Normal vector

Definition

Let $H$ be a line in $\mathbb{R}^2$ or a plane in $\mathbb{R}^3$. Let $\cdot$ be the dot product. A normal vector of $H$ is a nonzero vector $n$ such that $n \cdot (x - y) = 0$ for all points $x$ and $y$ of $H$.

Remarks

If $n$ is a normal vector of $H$, then the normal vectors of $H$ are exactly the vectors $c n$ with $c \neq 0$. The plane $\{(x, y, z) : a x + b y + c z = d\}$ with $(a, b, c) \neq (0, 0, 0)$ has the normal vector $(a, b, c)$. The line $\{(x, y) : a x + b y = c\}$ with $(a, b) \neq (0, 0)$ has the normal vector $(a, b)$. For the plane $\{p + s u + t v : s, t \in \mathbb{R}\}$, the cross product $u \times v$ is a normal vector. A unit normal vector is a normal vector of length 1.

Explanations

L1 human

A normal vector sticks straight out of a flat object. Think of a pencil standing upright on a table: it is normal to the table. Each arrow that lies in the table makes a right angle with the pencil, so its dot product with the pencil is $0$.