# Normal Lines to Level Surfaces Examples 2

Recall from the Normal Lines to Level Surfaces page that if we have a level surface $f(x, y, z) = k$ then we can determine the normal line that passes through the point $P(x_0, y_0, z_0)$ with the following parametric equations:

(1)

Let’s look at some examples of computing normal lines to level surfaces.

## Example 1

**Find parametric equations for the normal line to the sideways paraboloid $y = x^2 + z^2$ at the point $(1, 2, 1)$.**

Let $f(x, y, z) = y – x^2 – z^2 = 0$. The partial derivatives of this function are:

(2)

The partial derivatives of $f$ evaluated at the point $(1, 2, 1)$ are:

(3)

Therefore the normal line to the paraboloid $y = x^2 + z^2$ at the point $(1, 2, 1)$ is given parametrically by:

(4)

## Example 2

**Find parametric equations for the normal line to the surface $y^3 = (x – z)^2$ at the point $(2, 1, 1)$.**

Let $f(x, y, z) = y^3 – (x – z)^2 = y^3 – (x^2 – 2xz + z^2) = 0$. The partial derivatives of this function are:

(5)

The partial derivatives of $f$ evaluated at the point $(2, 1, 1)$ are:

(6)

Therefore the normal line to the surface $y^3 = (x – z)^2$ at the point $(2, 1, 1)$ is given parametrically by:

(7)