Why is $\mathbf A$.d$\mathbf A$=|$\mathbf A$||d$\mathbf A$|cos $\alpha$=AdA?
$\mathbf A$.d$\mathbf A$=|$\mathbf A$||d$\mathbf A$|cos $\alpha$ is fine. It comes from the dot product rule. But why is it furthur equal to A.dA?
Here, $\mathbf A$ is a vector.
It is so because $ d \vec A$ is taken along the same direction as that of $ \vec A$ i.e., normal to the plane. So ,the angle between the two is $ 0$