Question

Principal unit normal vectors: Find the principal unit normal vector for the given function at the specified value of t.

Short answer

Ans: Thus the principal unit normal vector of at $t=2$is$\left(\frac{-12\sqrt{145}}{145},\frac{\sqrt{145}}{145}\right)$

Step-by-step solution

Step 1. Given information:

Step 2. Simplifying the principal unit normal vector:

We start by computing $r^{\text{'}}\left(t\right)$first:

The unit tangent vector of $r\left(t\right)$

Step 3. Finding the principal unit normal vector:

The principal unit normal vector

$N\left(T\right)=\frac{T^{\text{'}}\left(t\right)}{||T^{\text{'}}\left(t\right)||}$

$\mathrm{Att}=2$

Thus the principal unit normal vector of $r\left(t\right)$at $t=2$is $\left(\frac{-12\sqrt{145}}{145},\frac{\sqrt{145}}{145}\right)$