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Let's say I have two different vectors, which are $\vec v_1=Ae^{i(kz-wt)}\hat y$ and $\hat v_2 =Be^{i(kz-wt)}$ and I want to take the cross product of the two of them, can I say that the following is true?

$$\vec v_1 \times \vec v_2=\hat x\ \times \hat y \ (v_1v_2)$$

I've been trying to figure this out on my own but I haven't been able to. I was told this is true but it doesn't make sense to me because it doesn't seem to hold when I use the definition of the cross product using it's sine form.

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Using the definition you've alluded to at the end of your post, $$\vec{v_1} \times \vec{v_2} = \|\vec{v_1}\| \|\vec{v_2}\| \sin(\theta) \vec{n} = \|\vec{v_1}\| \|\vec{v_2}\| (\hat{x} \times \hat{y})$$ where $\hat{x} = \vec{v_1}/\|\vec{v_1}\|$ and $\hat{y} = \vec{v_2}/\|\vec{v_2}\|$, where $\theta$ is the angle between $\vec{v_1}$ and $\vec{v_2}$, and where $\vec{n}$ is the normal vector to $\vec{v_1}$ and $\vec{v_2}$.

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