Is this a valid dot product?

Question

So I have a homework problem, where $$ F(x,y) = \sin(y) (4 x ~ \mathbf{i} + \mathbf{j}) ~ \text{and} ~ \mathrm{d}{S} = - x ~ \mathbf{i} + y ~ \mathbf{j} + \mathbf{k}, $$ and I need to find the dot product of $ F(x,y) $ and $ \mathrm{d}{S} $, but I’m confused about how to do that when both the $ \mathbf{i} $ and $ \mathbf{j} $ components are inside of $ \sin $. I’ve written it exactly the same way it’s notated on my homework.

Answer 1

Note that $F(x,y)=4x\ \mathrm{sin}(y)\ \mathbf{i}+ \mathrm{sin}(y)\mathbf{j}\ $ is a completely equivalent way of writing your function.