Question

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

$\mathbf{r}\left(t\right)=(t,2t\sin t,2t\cos t),t=\pi$

Short answer

The principal unit normal vector to $r\left(t\right)$is,

at$t=\pi$.

Step-by-step solution

Step 1. Given Information

We are given an function,

$\mathbf{r}\left(t\right)=(t,2t\sin t,2t\cos t)$

And,

role="math" $t=\pi$

Step 2. Finding principal unit normal vector 

Principal unit normal vector:

If $r\left(t\right)$is a twice differentiable vector function, then the principal unit vector to $r\left(t\right)$is

localid="1649652059482" $N\left(t\right)=\frac{T^{\text{'}}\left(t\right)}{\left|T^{\text{'}}\left(t\right)\right|}$

$$\begin{gathered} r\left(t\right)=⟨t,2t\sin t,2t\cos t⟩ \\ {r}^{\text{'}}\left(t\right)=⟨1,2(t\cos t+\sin t),2(-t\sin t+\cos t)⟩ \\ \left|r^{\text{'}}\left(t\right)\right|=\sqrt{1+4(t\cos t+\sin t{)}^2+4(-t\sin t+\cos t{)}^2} \\ =\sqrt{1+4\left(t^2+1\right)} \\ =\sqrt{4t^2+5} \end{gathered}$$

Step 3. Finding principal unit normal vector 

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

role="math" localid="1649653047945"

Using the Quotient rule for derivatives,

Step 4. Finding principal unit normal vector 

We need to evaluate N(t) at $t=\pi$,

So,

role="math" localid="1649654290745"

Thus, the principal unit normal vector is,