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I'm asked to write the equation of an oscillating EF that is propagating along the direction $\hat{i} + \hat{j}\over \sqrt{2}$. Sufficient parameters are given such that I can find the value of $E_0$, $k$ (angular wavenumber) and $\omega$ (angular frequency).

$$\vec{E} = E_0 \sin(\omega t-k(\space \space \space \space))\frac{-\hat{i} + \hat{j}}{\sqrt2}$$

But what goes into the empty bracket? My initial thought was $x+y\over \sqrt2$, drawing a parallel with the way the vectors work. But I don't think this makes any sense.

Any help is appreciated! :)

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The wave vector $\vec{k}$ is in the same direction as the wave is travelling. Since you have been given a unit vector for the direction, then $\vec{k}$ is just $k$ multiplied by that unit vector.

The general expression you are attempting to write is $$\vec{E}= E_0 \sin(\omega t - \vec{k}\cdot \vec{r}) \hat{u} \ , $$ where $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ in Cartesian coordinates. Thus, knowing $\vec{k}$ you can complete the expression.

However, $\hat{u}$ is a unit vector expressing the polarisation direction of the wave. This is perpendicular to the wave vector (i.e. $\vec{k}\cdot \hat{u}=0$) and this is not uniquely defined by the information in your question. You have chosen a possible but not unique value for $\hat{u}$.

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