In an electromagnetic wave in free space, the direction of electric field vector \(E^{-}\) is along \(y\) axis and magnetic field vector \(\mathrm{B}^{-}\) is along \(\mathrm{z}\) axis then which of the following is true (A) $\left(\mathrm{E}^{-} \times \mathrm{B}^{-}\right) \times \mathrm{E}^{-}=1$ (B) $\left(\mathrm{E}^{-} \times \mathrm{B}^{-}\right) \times \mathrm{B}^{-}=1$ (C) $\left(\mathrm{E}^{-} \times \mathrm{B}^{-}\right) \times \mathrm{B}^{-}=0$ (D) none of these

The electric field vector \(E^{-}\) is along the \(y\) axis, so it can be represented as: \[E^{-} = E_y \hat{j}\] The magnetic field vector \(B^{-}\) is along the \(z\) axis, so it can be represented as: \[B^{-} = B_z \hat{k}\]

We need to calculate the cross product of the electric field vector and the magnetic field vector: \[(E^{-} \times B^{-}) = (E_y\hat{j}) \times (B_z\hat{k})\] To calculate the cross product of these vectors, we can use the following rule for cross product of unit vectors: \[\hat{i} \times \hat{j} = \hat{k}, \hat{j} \times \hat{k} = \hat{i}, \text{ and } \hat{k} \times \hat{i} = \hat{j}\] Using this rule, we find the cross product: \[(E^{-} \times B^{-}) = E_yB_z(\hat{j} \times \hat{k}) = E_yB_z\hat{i}\]

Now, let's evaluate the options one by one using the cross product of the given vectors: (A) \[\left(E^{-} \times B^{-}\right) \times E^{-}\] \[(E_yB_z\hat{i}) \times (E_y\hat{j})\] The above expression will not be equal to 1. (B) \[\left(E^{-} \times B^{-}\right) \times B^{-}\] \[(E_yB_z\hat{i}) \times (B_z\hat{k})\] The above expression will not be equal to 1. (C) \[\left(E^{-} \times B^{-}\right) \times B^{-}\] \[(E_yB_z\hat{i}) \times (B_z\hat{k})\] In this case, we have the cross product of two perpendicular unit vectors, which is equal to the third unit vector, so the expression becomes: \[E_yB_z^2\hat{j}\] Now, let's calculate the cross product with the magnetic field vector: \[E_yB_z^2(\hat{j} \times \hat{k}) = E_yB_z^2\hat{i}\] The above expression will be equal to 0. (D) none of these. Therefore, the correct answer is (C).