Why is area vector perpendicular to the plane of the area?

Question

Is there any specific reason for defining area vector as perpendicular to the plane of the area? Or will I ever go wrong if I define it in any arbitrary direction?

Answer 1

If you have a parallelogram spanned by two vectors, and an area vector for the parallelogram that is perpendicular to both of those vectors, then the volume of the prism spanned by that parallelogram and a third vector is equal to the dot product of the area vector and the third vector, which is nice. In a very similar manner, calculating movement of a fluid through a surface can be phrased in terms of a normal vector scaled relative to the area of the surface (in other words, a perpendicular area vector).

Long story short, if you want to encode information about your surface into a single vector, then a perpendicular area vector is better than any other area vector because it also carries explicit information about the orientation of the surface.