the right-hand rule, if \(\vec{E}\) is in the x-direction and \(\vec{B}\) is in the y-direction, then \(\vec{K}\) will be in the z-direction.

\[ \vec{K} \parallel \hat{k} \quad (\text{where } \hat{k} \text{ is the unit vector in the z-direction}) \]

4. **Conclusion**:
- We have determined the relationships:
- \(\vec{E} \parallel \hat{i}\) (x-direction)
- \(\vec{B} \parallel \hat{j}\) (y-direction)
- \(\vec{K} \parallel \hat{k}\) (z-direction)
- Thus, the final relationships can be summarized as:
- \(\vec{X} \parallel \vec{B}\)
- \(\vec{K} \parallel \vec{E} \times \vec{B}\)

### Final Answer:
- The direction of polarization \(\vec{X}\) is parallel to \(\vec{B}\) and the direction of wave propagation \(\vec{K}\) is parallel to \(\vec{E} \times \vec{B}\).