Normal Line

Let $f: \mathbb{R}^2 \to \mathbb{R}$. Normal line at point $P$, is the line passing through $P$ and perpendicular to the tangent plane. The equation of the normal line is:

$$\frac{x − x_0}{F_x(x_0,y_0,z_0)}= \frac{y − y_0}{F_y(x_0,y_0,z_0)}$$

To a surface level

Let

• $F(x,y,z)$ is a 3-variable function
• $S$ is the level surface of $F$ at $F(x,y,z)=k$
• $P = (x_0,y_0,z_0)$ be a point on $S$

Normal line to the surface $S$ at point $P$, is the line passing through $P$ and perpendicular to the tangent plane. The equation of the normal line is:

$$\frac{x − x_0}{F_x(x_0,y_0,z_0)}= \frac{y − y_0}{F_y(x_0,y_0,z_0)}= \frac{z − z_0}{F_z(x_0,y_0,z_0)}$$