Normal Line — Definition, Formula & Examples

A normal line is the line that passes through a point on a curve and is perpendicular to the tangent line at that point. It points directly away from the curve's surface.

Given a differentiable function $f$ at a point $(a, f(a))$ where $f'(a) \neq 0$ , the normal line is the line through $(a, f(a))$ with slope $-\frac{1}{f'(a)}$ , the negative reciprocal of the tangent slope. If $f'(a) = 0$ , the normal line is the vertical line $x = a$ .

Key Formula

y - f(a) = -\frac{1}{f'(a)}\bigl(x - a\bigr)

Where:

$a$ = The x-coordinate of the point on the curve
$f(a)$ = The y-coordinate of the point on the curve
$f'(a)$ = The derivative of f evaluated at x = a (slope of the tangent line)
$-\frac{1}{f'(a)}$ = The negative reciprocal of the tangent slope, giving the normal slope

Worked Example

Problem: Find the equation of the normal line to

f(x) = x^2

at the point where

x = 3

Find the point: Evaluate the function at

x = 3

f(3) = 3^2 = 9 \quad \Rightarrow \quad \text{Point: } (3,\, 9)

Find the tangent slope: Differentiate and evaluate at

x = 3

f'(x) = 2x \quad \Rightarrow \quad f'(3) = 6

Find the normal slope and write the equation: The normal slope is the negative reciprocal of 6. Then use point-slope form.

m_{\text{normal}} = -\frac{1}{6} \quad \Rightarrow \quad y - 9 = -\frac{1}{6}(x - 3)

Answer:

y = -\dfrac{1}{6}x + \dfrac{19}{2}

Why It Matters

Normal lines appear in physics when calculating reflection angles off curved surfaces and in engineering when determining forces perpendicular to a surface. In multivariable calculus, the concept extends to normal vectors, which are essential for defining planes tangent to surfaces.

Common Mistakes

Mistake: Using the tangent slope directly instead of its negative reciprocal for the normal line.

Correction: Perpendicular lines have slopes that are negative reciprocals of each other. If the tangent slope is

m

, the normal slope is

-1/m

, not

m

1/m