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

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To express a P-State lightwave propagating at a 45-degree angle in the xy-plane, the wave equation can be formulated using the dot product of the wave vector and position vector. The general form of the wave is A*cos(k·r - ωt - φ₀), where k is the wave vector and r is the position vector. The challenge lies in determining the appropriate components of the wave vector for the specified angle. Understanding the relationship between the wave vector and its direction is crucial for solving the problem. A diagram may aid in visualizing the wave propagation and facilitate comprehension of the concept.
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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} propogating 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.
 
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No takers on this question?
I know I have shown no work, but I am completely stuck.
Can anyone gander atleast a starting point?
 
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.
 
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