1.) All non-negative $t$ except $t=1$. Also written $\{ t \in \mathbb{R} \colon t \geq 0$ and $t \neq 1 \}$.

2.) $< 0, 2, 1 >$

3.) $\vec{r}(t) = t < 0,0,2 > + (1-t) < -1,2,3 >$

4.) Parametrize the equation of the cylinder as you normally would a circle(think polar!) and then solve the equation for the plane to get $z = \frac{5-x}{2}$. Substitute in what $x$ is in your circle parametrization.

5.) $\vec{r}'(t) = < e^t \sin t + e^t \cos t, -\sin t, 5t^4 >$

6.) Compute $\vec{r}'(t) = < e^{3t} + 3te^{3t}, \cos t, 14t >$ and then use the formula in the book for $\vec{T}(t)$.

7.) $\displaystyle\int_2^3 \vec{r}(t) dt = \left< \log(3) - \log(2), 3\log(3) - 3 -(2 \log(2) - 2), 5 \left[ \frac{3^2}{2} - \frac{2^2}{2} \right] \right>$

8.) Use the formula $\frac{d}{dt}[\vec{u}(t) \times \vec{v}(t)] = \vec{u}'(t) \times \vec{v}(t) + \vec{u}(t) \times \vec{v}'(t)$.