91影视

$\begin{array}{l}鈭�r^{\mathrm{鈥�}}\left(t\right)鈭�=鈭�鉄�3\cos 鈦�t,鈭�5\sin 鈦�t,4\cos 鈦�t鉄�鈭�\\ =\sqrt{(3\cos 鈦�t{)}^2+(鈭�5\sin 鈦�t{)}^2+(4\cos 鈦�t{)}^2}\\ =\sqrt{9{\cos }^2鈦�t+25{\sin }^2鈦�t+16{\cos }^2鈦�t}\\ =\sqrt{25{\cos }^2鈦�t+25{\sin }^2鈦�t}\\ =\sqrt{25\left({\cos }^2鈦�t+{\sin }^2鈦�t\right)}\\ =5\sin ce{\sin }^{2}t+{\cos }^{2}t=1\\ \text{The unit tangent vector to}r\left(t\right)\text{is}\\ \begin{array}{l}T\left(t\right)=\frac{r^{\mathrm{鈥�}}\left(t\right)}{鈭�r^{鈥测赌�}\left(t\right)鈭�}\\ =\frac{鉄�3\cos 鈦�t,鈭�5\sin 鈦�t,4\cos 鈦�t鉄�}{5}\\ =\frac{1}{5}鉄�3\cos 鈦�t,鈭�5\sin 鈦�t,4\cos 鈦�t鉄�\\ \text{Thus the unit tangent vector to the given vector function is}\\ \frac{1}{5}鉄�3\cos 鈦�t,鈭�5\sin 鈦�t,4\cos 鈦�t鉄�\end{array}\end{array}$