91Ó°ÊÓ

Make the required change in the given equation. \(\rho \sin \phi=1\) to Cartesian coordinates

The parabola \(z=2 x^{2}\) in the \(x z\) -plane is revolved about the \(z\) -axis. Write the equation of the resulting surface in cylindrical coordinates.

Show that if the speed of a moving particle is constant its acceleration vector is always perpendicular to its velocity vector.

Consider a horizontal triangular table with each vertex angle less than \(120^{\circ}\). At the vertices are frictionless pulleys over which pass strings knotted at \(P\), each with a weight \(W\) attached as shown in Figure \(20 .\) Show that at equilibrium the three angles at \(P\) are equal; that is, show that \(\alpha+\beta=\alpha+\gamma=\beta+\gamma=120^{\circ}\).

, find the curvature \(\kappa\), the unit tangent vector \(\mathbf{T}\), the unit normal vector \(\mathbf{N}\), and the binormal vector \(\mathbf{B}\) at \(t=t_{1} .\) $$ \mathbf{r}(t)=3 \cosh (t / 3) \mathbf{i}+t \mathbf{j} ; t_{1}=1 $$

Find each of the given projections if \(\mathbf{u}=3 \mathbf{i}+2 \mathbf{j}+\mathbf{k}, \mathbf{v}=2 \mathbf{i}-\mathbf{k}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}-3 \mathbf{k}\). \(\operatorname{proj}_{\mathbf{u}} \mathbf{w}\)

Prove Lagrange's Identity, $$ \|\mathbf{u} \times \mathbf{v}\|^{2}=\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2}-(\mathbf{u} \cdot \mathbf{v})^{2} $$ without using Theorem \(\mathrm{A}\).

Find the arc length of the given curve. \(x=2 \cos t, y=2 \sin t, z=3 t ;-\pi \leq t \leq \pi\)

Find the coordinates of the foci of the ellipse that is the intersection of \(z=x^{2} / 4+y^{2} / 9\) with the plane \(z=4\).

Find the arc length of the given curve. \(x=2 \cos t, y=2 \sin t, z=t / 20 ; 0 \leq t \leq 8 \pi\)