Overview
On this page, we discuss a different way to describe lines and planes in 3dimensional space, using the concept of a normal vector.
Important
The basic and advanced learning objectives listed below are meant to give you an idea of the material you should learn about this section. These are mainly intended to be used in a course which uses an Active Learning approach, where students are required to “read ahead” before each class  but can equally be used in a more traditional course setting.
Unless your teacher gives you specific instructions, it is up to you to decide how much of the listed resources you need to read or watch  you probably do not need to go through all of it. You might also want to look at the General Study Tips & Tricks page for some recommendations on how to effectively study with a math textbook and videos.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

Determine whether two vectors are orthogonal.

Find a normal equation for a plane in \(\mathbb{R}^3\), given a normal vector and a point on the plane.

Find a normal equation for a lines in \(\mathbb{R}^2\), given a normal vector and a point on the line.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

Given the normal equation of a plane, read off a normal vector and a point on the plane.

Determine whether two planes are parallel, perpendicular or neither.

Determine whether a given line and plane are parallel, perpendicular or neither.

Prove results related to orthogonality.
To prepare for class

Watch these two videos explaining how to find the normal equation for a plane in 3space:

Watch this video about the two 2 ways of writing the normal equation of a plane, and how to read off the normal vector and find a (random) point in the plane (note especially the examples at 3:36 and at 5:55):