Question

what are you finding when you do the cross product of two vectors i.e. when you do an integral you are finding the area under he curve.

Answer 1

you are finding a vector perpendicular to the two vectors you are crossing, whose length is proportional to "how perpendicular" the two original vectors are to each other.

Answer 2

It's basically another vector that is perpendicular to both vectors.

Answer 3

and if they are u and v are parallel, then u cross v is zero