91Ó°ÊÓ

Sketch the following planes in the window \([0,5] \times[0,5] \times[0,5]\) $$z=y$$

Find a tangent vector at the given value of \(t\) for the following curves. $$\mathbf{r}(t)=\langle 2 \sin t, 3 \cos t, \sin (t / 2)\rangle, t=\pi$$

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

Arc length calculations Find the length of the following two and three- dimensional curves. $$\mathbf{r}(t)=\left\langle t^{2} / 2,(2 t+1)^{3 / 2} / 3\right\rangle, \text { for } 0 \leq t \leq 2$$

Find the unit tangent vector \(\mathbf{T}\) and the curvature \(\kappa\) for the following parameterized curves. $$\mathbf{r}(t)=\left\langle\cos ^{3} t, \sin ^{3} t\right\rangle$$

Find equations of the following lines. The line through (1,-3,4) that is parallel to the line \(\mathbf{r}(t)=\langle 3+4 t, 5-t, 7\rangle\)

Arc length calculations Find the length of the following two and three- dimensional curves. $$\mathbf{r}(t)=\left\langle e^{2 t}, 2 e^{2 t}+5,2 e^{2 t}-20\right\rangle, \text { for } 0 \leq t \leq \ln 2$$

Find the unit tangent vector \(\mathbf{T}\) and the curvature \(\kappa\) for the following parameterized curves. $$\mathbf{r}(t)=\left\langle\int_{0}^{t} \cos \left(\pi u^{2} / 2\right) d u, \int_{0}^{t} \sin \left(\pi u^{2} / 2\right) d u\right\rangle, t>0$$

The plane that passes through \((2,0,0),(0,3,0),\) and (0,0,4)

Compute the dot product of the vectors \(\mathbf{u}\) and \(\mathbf{v},\) and find the angle between the vectors. \(\mathbf{u}=4 \mathbf{i}+3 \mathbf{j}\) and \(\mathbf{v}=4 \mathbf{i}-6 \mathbf{j}\)