# Normal Lines on a Surface Examples 1

Recall from the Normal Lines on a Surface page that if we have a two variable real-valued function $z = f(x, y)$ and $P(x_0, y_0, z_0)$ is a point on the surface generated by this function, then the normal line at $P$ is the line that passes through $P$ and is perpendicular to the tangent plane at $P$. This normal line is given parametrically by:

(1)We will now look at some examples of computing the normal line to a surface at a specific point.

## Example 1

**Find parametric equations for the normal line to the paraboloid $f(x, y) = 4 - x^2 - y^2$ at the point $(1, 1, 2)$.**

We first calculate the partial derivatives of $f$. We have that:

(2)Therefore, $\frac{\partial}{\partial x} f(1, 1) = -2$ and $\frac{\partial}{\partial y} f(1, 1) = -2$. Therefore the normal line to the paraboloid at $(1, 1, 2)$ is given parametrically by:

(3)## Example 2

**Find parametric equations for the normal line to the circular cylinder $x^2 + y^2 = 1$ at $(1, 0, 2)$ ( Hint: You do not need to calculate partial derivatives in order to find the parametric equations for this line).**

Notice that the circular cylinder $x^2 + y^2 = 1$ is formed from vertical rulings that are parallel to the $z$-axis. The vector $(0, 0, 1)$ is parallel to the $z$-axis, and so any vector of the form $(a, b, 0)$ is perpendicular to the $z$ axis since by the dot product we have that:

(4)We want to choose a vector that is also in the same direction as the point $(1, 0, 5)$, so choose the vector $(1, 0, 0)$. Therefore we have that the parametric equations of the normal line at $(1, 0, 2)$ is given by:

(5)