91影视

We are given an function,

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

And,

role="math" $t=蟺$

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

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

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

role="math" localid="1649653047945"

Using the Quotient rule for derivatives,

We need to evaluate N(t) at $t=蟺$,

So,

role="math" localid="1649654290745"

Thus, the principal unit normal vector is,