Final answer:
To determine if the given electric field vector satisfies the wave equation in free space, it must conform to the form and derivatives specified by the wave equation for electric fields in a vacuum.
Step-by-step explanation:
The given electric field vector E₅ = (x + z/√2 ) E₀ cos(ωt+ky) can be tested against the wave equation in free space to determine if it satisfies the required conditions for a wave propagating in space. In free space, the wave equation for an electric field E is given by ∇²E = (μ₀ε₀)×(∂²E/∂t²), where ∇² is the Laplacian operator, μ₀ is the permeability of free space, ε₀ is the permittivity of free space, and (∂²E/∂t²) represents the second time derivative of the electric field. The given electric field equation will satisfy this wave equation if it is a function of x-ct (where c is the speed of light in a vacuum) and all derivatives with respect to space and time are consistent with the wave equation.