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

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

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

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