# Definition:Vector Area

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## Definition

Let $S$ be a plane surface embedded in Cartesian $3$-space.

The **vector area** of $S$ is a technique of representing $S$ using a vector quantity $\mathbf S$.

The magnitude of the vector is determined by the area of $S$, while its direction is defined as the unit normal $\mathbf {\hat n}$ to the plane of $S$.

The sign of $\mathbf S = S \, \mathbf {\hat n}$ is determined by the right-hand rule according to the direction of rotation around $S$ when describing it.

The shape of $S$ is arbitrary.

While in the diagram above it is seen to be circular, it is usual to consider rectangles or whatever other shapes can be aligned conveniently with coordinate axes.

## Also see

- Results about
**vector area**can be found**here**.

## Sources

- 1951: B. Hague:
*An Introduction to Vector Analysis*(5th ed.) ... (previous) ... (next): Chapter $\text {II}$: The Products of Vectors: $5$. Vector Area