Vector area

In geometry, for a finite planar surface of scalar area $S$, the vector area

:$mathbf\{S\}$

is defined as a vector whose magnitude is $S$ and whose direction is perpendicular to the plane, as determined by the right hand rule on the rim (moving one's right hand counterclockwise around the rim, when the palm of the hand is "touching" the surface, and the straight thumb indicate the direction).

:$mathbf\{S\}\; =\; mathbf\{hat\{nS$

This can only be defined for flat surfaces, or for regions of curved surfaces which are sufficiently small that they can be considered flat.

The concept of an area vector simplifies the equation for determining the flux through the surface. Consider a planar surface in a uniform field. The flux can be written as the dot product of the field and area vector. This is much simpler than multiplying the field strength by the surface area and the cosine of the angle between the field and the surface normal.

Projection of area onto planes

The projected area onto (for example) the "x"-"y" plane is equivalent to the "z"-component of the vector area, and is given by

:$mathbf\{S\_z\}\; =\; left|\; mathbf\{S\}\; ight|\; cos\; heta$

where $heta$ is the angle between the plane normal and the "z"-axis.

ee also

*Cross product
*Surface normal
*Surface integral