Definition:Normal Vector

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.