# 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)

### Related post:

- New technique developed to detect autism in children – EurekAlert
- Gujarat Board to offer Mathematics to Non-Science Students from this year onwards – Jagran Josh
- Institute of Mathematics & Application (IMA) Recruitment 2019 for Professor Posts – Jagran Josh
- Help with primary school mathematics — September is just nine weeks away – Galway Advertiser
- Government Launches Effort to Strengthen Math Skills & Improve Job Prospects – Government of Ontario News
- Chaiwalla to Doctor: 5 Teachers Providing Free JEE/NEET Coaching to Needy Students – The Better India
- A celebration of Science, Technology, Engineering, and Mathematics (STEM) – Daily Trust
- Sum of a life – THE WEEK
- Assistant Professor (Tenure Track) in Mathematics in South Holland, Delft – IamExpat in the Netherlands
- Standing in Galileo’s shadow: Why Thomas Harriot should take his place in the scientific hall of fame – OUPblog