Normal Line: Definition & Equation

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Point of Intersection: Definition & Formula

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
Your next lesson will play in 10 seconds
  • 0:04 What Is a Normal Line?
  • 0:35 Vocabulary
  • 1:37 Definition of a Normal Line
  • 1:59 Equation for a Normal Line
  • 3:44 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Speed Speed

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: David Karsner

David holds a Master of Arts in Education

In calculus, a normal line is a line that goes through a point on a curve and is perpendicular with the tangent line at that point. In this lesson, you'll learn more about normal lines and how to find their equations.

What Is a Normal Line?

You are holding a bow. You have the string drawn back and are about to launch an arrow at the bulls-eye target. This picture illustrates a normal line. The bow is the curve and the arrow is the normal line at a point on the curve. Normal lines are associated with the field of calculus.

This lesson includes the definition and how to find the equation of a normal line. Because the definition of a normal line has many terms that may be unfamiliar, the lesson will begin with a vocabulary list of these terms.


Let's cover some of the basic vocabulary terms we need to have in mind for this:

  • A tangent line is a line that intersects a curve at only one point

  • A slope can be interchangeable with derivative; it's how much a function is changing at a certain point

  • A derivative, conversely, can be interchangeable with slope; it's a process in calculus that allows one to find the slope of a function

  • A set of lines that are perpendicular is when two intersecting lines form four right angles: these four right angles are said to be perpendicular

  • The opposite reciprocals are when two numbers are opposite reciprocals to each other if they have different signs and the numerator and denominator have been switched: for example -2/3 and 3/2 are opposite reciprocals

  • The slope intercept form is the equation of a line, and it can always be written in the form of y = mx + b where m is the slope, b is the y-intercept, and (x,y) are any point on the line

Definition of a Normal Line

A normal line to a point (x,y) on a curve is the line that goes through the point (x,y) and is perpendicular to the tangent line. Since the normal line and tangent line are perpendicular, they will have slopes that are opposite reciprocals of each other. To find the slope of the tangent line at the point (x,y), take the derivative of the function at that point.

Equation for a Normal Line

To find the equation of any line, you need to know the slope of the line and one point on the line. By definition, the slope of the normal line is the opposite reciprocal of the slope of the tangent line. The slope of the tangent line is the derivative of the curve (function) at that point.

For example, find the equation of the normal line for the function f(x) = -3x^2 + 2x + 6 at the point x = 1.

To unlock this lesson you must be a Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use

Become a member and start learning now.
Become a Member  Back
What teachers are saying about
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account