For an electromagnetic wave propagating through vacuum, \(\vec{k}\), \(\vec{E}\) and \(\omega\) represent propagation vector, electric field and angular frequency, respectively. The magnetic field associated with this wave is represented by :

For an electromagnetic wave, the direction of wave propagation \(\hat{k}\) is along the direction of \(\vec{E} \times \vec{B}\).

Since \(\vec{E}\), \(\vec{B}\), and \(\vec{k}\) form a mutually perpendicular right-handed system, the direction of the magnetic field is given by \(\hat{B} = \hat{k} \times \hat{E}\).

The magnitudes of the electric and magnetic fields are related by \(B = \dfrac{E}{c}\).

Using the relation for the speed of light \(c = \dfrac{\omega}{k}\), we get \(B = \dfrac{kE}{\omega}\).

Combining the magnitude and direction gives:

\(\vec{B} = B\hat{B} = \dfrac{kE}{\omega} (\hat{k} \times \hat{E})\)

Since \(\vec{k} = k\hat{k}\) and \(\vec{E} = E\hat{E}\), this simplifies to:

\(\vec{B} = \dfrac{\vec{k} \times \vec{E}}{\omega}\)

Answer: \(\dfrac{\vec{k} \times \vec{E}}{\omega}\)