91Ó°ÊÓ

Determine whether the following curves use arc length as a parameter. If not, find a description that uses arc length as a parameter. $$\mathbf{r}(t)=\langle 1, \sin t, \cos t\rangle, \text { for } t \geq 1$$

Evaluate the following limits. $$\lim _{t \rightarrow \pi / 2}\left(\cos 2 t \mathbf{i}-4 \sin t \mathbf{j}+\frac{2 t}{\pi} \mathbf{k}\right)$$

Answer the following questions about torque. Let \(\mathbf{r}=\overrightarrow{O P}=\mathbf{i}+\mathbf{j}+\mathbf{k} .\) A force \(\mathbf{F}=\langle 20,0,0\rangle\) is applied at P. Find the torque about \(O\) that is produced.

For the given vectors u and \(\mathbf{v},\) evaluate the following expressions. a. \(3 \mathbf{u}+2 \mathbf{v}\) b. \(4 \mathbf{u}-\mathbf{v}\) 3 c. \(|\mathbf{u}+3 \mathbf{v}|\) $$\mathbf{u}=\langle-2,1,-2\rangle, \mathbf{v}=\langle 1,1,1\rangle$$

Evaluate the following limits. $$\lim _{t \rightarrow \ln 2}\left(2 e^{t} \mathbf{i}+6 e^{-t} \mathbf{j}-4 e^{-2 t} \mathbf{k}\right)$$

Compute \(\mathbf{r}^{\prime \prime}(t)\) and \(\mathbf{r}^{\prime \prime \prime}(t)\) for the following functions. $$\mathbf{r}(t)=\left\langle 3 t^{12}-t^{2}, t^{8}+t^{3}, t^{-4}-2\right\rangle$$

Define the points \(P(-4,1), Q(3,-4),\) and \(R(2,6)\) Express \(\overrightarrow{P Q}\) in the form \(a \mathbf{i}+b \mathbf{j}\)

Answer the following questions about torque. Let \(\mathbf{r}=\overrightarrow{O P}=\mathbf{i}-\mathbf{j}+2 \mathbf{k} .\) A force \(\mathbf{F}=\langle 10,10,0\rangle\) is applied at \(P .\) Find the torque about \(O\) that is produced.

For the given vectors u and \(\mathbf{v},\) evaluate the following expressions. a. \(3 \mathbf{u}+2 \mathbf{v}\) b. \(4 \mathbf{u}-\mathbf{v}\) 3 c. \(|\mathbf{u}+3 \mathbf{v}|\) $$\mathbf{u}=\langle-5,0,2\rangle, \mathbf{v}=\langle 3,1,1\rangle$$

Calculate the work done in the following situations. A constant force \(\mathbf{F}=\langle 2,4,1\rangle\) (in newtons) moves an object from (0,0,0) to (2,4,6) . (Distance is measured in meters.)