Normal Vector¶

Definition¶

A normal vector to a plane is a nonzero vector perpendicular to every direction vector in the plane.

Interpretation¶

A normal vector records which way a plane faces. The direction vectors say how one can move inside the plane; the normal vector points straight out of it, so it fixes the tilt of the plane but not its position.

A shaded plane with two direction vectors v and w from a point A and a normal vector n standing perpendicular to both.

In the figure, $\boldsymbol{n}$ stands at a right angle to both direction vectors at the point $\boldsymbol{A}$ , so $\boldsymbol{n}\cdot\boldsymbol{v}=\boldsymbol{n}\cdot\boldsymbol{w}=0$ $0$ . Every displacement inside the plane is a linear combination $s\boldsymbol{v}+t\boldsymbol{w}$ $t w$ , and $\boldsymbol{n}\cdot(s\boldsymbol{v}+t\boldsymbol{w})=s\,\boldsymbol{n}\cdot\boldsymbol{v}+t\,\boldsymbol{n}\cdot\boldsymbol{w}=0$ $0$ . Checking the two independent directions is therefore enough to be perpendicular to the whole plane.

To find a normal vector $\boldsymbol{n}=(a,b,c)$ $(a, b, c)$ , solve $\boldsymbol{n}\cdot\boldsymbol{v}=0$ and $\boldsymbol{n}\cdot\boldsymbol{w}=0$ $0$ , as in the second example of the point-normal equation. In $\mathbb{R}^3$ one can also take the cross product $\boldsymbol{n}=\boldsymbol{v}\times\boldsymbol{w}=(v_2w_3-v_3w_2,\,v_3w_1-v_1w_3,\,v_1w_2-v_2w_1)$ $v_{2} w_{1})$ . Any nonzero multiple of a normal vector is again normal, and in $\mathbb{R}^3$ all normals of one plane are multiples of each other.

Two planes in $\mathbb{R}^3$ are parallel or equal exactly when their normals are multiples of each other. The same idea gives the normal $(a_1,\ldots,a_n)$ of a hyperplane $a_1x_1+\cdots+a_nx_n=d$ in $\mathbb{R}^n$ .

A common slip is to expect the normal vector to be orthogonal to the points of the plane. It is orthogonal to differences $\boldsymbol{B}-\boldsymbol{A}$ of points, as in the Key Results; $\boldsymbol{n}\cdot\boldsymbol{B}$ itself is the same for every point $\boldsymbol{B}$ of the plane, and it is $0$ only when the plane passes through the origin.

Key Results¶

A point and a normal vector determine a plane in $\mathbb{R}^3$ .
If $\boldsymbol{n}$ is normal to a plane and $\boldsymbol{A}$ is a point of the plane, then $\boldsymbol{n}\cdot(\boldsymbol{B}-\boldsymbol{A})=0$ for every point $\boldsymbol{B}$ on the plane.

Examples¶

For the $xy$ -plane, $(0,0,1)$ is a normal vector.
If a plane has direction vectors $\boldsymbol{v},\boldsymbol{w}$ , then a normal vector is orthogonal to both $\boldsymbol{v}$ and $\boldsymbol{w}$ .

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 5

Aliases: normal Keywords: surface normal, vector perpendicular to a plane Symbols: real numbers

Last updated: 20 October 2026 19:41