Normal Line Calculator

The Normal Line Calculator finds the line perpendicular to a curve at a chosen point. In calculus, the normal line is closely related to the tangent line because the two lines meet at a right angle when their slopes are finite and nonzero.

Normal Line Formula

If the tangent slope at a point is m, the normal slope is −1/m. Once the normal slope and the point (x₁,y₁) are known, the line can be written in point-slope form: y − y₁ = mₙ(x − x₁).

Using the Calculator

Provide the function or the information requested by the calculator, along with the point on the curve when required. Make sure the point actually lies on the function. The calculator can then determine the tangent slope, convert it to the normal slope, and form the line.

Example

For y = x² at x = 1, the tangent slope is 2. The normal slope is therefore −1/2. Since the point is (1,1), the normal line is y − 1 = −1/2(x − 1).

Special Cases

A horizontal tangent has slope zero, so its normal line is vertical. A vertical tangent has a vertical normal relationship in the opposite direction. These cases are better handled from the geometric definition than by blindly applying −1/m.

Check the Result

Differentiate the original function at the supplied point and verify the tangent slope. Then confirm that the normal and tangent slopes have the expected perpendicular relationship.