91影视

$\mathbf{r}\left(t\right)=鉄�\sin 鈦�2t,\cos 鈦�2t,t鉄�\text{at}t=\frac{蟺}{2}$

$\begin{array}{r}r\left(t\right)=鉄�\sin 鈦�2t,\cos 鈦�2t,t鉄�\\ r^{\mathrm{鈥�}}\left(t\right)=鉄�2\cos 鈦�2t,鈭�2\sin 鈦�2t,1鉄�\\ 鈭�r^{\mathrm{鈥�}}\left(t\right)鈭�=\sqrt{(2\cos 鈦�2t{)}^2+(鈭�2\sin 鈦�2t{)}^2+1}\\ =\sqrt{4\left({\cos }^2鈦�2t+{\sin }^2鈦�2t\right)+1}\\ =\sqrt{5}\\ T\left(t\right)=\frac{r^{\mathrm{鈥�}}\left(t\right)}{鈭�r^{\mathrm{鈥�}}\left(t\right)鈭�}\\ =\frac{鉄�2\cos 鈦�2t,鈭�2\sin 鈦�2t,1鉄�}{\sqrt{5}}\end{array}$

$t=\frac{蟺}{2},T\left(t\right)=T\left(\frac{蟺}{2}\right)=\frac{1}{\sqrt{5}}鉄�2\cos 鈦�蟺,鈭�2\sin 鈦�蟺,1鉄�$

$\begin{array}{r}{T}^{\mathrm{鈥�}}\left(t\right)=\frac{1}{\sqrt{5}}鉄�鈭�4\sin 鈦�2t,鈭�4\cos 鈦�2t,0鉄�\\ 鈭�T^{\mathrm{鈥�}}\left(t\right)鈭�=\sqrt{\frac{16}{5}\left({\sin }^2鈦�2t+{\cos }^2鈦�2t\right)}=\frac{4}{\sqrt{5}}\\ N\left(t\right)=\frac{T^{\mathrm{鈥�}}\left(t\right)}{鈭�T^{\mathrm{鈥�}}\left(t\right)鈭�}\\ =\frac{1}{\sqrt{5}}鉄�鈭�4\sin 鈦�2t,鈭�4\cos 鈦�2t,0鉄�\\ =鉄�鈭�\sin 鈦�2t,鈭�\cos 鈦�2t,0鉄�\\ \text{At}t=\frac{蟺}{2},N\left(t\right)=N\left(\frac{蟺}{2}\right)=鉄�鈭�\sin 鈦�蟺,鈭�\cos 鈦�蟺,0鉄�\\ =鉄�0,1,0鉄�\end{array}$

First calculating $N\left(\frac{蟺}{2}\right)脳B\left(\frac{蟺}{2}\right)$

$\left(T\left(\frac{蟺}{2}\right)脳\mathrm{B}\left(\frac{蟺}{2}\right)\right)鈰�鉄�x鈭�x\left(0\right),y鈭�y\left(0\right),z鈭�z\left(0\right)鉄�=0$

Now,