Question

Let Cbe the graph of a vector-valued function r. The plane determined by the vectors $N\left(t_0\right),B\left(t_0\right)$and containing the point $\mathit{r}\left(t_0\right)$ is called the normal plane forC at$\mathit{r}\left(t_0\right)$. Find the equation of the normal plane to the helix determined by$\mathbf{r}\mathbf{\left(}\mathbf{t}\mathbf{\right)}=<\cos \mathrm{t},\sin \mathrm{t},\mathrm{t}>$for$t=\mathrm{\pi }$.

Short answer

The equation of the normal plane for C at$r\left(t=\mathrm{\pi }\right)$is$-y+z=\mathrm{\pi }$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

$$\begin{gathered} \mathit{r}\left(t\right)=<\cos \mathrm{t},\sin \mathrm{t},\mathrm{t}>\mathbf{,} \\ t=\mathrm{\pi } \end{gathered}$$

Step 2. Find the equation of the normal plane to rt.

On calculating $r\text{'}\left(t\right)$,

role="math" localid="1649579224387" $$\begin{gathered} \mathit{r}\mathbf{\text{'}}\left(t\right)=<-\sin \mathrm{t},cos\mathrm{t},1> \\ ||\mathit{r}\mathbf{\text{'}}\left(t\right)||=\sqrt{{\sin }^{2}t+{\cos }^{2}t+1} \\ =\sqrt{2} \\ T\left(t\right)=\frac{\mathit{r}\text{'}\left(t\right)}{||\mathit{r}\mathbf{\text{'}}\left(t\right)||} \\ T\left(t\right)=\frac{<-\sin \mathrm{t},cos\mathrm{t},1>}{\sqrt{2}}......\left(i\right) \end{gathered}$$

Substitute $t=\mathrm{\pi }$in equation (i),

Step 3. Find the principal unit normal vector to rt.

On calculating $T\text{'}\left(t\right)$,

role="math" localid="1649579517209"

Substitute $t=\mathrm{\pi }$in equation (ii),

role="math" localid="1649579559027"

Step 4. Find the value of Bπ.

On calculating $B\left(\mathrm{\pi }\right)$,

role="math" localid="1649579665264"

Step 5. Consider the normal plane.

The plane determined by the vectors $N\left(t_0\right),B\left(t_0\right)$and containing the point $r\left(t_0\right)$ is called the normal plane for C at $r\left(t_0\right)$.

Thus, the equation of the normal plane to $r\left(t\right)$at $t=\mathrm{\pi }$is given below,

role="math" localid="1649579845696"

First computing $N\left(\mathrm{\pi }\right)\times B\left(\mathrm{\pi }\right)$,

Step 6. Evaluate equation (iii).

On evaluating equation (iii),

Thus, the equation of the normal plane for c at $r\left(t=\mathrm{\pi }\right)$is given below,

$-y+z=\mathrm{\pi }$