91Ó°ÊÓ

A projectile is fired with an initial speed of 200 \(\mathrm{m} / \mathrm{s}\) and angle of elevation \(60^{\circ} .\) Find (a) the range of the projectile, (b) the maximum height reached, and (c) the speed at impact.

What is the angle between the given vector and the positive direction of the \(x\) -axis? What is the angle between the given vector and the positive direction of the \(x\) -axis? $$\mathbf{i}+\sqrt{3} \mathbf{j}$$

Find equations of the spheres with center \((2,-3,6)\) that touch (a) the \(x y\) -plane, (b) the \(y z\) -plane, (c) the \(x z\) -plane.

\(17-20=\) Determine whether the lines \(L_{1}\) and \(L_{2}\) are parallel, skew, or intersecting. If they intersect, find the point of intersection. $$ \begin{array}{l}{L_{1} : \frac{x-2}{1}=\frac{y-3}{-2}=\frac{z-1}{-3}} \\\ {L_{2} : \frac{x-3}{1}=\frac{y+4}{3}=\frac{z-2}{-7}}\end{array} $$

\(17-20=\) Determine whether the lines \(L_{1}\) and \(L_{2}\) are parallel, skew, or intersecting. If they intersect, find the point of intersection. $$ \begin{array}{l}{L_{1} : \frac{x}{1}=\frac{y-1}{-1}=\frac{z-2}{3}} \\ {L_{2} : \frac{x-2}{2}=\frac{y-3}{-2}=\frac{z}{7}}\end{array} $$

What is the angle between the given vector and the positive direction of the \(x\) -axis? $$8 \mathbf{i}+6 \mathbf{j}$$

Use traces to sketch and identify the surface. \(x=y^{2}-z^{2}\)

\(19-20\) Determine whether the given vectors are orthogonal, parallel, or neither. $$\begin{array}{ll}{\text { (a) } \mathbf{u}=\langle- 3,9,6\rangle,}\quad {\mathbf{v}=\langle 4,-12,-8\rangle} \\ {\text { (b) } \mathbf{u}} {=\mathbf{i}-\mathbf{j}+ 2 \mathbf{k}, \quad \mathbf{v}=2 \mathbf{i}-\mathbf{j}+\mathbf{k}} \\ {(\mathrm{c}) \mathbf{u}} {=\langle a, b, c\rangle, \quad \mathbf{v}=\langle- b, a, 0\rangle}\end{array}$$

Find two unit vectors orthogonal to both \(\mathbf{j}-\mathbf{k}\) and \(\mathbf{i}+\mathbf{j}\).

Find the curvature of \(\mathbf{r}(t)=\left\langle t^{2}, \ln t, t \ln t\right\rangle\) at the point \((1,0,0) .\)