normal line

A normal line (or simply normal or perpendicular) 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 $P$.  The point $P$ is the foot of the normal.

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  $f^{\prime }(x_0)$  and the slope of the normal at that point is  $-{\displaystyle \frac{1}{f^{\prime }(x_0)}}$.  The equation of the normal is thus

$$y-f(x_0)=-\frac{1}{f^{\prime }(x_0)}(x-x_0).$$

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 $P$.