# Normal Line Calculator

## Find normal lines step by step

The calculator will find the normal line to the explicit, polar, parametric and implicit curve at the given point, with steps shown.

It can handle horizontal and vertical normal lines as well.

The normal line is perpendicular to the tangent line.

Related calculator: Tangent Line Calculator

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

The Normal Line Calculator offers an efficient and accurate method for determining the normal line of a curve at a specific point. You're in the right spot if you're seeking clarity on the equation of the normal line, its definition, and the derivation of its formula.

## How to Use the Normal Line Calculator?

• ### Input

Start by choosing the type of a curve: explicit, implicit, parametric, and polar. Then enter the equation of the curve in the specified input field. Ensure the equation is input correctly using standard mathematical notation. In the provided space, enter the coordinates of the point where you wish to find the normal line.

• ### Calculation

After entering the necessary information, click the "Calculate" button.

• ### Result

The calculator will promptly display the equation of the normal line at the given point on the curve.

## What Is a Normal Line?

A normal line is a line that stands at a right angle (90 degrees) to the tangent of a curve at a particular point where they intersect. If you plot a curve and the tangent to it, the line that intersects this tangent perpendicularly at that specific point is the normal line.

To find the slope of the tangent for any curve, one uses the derivative of the function representing the curve. Given the perpendicular relationship between the tangent and the normal line, the slope of the normal line is the negative reciprocal of the slope of the tangent.

In essence, the normal line is the line that intersects the curve's tangent at a right angle at their point of contact.

## What Is the Equation of a Normal Line?

To define the equation of a normal line, it's important to understand its relationship with the tangent to a curve at a given point.

• Slope of the Tangent Line: For a function $f(x)$, the slope of the tangent line at a point $x_0$ is given by the derivative of the function evaluated at that point:

$$m_t=f^{\prime}\left(x_0\right)$$
• Slope of the Normal Line: Since the normal line is perpendicular to the tangent line, the slope of the normal line $m_n$ ​is the negative reciprocal of the slope of the tangent line. Thus, if $m_t$ is the the slope of the tangent line, then the slope of the normal line is

$$m_n=-\frac{1}{m_t}=-\frac{1}{f^{\prime}\left(x_0\right)}$$
• Equation of the Normal Line: Using the point-slope form of a linear equation and the slope of the normal line, the equation of the normal line at the point $\left(x_0,f\left(x_0\right)\right)$ can be written as

$$y-f\left(x_0\right)=m_n\left(x-x_0\right),$$

where:

• $y$ is the dependent variable.
• $x$ is the independent variable.
• $m_n=-\frac{1}{f^{\prime}\left(x_0\right)}$ is the slope of the normal line.
• $x_0$ is the x-coordinate of the given point.
• $f\left(x_0\right)$ is the y-coordinate of the given point.

## Why Choose Our Normal Line Calculator?

• ### Accuracy

Our Normal Line Calculator uses advanced algorithms to ensure that the results you obtain are precise every time.

• ### User-Friendly Interface

Designed with the user in mind, our calculator offers a straightforward interface that's easy to navigate, even if you're new to the concept of normal lines.

• ### Fast Results

Our calculator provides instant results, eliminating waiting and traditional manual calculations.

• ### Versatility

Whether you're working with simple quadratic curves or more complex functions, our calculator is capable of handling a wide range of mathematical equations.

### FAQ

#### Who can benefit from using this calculator?

The Normal Line Calculator is valuable for students, teachers, researchers, and anyone with an interest in calculus or who needs to determine the normal line to a curve quickly and efficiently.

#### How is the normal line different from the tangent line?

The normal line is perpendicular to the tangent of a curve at the point of tangency. In essence, if you have the tangent line at a specific point on a curve, the normal line will form a right angle (90 degrees) with this tangent line at that very point.

#### What is the Normal Line Calculator used for?

The Normal Line Calculator is a specialized tool designed to find the equation of the normal line to a given curve at a specified point. This tool simplifies the process, providing accurate results instantaneously.