Determine normal vectors to planes

A normal vector to a plane is a nonzero vector perpendicular to every direction lying in the plane. In three-dimensional Cartesian geometry, it can be read from the coefficients of a plane equation $ax+by+cz=d$ as $\langle a,b,c\rangle$ , or found by taking the cross product of two nonparallel direction vectors in the plane; any nonzero scalar multiple represents the same normal direction. This understanding supports plane equations, parallelism, perpendicularity, intersections, and distance calculations, without extending to abstract higher-dimensional spaces.

Detailed Explanation: 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.

Learn by doing: Determine normal vectors to planes

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Practice with unlimited practice problems

3D Space - Plane Notation - Scalar Equation to Normal

Earned ?