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• Notes

$Q_{1}: \vec{r}(t)=\left\langle\sqrt{t - 2}, 3,1 / t^{2}\right\rangle$

1- $\overrightarrow{r^{\prime}(t)} :\left\langle\frac{d(\sqrt{t-2})}{d t}, \frac{d(3)}{d t}, \frac{d\left(1 / t^{2}\right)}{d t}\right\rangle$

2- $\vec{r\prime}(t)=\left\langle\frac{1}{2 \sqrt{t-2}}, 0, \frac{-2}{t^{3}}\right\rangle$

$Q_{2}:$ Unit tangent vector??

$\vec{r}(t)=\left\langle t^{2}-2 t, 1+3 t, \frac{1}{3} t^{3}+\frac{1}{2} t^{2}\right\rangle$

1- $\overrightarrow{r^{\prime}(t)} ??$

$\overrightarrow{r^{\prime}(t)}=\left\langle{2 t-2,3, t^{2}+t}\right\rangle$

2- $\overrightarrow{r^\prime}(2)=\langle 2,3,6\rangle$

3- $|\vec{r^\prime}(2)|=\sqrt{4+9+36} = \sqrt{49}=7$

4- $\vec{T}(2)=\frac{\vec r^{\prime}(2)}{\left|\vec r^{\prime}(2)\right|}$

$=\frac{\langle 2,3,6\rangle}{7}$

$=\left\langle\frac{2}{7}, \frac{3}{7}, \frac{6}{7}\right\rangle$

Q3: $\vec{r}_{1}(s)=\left\langle 3-s , s-2, s^{2}\right\rangle$

Req: 1- point of intersection

2- Angle bet two curves at point of intersection

solution:

(1) $\overrightarrow{r_{1}}(t)=\overrightarrow{r_{2}}(s)$

$t=3-s \rightarrow (1)$

$1-t=s-2 \rightarrow (2)$
$3-t^{2}=s^{2} \rightarrow (3)$

(2) $3-(3-s)^{2}=s^{2}$

$\Rightarrow 3+\left(9+s^{2}-6s\right)=s^{2}$

$6s=12 \Rightarrow s=2 \Rightarrow t=1$

(3) $\vec{r}_{1}(1)=\langle 1,0,4\rangle=\vec{r}_{2}(s)$

(1) $\vec{r^\prime}(t)=\langle 1,-1,2 t\rangle \Rightarrow \vec{r^\prime}(1)=\langle 1,-1,2\rangle$

$\overrightarrow{r^\prime(s)}=\langle- 1,1,2s\rangle \Rightarrow \vec{ r^{\prime}}(2)=\langle- 1,1,4\rangle$

$\cos \theta=\frac{\vec{r}_{1}^{\prime}(t) \cdot \vec{r_{2}}^{\prime}(2)}{\left|\vec{r}_{1}^{\prime}(1)\right|\left|\vec{r}_{2}^{\prime}(2)\right|}=\frac{-1-1+8}{\sqrt{6} \cdot \sqrt{18}}$

$\cos \theta=\frac{6}{6 \sqrt{3}}$

$\theta=54.73^{\circ}$

Q4: Integration

$\int_{0}^{4}\left(2 t^{\frac{3}{2}}\right) \hat{i}+(t+1) \sqrt{t} \hat{k} dt$

(1) $=\left[\frac{4}{5} t^{\frac{5}{2}} \hat{i}\right]_{0}^{4}+\left[\left(\frac{2}{5} t^{\frac{5}{2}}+\frac{2}{3} t^{\frac{3}{2}}\right) \hat{k}\right]_{0}^{4}$

(2) $=\frac{4}{5} \cdot(4)^{\frac{5}{2}} \hat{i}+\left(\frac{2}{5} \cdot(4)^{\frac{5}{2}}+\frac{2}{3} \cdot(4)^{\frac{3}{2}}\right)\hat k$

$=\frac{124}{5} \hat{i}+\frac{256}{15} \hat{k}$