= Normal vector
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A normal vector at a point of an embedded <smooth manifold> is <orthogonal> to every <tangent vector> in its <tangent space>, using the surrounding <dot product>. Its direction describes departure from the local tangent geometry. For a <regular level set> $F(x)=c$ with nonzero <gradient>, $\nabla F$ is a <normal vector>: differentiation of $F$ along any curve on the <level set> gives $\nabla F\cdot r'=0$.
For a <plane curve>, the normal direction is a <straight line> perpendicular to its <unit tangent vector>; the two <unit normals> differ by sign. For a <curve> in three-dimensional <Euclidean space>, the orthogonal directions form a two-dimensional plane. A <principal normal vector> selects one of them using <curvature>, so it must be distinguished from an arbitrary <normal vector>.
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