How can area be represented as a vector in three dimensions?

[kay]

How is Area a vector? How does it have direction? I thought it was basically a scalar quantity because it only had magnitudes, e.g. 4m^2, 7m^2 etc.
Please help. I can't understand. :(

 

[verty]

I think area is not a vector but $d\vec{A}$ is a vector? The surface area of a sphere for example won't have a definite direction, but it is certainly not the zero vector, that would make no sense to me.

 

[HallsofIvy]

Simple answer- area is NOT a vector. But it can be "represented" by one. If you are dealing with planar regions in three dimensions, then it can be convenient to represent the "area" as a vector whose length is the actual scalar area and whose direction is perpendicular to the plane. For example, the area of a parallelogram in three dimensions, two adjacent sides of which have length a and b and have angle \theta, is given by ab sin(\theta). If that parallelogram lies in three dimensions, we can think of the sides as given by the vectors \vec{u} and \vec{v}. In that case, the lengths of the sides are |\vec{u}| and |\vec{v}| so the area is |\vec{u}||\vec{v}|sin(\theta)|. Notice that, here, since we are "given" the sides as vectors we have not only the lengths but the angle between them, \theta given as part of the vector information. And, in fact, the cross product of the two vectors, \vec{u}\times\vec{v} is a vector whose length is equal to the area of the parallelogram and which is perpendicular to the plane the parallelogram lies in.

We can extend this to non-planar figures in three dimensions by taking the "differential of surface area" at each point to be the "vector" whose "length" is the differential of area, dxdy, and is perpendicular to the surface at each point.