How Do You Express a P-State Lightwave Propagating at an Angle in the XY-Plane?

[paul11273]

I do not even really know where to begin with this problem.
Any help would be great.

Q. Write an expression for a P-State (linearly polarized) lightwave of angular frequency $$\omega$$ and amplitude $$E_{0}$$ propagating along a line in the xy-plane at 45 degress to the x-axis and having its plane of vibration corresponding to the xy-plane. At t=0, y=0 and x=0 the field is zero.

Like I said, I don't even know where to start. This prof is miserable, and the book is light on examples and explanations.

I am figuring the equation will be of the form $$\vec{E}=(\tilde{i}E_{0x}+\tilde{j}E_{0y})cos(kz-\omega t)$$
This would be a wave traveling along the z-axis, so I expect that I have to change this term with a vector specifying the path 45 degrees from the x-axis. But how to do this? The prof did give a hint that we need to perform an operation of taking the dot product of two vectors, say k dot r. He was very vague, in fact down right confusing after that.

Thanks in advance.

 

[paul11273]

No takers on this question?
I know I have shown no work, but I am completely stuck.
Can anyone gander atleast a starting point?

 

[Galileo]

A plane wave with wavenumber k traveling in an arbitrary direction given by the unit vector $\hat k$ can be represented as:

$$A\cos(\vec k \cdot \vec r - \omega t - \varphi_0)$$
where $|\vec k|=k$.

Try to see why this is true. Draw a diagram or so. I think you'll learn the most by understanding this general case. Solving your problem is then easy.