# How is Area a vector?

## Main Question or Discussion Point

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.

verty
Homework Helper
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.