91Ó°ÊÓ

Find equations of the following lines. The line through (0,2,1) that is perpendicular to both \(\mathbf{u}=\langle 4,3,-5\rangle\) and the \(z\) -axis

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

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

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

Arc length calculations Find the length of the following two and three- dimensional curves. $$\mathbf{r}(t)=\langle 3 \cos t, 4 \cos t, 5 \sin t\rangle, \text { for } 0 \leq t \leq 2 \pi$$

Compute the dot product of the vectors \(\mathbf{u}\) and \(\mathbf{v},\) and find the angle between the vectors. \(\mathbf{u}=\langle 3,-5,2\rangle\) and \(\mathbf{v}=\langle-9,5,1\rangle\)

Sketch the plane parallel to the \(y z\) -plane through (2,4,2) and find its equation.

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

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

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