Why is $d\mathbf{a}$ a vector when doing surface integrals?

Question

I'm just learning about surface integrals, and I'm doing a problem where I'm supposed to calculate the surface integral of some vector function $\mathbf{v}$ over five sides of a cube. For the first side, $x = 1$, so $dx = 0$. I get this. But then the book says, "so $d\mathbf{a} = dydz\mathbf{\hat{x}}$". This I can't seem two wrap my head around. I know that $d\mathbf{a}$ is supposed to be an infinitesimal area, so why is it a vector? Shouldn't it be a scalar?

Answer 1

I think everything works out if we just say $\mathbf{\hat{n}}$ is the unit normal vector to the surface, and then $dA$ is a scalar quantity, e.g. $dA = dxdy$ when $\mathbf{\hat{n}}$ is in the $z$ direction.