Chapter 10

The magnitude of a velocity vector is called speed. Suppose that a wind is blowing from the direction \(\mathrm{N} 45^{\circ} \mathrm{W}\) at a speed of 50 \(\mathrm{km} / \mathrm{h} .\) (This means that the direction from which the wind blows is \(45^{\circ}\) west of the northerly direction.) A pilot is steering a plane in the direction \(\mathrm{N6} 0^{\circ} \mathrm{E}\) at an airspeed (speed in still air) of 250 \(\mathrm{km} / \mathrm{h}\) . The true course, or track, of the plane is the direction of the resultant of the velocity vectors of the plane and the wind. The ground speed of the plane is the magnitude of the resultant. Find the true course and the ground speed of the plane.

\(25-26\) Find the acute angle between the lines. $$x+2 y=7, \quad 5 x-y=2$$

\(21-32=\) Find an equation of the plane. The plane through the origin and the points \((2,-4,6)\) and \((5,1,3)\)

A ball is thrown eastward into the air from the origin (in the direction of the positive \(x\) -axis). The initial velocity is \(50 \mathbf{i}+80 \mathbf{k},\) with speed measured in feet per second. The spin of the ball results in a southward acceleration of \(4 \mathrm{ft} / \mathrm{s}^{2},\) so the acceleration vector is \(\mathbf{a}=-4 \mathrm{j}-32 \mathrm{k}\) Where does the ball land and with what speed?

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

Show that the curve with parametric equations \(x=t^{2}\) , \(y=1-3 t, z=1+t^{3}\) passes through the points \((1,4,0)\) and \((9,-8,28)\) but not through the point \((4,7,-6)\)

\(21-32=\) Find an equation of the plane. The plane that passes through the point \((6,0,-2)\) and contains the line \(x=4-2 t, y=3+5 t, z=7+4 t\)

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

Find the area of the parallelogram with vertices \(A(-2,1),\) \(B(0,4), C(4,2),\) and \(D(2,-1)\) .

Describe in words the region of \(\mathbb{R}^{3}\) represented by theequations or inequalities. \(x^{2}+y^{2}+z^{2} \leqslant 3\)

A woman walks due west on the deck of a ship at 3 \(\mathrm{mi} / \mathrm{h}\) . The ship is moving north at a speed of 22 \(\mathrm{mi} / \mathrm{h}\) . Find the speed and direction of the woman relative to the surface of the water.

\(27-28\) Find the acute angles between the curves at their points of intersection. (The angle between two curves is the angle between their tangent lines at the point of intersection.) $$y=x^{2}, \quad y=x^{3}$$

\(28-30=\) Find a vector function that represents the curve of intersection of the two surfaces. The cylinder \(x^{2}+y^{2}=4\) and the surface \(z=x y\)

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

Find the area of the parallelogram with vertices \(K(1,2,3),\) \(L(1,3,6), M(3,8,6),\) and \(N(3,7,3) .\)

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

Ropes 3 \(\mathrm{m}\) and 5 \(\mathrm{m}\) in length are fastened to a holiday deco- ration that is suspended over a town square. The decoration has a mass of 5 \(\mathrm{kg}\) . The ropes, fastened at different heights, make angles of \(52^{\circ}\) and \(40^{\circ}\) with the horizontal. Find the tension in each wire and the magnitude of each tension.

\(27-28\) Find the acute angles between the curves at their points of intersection. (The angle between two curves is the angle between their tangent lines at the point of intersection.) $$y=\sin x, \quad y=\cos x, \quad 0 \leqslant x \leqslant \pi / 2$$

Water traveling along a straight portion of a river normally flows fastest in the middle, and the speed slows to almost zero at the banks. Consider a long straight stretch of river flowing north, with parallel banks 40 \(\mathrm{m}\) apart. If the maximum water speed is \(3 \mathrm{m} / \mathrm{s},\) we can use a quadratic function as a basic model for the rate of water flow \(x\) units from the west bank: \(f(x)=\frac{3}{400} x(40-x)\) $$\begin{array}{l}{\text { (a) A boat proceeds at a constant speed of } 5 \mathrm{m} / \mathrm{s} \text { from a }} \\ {\text { point } A \text { on the west bank while maintaining a heading }} \\ {\text { perpendicular to the bank. How far down the river on }}\end{array}$$$$ \begin{array}{l}{\text { the opposite bank will the boat touch shore? Graph the }} \\ {\text { path of the boat. }}\end{array}$$ $$\begin{array}{l}{\text { (b) Suppose we would like to pilot the boat to land at the }} \\ {\text { point } B \text { on the east bank directly opposite } A . \text { If we }} \\ {\text { maintain a constant speed of } 5 \mathrm{m} / \mathrm{s} \text { and a constant }} \\ {\text { heading, find the angle at which the boat should head. }} \\ {\text { Then graph the actual path the boat follows. Does the }} \\ {\text { path seem realistic? }}\end{array}$$

\(28-30=\) Find a vector function that represents the curve of intersection of the two surfaces. The cone \(z=\sqrt{x^{2}+y^{2}}\) and the plane \(z=1+y\)

\(21-32=\) Find an equation of the plane. The plane that passes through the point \((-1,2,1)\) and contains the line of intersection of the planes \(x+y-z=2\) and \(2 x-y+3 z=1\)

(a) Find a nonzero vector orthogonal to the plane through the points \(P, Q,\) and \(R,\) and (b) find the area of triangle \(P Q R .\) $$P(1,0,1), \quad Q(-2,1,3), \quad R(4,2,5)$$

Sketch the region bounded by the surfaces \(z=\sqrt{x^{2}+y^{2}}\) and \(x^{2}+y^{2}=1\) for 1\(\leqslant z \leqslant 2\)

Describe in words the region of \(\mathbb{R}^{3}\) represented by theequations or inequalities. \(x^{2}+z^{2} \leqslant 9\)

A clothesline is tied between two poles, 8 \(\mathrm{m}\) apart. The line is quite taut and has negligible sag. When a wet shirt with a mass of 0.8 \(\mathrm{kg}\) is hung at the middle of the line, the mid- point is pulled down 8 \(\mathrm{cm} .\) Find the tension in each half of the clothesline.

\(29-32\) Find the scalar and vector projections of b onto a. $$\mathbf{a}=\langle- 5,12\rangle, \quad \mathbf{b}=\langle 4,6\rangle$$

Find the tangential and normal components of the acceleration vector. $$\mathbf{r}(t)=(1+t) \mathbf{i}+\left(t^{2}-2 t\right) \mathbf{j}$$

Use a graphing calculator or computer to graph both the curve and its curvature function \(\kappa(x)\) on the same screen. Is the graph of \(\kappa\) what you would expect? $$y=x^{4}-2 x^{2}$$

\(21-32=\) Find an equation of the plane. The plane that passes through the points \((0,-2,5)\) and \((-1,3,1)\) and is perpendicular to the plane \(2 z=5 x+4 y\)

(a) Find a nonzero vector orthogonal to the plane through the points \(P, Q,\) and \(R,\) and (b) find the area of triangle \(P Q R .\) $$P(0,0,-3), \quad Q(4,2,0), \quad R(3,3,1)$$

Sketch the region bounded by the paraboloids \(z=x^{2}+y^{2}\) and \(z=2-x^{2}-y^{2}\)

Describe in words the region of \(\mathbb{R}^{3}\) represented by theequations or inequalities. \(x^{2}+y^{2}+z^{2}>2 z\)

\(29-32\) Find the scalar and vector projections of b onto a. $$\mathbf{a}=\langle 1,4\rangle, \quad \mathbf{b}=\langle 2,3\rangle$$

Find the tangential and normal components of the acceleration vector. $$\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+t \mathbf{k}$$

Use a graphing calculator or computer to graph both the curve and its curvature function \(\kappa(x)\) on the same screen. Is the graph of \(\kappa\) what you would expect? $$y=x^{-2}$$

The semiellipsoid $$x^{2}+y^{2}+4 z^{2}=4, y \geqslant 0,\( and the cylinder \)x^{2}+z^{2}=1$$

\(21-32=\) Find an equation of the plane. The plane that passes through the point \((1,5,1)\) and is perpendicular to the planes \(2 x+y-2 z=2\) and \(x+3 z=4\)

(a) Find a nonzero vector orthogonal to the plane through the points \(P, Q,\) and \(R,\) and (b) find the area of triangle \(P Q R .\) $$P(0,-2,0), \quad Q(4,1,-2), \quad R(5,3,1)$$

Find an equation for the surface consisting of all points that are equidistant from the point \((-1,0,0)\) and the plane \(x=1 .\) Identify the surface.

A boatman wants to cross a canal that is 3 \(\mathrm{km}\) wide and wants to land at a point 2 \(\mathrm{km}\) upstream from his starting point. The current in the canal flows at 3.5 \(\mathrm{km} / \mathrm{h}\) and the speed of his boat is 13 \(\mathrm{km} / \mathrm{h}\) . (a) In what direction should he steer? (b) How long will the trip take?

\(29-32\) Find the scalar and vector projections of b onto a. $$\mathbf{a}=\langle 3,6,-2\rangle, \quad \mathbf{b}=\langle 1,2,3\rangle$$

The region between the \(y z\) -plane and the vertical plane \(x=5\)

Find the tangential and normal components of the acceleration vector. $$\mathbf{r}(t)=t \mathbf{i}+\cos ^{2} t \mathbf{j}+\sin ^{2} t \mathbf{k}$$

Plot the space curve and its curvature function \(\kappa(t)\). Comment on how the curvature reflects the shape of the curve. $$\mathbf{r}(t)=\langle t-\sin t, 1-\cos t, 4 \cos (t / 2)\rangle, \quad 0 \leqslant t \leqslant 8 \pi$$

Try to sketch by hand the curve of intersection of the parabolic cylinder \(y=x^{2}\) and the top half of the ellipsoid \(x^{2}+4 y^{2}+4 z^{2}=16 .\) Then find parametric equations for this curve and use these equations and a computer to graph the curve.

\(21-32=\) Find an equation of the plane. The plane that passes through the line of intersection of the planes \(x-z=1\) and \(y+2 z=3\) and is perpendicular to the plane \(x+y-2 z=1\)

Find an equation for the surface consisting of all points \(P\) for which the distance from \(P\) to the \(x\) -axis is twice the distance from \(P\) to the \(y z\) -plane. Identify the surface.

(a) Find a nonzero vector orthogonal to the plane through the points \(P, Q,\) and \(R,\) and (b) find the area of triangle \(P Q R .\) $$P(-1,3,1), \quad Q(0,5,2), \quad R(4,3,-1)$$

The solid cylinder that lies on or below the plane \(z=8\) and on or above the disk in the \(x y\) -plane with center the origin and radius 2

Three forces act on an object. Two of the forces are at an angle of \(100^{\circ}\) to each other and have magnitudes 25 \(\mathrm{N}\) and 12 \(\mathrm{N} .\) The third is perpendicular to the plane of these two forces and has magnitude 4 \(\mathrm{N.}\) Calculate the magnitude of the force that would exactly counterbalance these three forces.