Equation of a Normal Line

Equation of a Normal Line

A normal line to a curve is the line perpendicular to the tangent line at a given point. To write its equation, you usually need the slope of the curve at that point.

1) Find the tangent slope

If the curve is given by a function, compute the derivative and evaluate it at the point of interest. This gives the tangent slope, often written as (m_t).

2) Use the perpendicular slope

The normal line slope is the negative reciprocal of the tangent slope: [ m_n = -\frac{1}{m_t} ] If the tangent slope is 0, the normal line is vertical. If the tangent line is vertical, the normal line is horizontal.

3) Write the line equation

Use point-slope form with the point ((x_0, y_0)): [ y - y_0 = m_n(x - x_0) ] Then simplify the result as requested.

4) Check your answer

Make sure the line passes through the given point and is perpendicular to the tangent line. A quick check is that the product of the tangent and normal slopes is (-1), when both slopes are defined.