91Ó°ÊÓ

Give a geometric description of the projection of \(\mathbf{u}\) onto \(\mathbf{v}\).

Use the shell method to find the volume of the solid below the surface of revolution and above the \(x y\) -plane. The curve \(z=\sin y(0 \leq y \leq \pi)\) in the \(y z\) -plane is revolved about the \(z\) -axis.

Find an equation of the surface satisfying the conditions, and identify the surface. The set of all points equidistant from the point \((0,2,0)\) and the plane \(y=-2\)

The top of a rubber bushing designed to absorb vibrations in an automobile is the surface of revolution generated by revolving the curve \(z=\frac{1}{2} y^{2}+1(0 \leq y \leq 2)\) in the \(y z\) -plane about the \(z\) -axis. (a) Find an equation for the surface of revolution. (b) All measurements are in centimeters and the bushing is set on the \(x y\) -plane. Use the shell method to find its volume. (c) The bushing has a hole of diameter 1 centimeter through its center and parallel to the axis of revolution. Find the volume of the rubber bushing.

Determine the intersection of the hyperbolic paraboloid \(z=y^{2} / b^{2}-x^{2} / a^{2}\) with the plane \(b x+a y-z=0\). (Assume \(a, b>0 .)\)

Find \(a\) and \(b\) such that \(v=a \mathbf{u}+b w\), where \(\mathbf{u}=\langle\mathbf{1}, \mathbf{2}\rangle\) and \(\mathbf{w}=\langle\mathbf{1},-\mathbf{1}\rangle\) $$ \mathbf{v}=\langle-1,7\rangle $$

Find the component form of \(v\) given the magnitudes of \(\mathbf{u}\) and \(\mathbf{u}+\mathbf{v}\) and the angles that \(\mathrm{u}\) and \(\mathrm{u}+\mathbf{v}\) make with the positive \(x\) -axis. $$ \begin{aligned} &\|\mathbf{u}\|=1, \theta=45^{\circ} \\ &\|\mathbf{u}+\mathbf{v}\|=\sqrt{2}, \theta=90^{\circ} \end{aligned} $$

Find the angle between a cube's diagonal and one of its edges.

(a) find the unit tangent vectors to each curve at their points of intersection and (b) find the angles \(\left(0 \leq \theta \leq 90^{\circ}\right)\) between the curves at their points of intersection. $$ y=x^{3}, \quad y=x^{1 / 3} $$

(a) find the unit tangent vectors to each curve at their points of intersection and (b) find the angles \(\left(0 \leq \theta \leq 90^{\circ}\right)\) between the curves at their points of intersection. $$ (y+1)^{2}=x, \quad y=x^{3}-1 $$