Chapter 10

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

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.

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? $$\mathbf{i}+\sqrt{3} \mathbf{j}$$

\(19-20\) Determine whether the given vectors are orthogonal, parallel, or neither. $$ \begin{array}{l}{\text { (a) } \mathbf{a}=\langle- 5,3,7\rangle, \quad \mathbf{b}=\langle 6,-8,2\rangle} \\ {\text { (b) } \mathbf{a}=\langle 4,6\rangle, \quad \mathbf{b}=\langle- 3,2\rangle} \\ {\text { (c) } \mathbf{a}=-\mathbf{i}+ 2 \mathbf{j}+5 \mathbf{k}, \quad \mathbf{b}=3 \mathbf{i}+4 \mathbf{j}-\mathbf{k}} \\ {\text { (d) } \mathbf{a}=2 \mathbf{i}+6 \mathbf{j}-4 \mathbf{k}, \quad \mathbf{b}=-3 \mathbf{i}-9 \mathbf{j}+6 \mathbf{k}}\end{array} $$

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

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

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

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

Find an equation of the largest sphere with center \((5,4,9)\) that is contained in the first octant.

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

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

A ball is thrown at an angle of \(45^{\circ}\) to the ground. If the ball lands 90 \(\mathrm{m}\) away, what was the initial speed of the ball?

Find the curvature of \(\mathbf{r}(t)=\left\langle t, t^{2}, t^{3}\right\rangle\) at the point \((1,1,1) .\)

Reduce the equation to one of the standard forms, classify the surface, and sketch it. \(y^{2}=x^{2}+\frac{1}{9} z^{2}\)

Show that \(0 \times \mathbf{a}=\mathbf{0}=\mathbf{a} \times \mathbf{0}\) for any vector \(\mathbf{a}\) in \(V_{3}\).

\(21-32=\) Find an equation of the plane. The plane through the point \(\left(-1, \frac{1}{2}, 3\right)\) and with normal vector \(\mathbf{i}+4 \mathbf{j}+\mathbf{k}\)

Describe in words the region of \(\mathbb{R}^{3}\) represented by thee quations or inequalities. \(x=5\)

If \(\mathbf{v}\) lies in the first quadrant and makes an angle \(\pi / 3\) with the positive \(x\) -axis and \(|\mathbf{v}|=4,\) find \(\mathbf{v}\) in component form.

Use vectors to decide whether the triangle with vertices \(P(1,-3,-2), Q(2,0,-4),\) and \(R(6,-2,-5)\) is right-angled.

Graph the curve with parametric equations \(x=\cos t\) \(y=\sin t, z=\sin 5 t\) and find the curvature at the point \((1,0,0)\)

Reduce the equation to one of the standard forms, classify the surface, and sketch it. \(4 x^{2}-y+2 z^{2}=0\)

Show that \((\mathbf{a} \times \mathbf{b}) \cdot \mathbf{b}=0\) for all vectors a and \(\mathbf{b}\) in \(V_{3}\).

\(21-32=\) Find an equation of the plane. The plane through the point \((2,0,1)\) and perpendicular to the line \(x=3 t, y=2-t, z=3+4 t\)

Describe in words the region of \(\mathbb{R}^{3}\) represented by theequations or inequalities. \(y=-2\)

If a child pulls a sled through the snow on a level path with a force of 50 \(\mathrm{N}\) exerted at an angle of \(38^{\circ}\) above the horizontal, find the horizontal and vertical components of the force.

Find the values of \(x\) such that the angle between the vectors \(\langle 2,1,-1\rangle,\) and \(\langle 1, x, 0\rangle\) is \( 45^{\circ} .\)

A gun has muzzle speed 150 \(\mathrm{m} / \mathrm{s}\) . Find two angles of elevation that can be used to hit a target 800 \(\mathrm{m}\) away.

Show that the curve with parametric equations \(x=t\) cos \(t\) \(y=t \sin t, z=t\) lies on the cone \(z^{2}=x^{2}+y^{2},\) and use this fact to help sketch the curve.

Reduce the equation to one of the standard forms, classify the surface, and sketch it. \(x^{2}+2 y-2 z^{2}=0\)

\(21-32=\) Find an equation of the plane. The plane through the point \((1,-1,-1)\) and parallel to the plane \(5 x-y-z=6\)

Describe in words the region of \(\mathbb{R}^{3}\) represented by theequations or inequalities. \(y<8\)

A quarterback throws a football with angle of elevation \(40^{\circ}\) and speed 60 \(\mathrm{ft} / \mathrm{s}\) . Find the horizontal and vertical components of the velocity vector.

Find a unit vector that is orthogonal to both \(\mathbf{i}+\mathbf{j}\) and \(\mathbf{i}+\mathbf{k}\)

Show that the curve with parametric equations \(x=\sin t\) \(y=\cos t, z=\sin ^{2} t\) is the curve of intersection of the surfaces \(z=x^{2}\) and \(x^{2}+y^{2}=1 .\) Use this fact to help sketch the curve.

Reduce the equation to one of the standard forms, classify the surface, and sketch it. \(y^{2}=x^{2}+4 z^{2}+4\)

\(21-32=\) Find an equation of the plane. The plane through the point \((2,0,1)\) and perpendicular to the line \(x=3 t, y=2-t, z=3+4 t\)

Describe in words the region of \(\mathbb{R}^{3}\) represented by theequations or inequalities. \(x \geqslant-3\)

Find two unit vectors that make an angle of \(60^{\circ}\) with \(\mathbf{v}=\langle 3,4\rangle\)

A batter hits a baseball 3 ft above the ground toward the center field fence, which is 10 ft high and 400 ft from home plate. The ball leaves the bat with speed 115 \(\mathrm{ft} / \mathrm{s}\) at an angle \(50^{\circ}\) above the horizontal. Is it a home run? (In other words, does the ball clear the fence?)

A medieval city has the shape of a square and is protected by walls with length 500 \(\mathrm{m}\) and height 15 \(\mathrm{m} .\) You are the commander of an attacking army and the closest you can get to the wall is 100 \(\mathrm{m} .\) Your plan is to set fire to the city by catapulting heated rocks over the wall (with an initial speed of 80 \(\mathrm{m} / \mathrm{s} ) .\) At what range of angles should you tell your men to set the catapult? (Assume the path of the rocks is perpendicular to the wall.)

At what points does the curve \(\mathbf{r}(t)=t \mathbf{i}+\left(2 t-t^{2}\right) \mathbf{k}\) inter- sect the paraboloid \(z=x^{2}+y^{2} ?\)

\(21-32=\) Find an equation of the plane. The plane through the points \((0,1,1),(1,0,1),\) and \((1,1,0)\)

Reduce the equation to one of the standard forms, classify the surface, and sketch it. \(4 x^{2}+y^{2}+4 z^{2}-4 y-24 z+36=0\)

Describe in words the region of \(\mathbb{R}^{3}\) represented by theequations or inequalities. 0\(\leqslant z \leqslant 6\)

\(25-26\) Find the acute angle between the lines. $$2 x-y=3, \quad 3 x+y=7$$

Show that a projectile reaches three-quarters of its maximum height in half the time needed to reach its maximum height.

At what point does the curve have maximum curvature? What happens to the curvature as \(x \rightarrow \infty ?\) $$y=\ln x$$

Graph the curve with parametric equations $$x=\sqrt{1-0.25 \cos ^{2}(10 t)} \cos t$$ $$y=\sqrt{1-0.25 \cos ^{2}(10 t)} \sin t$$ $$z=0.5 \cos (10 t)$$ Explain the appearance of the graph by showing that it lies on a sphere.

Reduce the equation to one of the standard forms, classify the surface, and sketch it. \(4 y^{2}+z^{2}-x-16 y-4 z+20=0\)

Describe in words the region of \(\mathbb{R}^{3}\) represented by theequations or inequalities. \(z^{2}=1\)