Normal vector
High school
Consider a point on a curve. A normal vector at the point is a vector that is perpendicular to the curve at that point. More precisely, a vector $v$ is normal to a curve at $P$ if the line spanned by $v$ from $P$ is perpendicular to the tangent line of the curve at $P$.
Undergraduate (differential geometry)
Consider a smooth surface $S$ in $\reals^n$ and a point $p \in S$. A vector $n$ is called a normal vector to $S$ at $p$ if $n$ is orthogonal to every tangent vector of $S$ at $p$:
\[\ip{n}{v} = 0 \quad \text{for every tangent vector } v \text{ of } S \text{ at } p.\]Examples
Examples
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The vector $(1, 1)$ and the line $x + y = 1$ in the plane, whose direction vector is $(1, -1)$
High school
The vector $(1, 1)$ is a normal vector to the line $x + y = 1$, because $(1, 1)$ is the gradient of $x + y$, and the gradient is perpendicular to each level set.
Properties
- For a level set $\setdef{x \suchthat f(x) = c}$, the gradient $\nabla f(p)$, if nonzero, is a normal vector at $p$.
- The normal line at $p$ is the line through $p$ in the direction of a normal vector.
- “Normal” means “perpendicular” here, and the word is unrelated to the normal distribution and to a normal space.
Distinctions involving this term
- Right angle vs. perpendicular vs. orthogonal vs. normal: Right angles, perpendicular lines, orthogonal vectors, and normal vectors all describe things that meet “squarely.” A right angle is an angle of $90^\circ$, so the term describes an angle. Perpendicular describes lines or planes that intersect and form right angles. Orthogonal describes vectors whose inner product is $0$, which extends the idea to vectors that do not meet, such as directions of skew lines, to the zero vector, and to vectors that are not arrows, such as functions. A normal vector is perpendicular to a line, curve, or surface at a point, so the term describes a vector relative to a shape.