# How to write equation of oscillating electric field?

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! :)

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 $$\hat{u}$$.