Skill: Determine normal vectors to planes

A normal vector is a nonzero vector perpendicular to a plane.

If a plane is written as

$$ax+by+cz=d,$$

then a normal vector is obtained directly from the coefficients of $x$, $y$, and $z$:

$$\mathbf n=\langle a,b,c\rangle.$$

The constant $d$ is not used.

Worked example

Find a normal vector to the plane

$$2x-3y+z=7.$$

Step 1: Identify the coefficients.

The coefficients are:

• coefficient of $x$: $2$
• coefficient of $y$: $-3$
• coefficient of $z$: $1$

Step 2: Write the normal vector.

Therefore, one normal vector is

$$\boxed{\mathbf n=\langle 2,-3,1\rangle}.$$

Any nonzero scalar multiple gives the same normal direction. For example,

$$\langle -2,3,-1\rangle$$

is also a normal vector because it is $-1$ times $\langle 2,-3,1\rangle$.

The number $7$ does not affect the normal vector; it only determines where the plane is located.

If two direction vectors in a plane are given instead, you can find a normal vector by taking their cross product. The cross product is perpendicular to both direction vectors, and therefore perpendicular to the plane.