the normal line to a curve $\displaystyle F(x,y,z)$ at the point $\displaystyle (x_0,y_0,z_0)$ is given by:
$\displaystyle \boxed{ x = x_0 + tF_x(x_0,y_0,z_0) \mbox { , }y = y_0 + tF_y(x_0,y_0,z_0) \mbox { , } z = z_0 + tF_z(x_0,y_0,z_0) }$ ----> Parametric equation of normal line
where $\displaystyle t$ is a parameter, and $\displaystyle F_x(x_0,y_0,z_0), F_y(x_0,y_0,z_0), \mbox { and } F_z(x_0,y_0,z_0)$ are the partial derivatives of $\displaystyle F$ with respect to $\displaystyle x,y, \mbox { and }z$ respectively, evaluated at the point $\displaystyle (x_0,y_0,z_0)$
or
$\displaystyle \boxed { \frac {x - x_0}{F_x(x_0,y_0,z_0)} = \frac {y - y_0}{F_y(x_0,y_0,z_0)} = \frac {z - z_0}{F_z(x_0,y_0,z_0)}}$ -----> Symmetric equation of normal line