91Ó°ÊÓ

Find the unit tangent vector for the following parameterized curves. $$\mathbf{r}(t)=\langle 8, \cos 2 t, 2 \sin 2 t\rangle, \text { for } 0 \leq t \leq 2 \pi$$

Find an equation or inequality that describes the following objects. A sphere with center (1,2,3) and radius 4

Use the alternative curvature formula \(\kappa=|\mathbf{v} \times \mathbf{a}| /|\mathbf{v}|^{3}\) to find the curvature of the following parameterized curves. $$\mathbf{r}(t)=\left\langle 4+t^{2}, t, 0\right\rangle$$

For the following trajectories, find the speed associated with the trajectory and then find the length of the trajectory on the given interval. $$\mathbf{r}(t)=\left\langle 2 t^{3},-t^{3}, 5 t^{3}\right\rangle, \text { for } 0 \leq t \leq 4$$

Compute the dot product of the vectors u and \(\mathbf{v},\) and find the angle between the vectors. $$\mathbf{u}=2 \mathbf{i}-3 \mathbf{k} \text { and } \mathbf{v}=\mathbf{i}+4 \mathbf{j}+2 \mathbf{k}$$

Find the area of the parallelogram that has two adjacent sides \(\mathbf{u}\) and \(\mathbf{v}\) $$\mathbf{u}=8 \mathbf{i}+2 \mathbf{j}-3 \mathbf{k}, \mathbf{v}=2 \mathbf{i}+4 \mathbf{j}-4 \mathbf{k}$$

Compute the dot product of the vectors u and \(\mathbf{v},\) and find the angle between the vectors. $$\mathbf{u}=\mathbf{i}-4 \mathbf{j}-6 \mathbf{k} \text { and } \mathbf{v}=2 \mathbf{i}-4 \mathbf{j}+2 \mathbf{k}$$

Find the unit tangent vector for the following parameterized curves. $$\mathbf{r}(t)=\langle\sin t, \cos t, \cos t\rangle, \text { for } 0 \leq t \leq 2 \pi$$

Find an equation or inequality that describes the following objects. A sphere with center (1,2,0) passing through the point (3,4,5)

Use the alternative curvature formula \(\kappa=|\mathbf{v} \times \mathbf{a}| /|\mathbf{v}|^{3}\) to find the curvature of the following parameterized curves. $$\mathbf{r}(t)=\langle\sqrt{3} \sin t, \sin t, 2 \cos t\rangle$$