Applying Maxwell's equations to plane waves

[cksoon11]

Homework Statement

For a harmonic uniform plane wave propagating in a simple medium, both \vec{E} and \vec{H} vary in accordance with the factor exp(-i \vec{k}.\vec{R})

Show that the four Maxwell’s equations
for a uniform plane wave in a source-free region reduce to the following:

\vec{k} \times\vec{E}= \omega\mu\vec{H}

\vec{k} \times\vec{H} = \omega\epsilon\vec{E}

\vec{k} \bullet \vec{E} = 0

\vec{k} \bullet \vec{H} = 0

Apparently "Vector k and Vector R are the the general forms of wave number and position vector(or direction of propagation)"

Question Source(no.4) :http://www.lib.yuntech.edu.tw/exam_new/96/de.pdf"

Homework Equations

You supposed to use Maxwell's Equations for a plane wave

\nabla x \vec{E} = -i\omega\mu\vec{H}\nabla x \vec{H} = i\omega\mu\vec{E}

\nabla . \vec{E} = 0

\nabla .\vec{H} = 0

The Attempt at a Solution

First off,I am confused as to how k can be a vector when it is the wave number(a scalar).
From what I can tell,the wave is propagating in the radial direction in spherical coordinates.
I then assumed the electric and magnetic fields to be orthogonal in the \thetaand \phi direction.

But just simply substituting the phasor form of the plane wave into Maxwell's equations:

E_{o} exp((-i \vec{k}.\vec{R}) into Maxwell equations doesn't seem to yield the desired results because I don't understand how they obtained the cross-products and dot products of \vec{k}with \vec{E} and
\vec{H}.

I just can't seem to grasp the concept of a vector as my wave number.Could someone please phrase the question in more concise terms?What am I misunderstanding here?

 

[gabbagabbahey]

The vector \textbf{k} is called the "wave vector". It's magnitude is the wavenumber and it points in the direction of propagation.

Just substitute \textbf{E}=\textbf{E}_0e^{i(\textfb{k}\cdot\textbf{r}-\omega t)}[/itex] and \textbf{H}=\textbf{H}_0e^{i(\textfb{k}\cdot\textbf{r}-\omega t)}[/itex] into Maxwell's equations and take the derivatives...what do you get?