Motion of objects

Exercise2.2

1.

Find the equation of straight line goes through the points $(-1,0,2),(1,4,3)$.

2.

Draw the plane curve $${\boldsymbol{r}(t) = \sin{t}\:\boldsymbol{i} + \sin{t}\:\boldsymbol{j}}$$.

3.

Find the equation of the tangent line to the curve $${\boldsymbol{r}(t) = \cos{t}\:\boldsymbol{i} + \sin{t}\:\boldsymbol{j}}$$ at $${t = \frac{\pi}{4}}$$．

4.

Find the arc length of $${\boldsymbol{r}(t) = \cos{t}\:\boldsymbol{i} + \sin{t}\:\boldsymbol{j} + t\:\boldsymbol{k}}$$ for $0 \leq t \leq 2\pi$.

5.

For

$$\boldsymbol{r}(t) = (\cos\pi t, \sin\pi t, t)$$

$t = 1$, find ${\bf v}(t),\boldsymbol{A}(t),v,{\bf t},\boldsymbol{n}$.

6.

For the curve $\boldsymbol{r} = 2a(\sin^{-1}{t} + t\sqrt{1- t^2})\:\boldsymbol{i} + 2at^2\:\boldsymbol{j} + 4at\:\boldsymbol{k}$，find the followings．where, $a$ is an arbitrary positive constant.

(a)

the arc length the curve for $t_{1} \leq t \leq t_{2}$

(b)

the unit tangent vector ${\bf t}$

(c)

the normal vector $\boldsymbol{n}$ and the curvature $\kappa$

(d)

the binormal vector $\boldsymbol{B}$ and the torsion $\tau$