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Normal Line: Definition & Equation

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  • 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
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Lesson Transcript
Instructor: David Karsner
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.

Vocabulary

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.

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