# Definition:Normal Vector

*This page is about Normal Vector. For other uses, see Normal.*

## Definition

Let $S$ be a surface in ordinary $3$-space.

Let $P$ be a point of $S$.

Let $\mathbf n$ be a vector whose initial point is at $P$ such that $\mathbf n$ is perpendicular to $S$ at $P$.

Then $\mathbf n$ is a **normal vector** to $S$ at $P$.

## Also defined as

Some introductory texts, in an attempt to keep concepts simple when first presented, define a **normal vector** with respect to a plane surface only.

While this definition is appropriate, and completely compatible with this more general case, it is important to note that this definition can (and should) be expanded to include surfaces which are not in fact plane.

## Also known as

A **normal vector** is usually referred to as just a **normal** (to $S$ at $P$).

It is usually clear from the context that it is a vector which is being referred to.

## Also see

- Results about
**normal vectors**can be found**here**.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**normal vector**(to the plane)

- Weisstein, Eric W. "Normal Vector." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/NormalVector.html