A normal line of a curve at one of its points $P$ is the line passing through this point and perpendicular to the tangent line of the curve at $P$ .

If the plane curve $y = f(x)$ has a skew tangent at the point $(x_0,\,f(x_0))$ , then the slope of the tangent at that point is $f'(x_0)$ and the slope of the normal at that point is $\displaystyle -\frac{1}{f'(x_0)}$ . The equation of the normal is thus $$y-f(x_0) = -\frac{1}{f'(x_0)}(x-x_0).$$ In the case that the tangent is horizontal, the equation of the vertical normal is $$x = x_0,$$ and in the case that the tangent is vertical, the equation of the normal is $$y = f(x_0).$$

The normal of a curve at its point $P$ always goes through the center of curvature belonging to the point $P$ .

In the picture below, the black curve is a parabola, the red line is the tangent at the point $P$ , and the blue line is the normal at the point $P$ .