Answer Key Chapter 2 - Calculus Volume 3

The traces parallel to the xy-plane are ellipses and the traces parallel to the xz- and yz-planes are hyperbolas. Specifically, the trace in the xy-plane is ellipse $\frac{x^{2}}{3^{2}} + \frac{y^{2}}{2^{2}} = 1,$ the trace in the xz-plane is hyperbola $\frac{x^{2}}{3^{2}} - \frac{z^{2}}{5^{2}} = 1,$ and the trace in the yz-plane is hyperbola $\frac{y^{2}}{2^{2}} - \frac{z^{2}}{5^{2}} = 1$ (see the following figure).

This figure has four images. The first image is an ellipse centered at the origin of a rectangular coordinate system. It intersects the x axis at -3 and 3. It intersects the y axis at -2 and 2. The second image is the graph of a hyperbola. It is two curves one opening in the negative x direction and a symmetric one in the positive x direction. The third image is the graph of a hyperbola in the y z plane. It is opening in the negative y direction and a symmetric curve opening in the positive y direction. The fourth image is a 3-dimensional surface. It top and bottom cross sections would be circular. A vertical intersection would be a hyperbola.

2.54

Hyperboloid of one sheet, centered at $(0, 0, 1)$

2.55

The rectangular coordinates of the point are $(\frac{5 \sqrt{3}}{2}, \frac{5}{2}, 4) .$

This figure is the 3-dimensional coordinate system. There is a point labeled “(5, pi/6, 4).” The point is located above a line segment in the x y-plane labeled r = 5 that is pi/6 degrees from the x-axis. The distance from the x y-plane to the point is labeled “z = 4.”

2.56

$(8 \sqrt{2}, \frac{3 π}{4}, −7)$

2.57

This surface is a cylinder with radius $6 .$

This figure is a right circular cylinder. It is upright with the z-axis through the center. It is on top of the x y plane.

2.58

This figure is of the 3-dimensional coordinate system. It has a point. There is a line segment from the origin to the point. The angle between this line segment and the z-axis is phi. There is a line segment in the x y-plane from the origin to the shadow of the point.The angle between the x-axis and rho is theta.

Cartesian: $(- \frac{\sqrt{3}}{2}, - \frac{1}{2}, \sqrt{3}),$ cylindrical: $(1, - \frac{5 π}{6}, \sqrt{3})$

2.59

a. This is the set of all points $13$ units from the origin. This set forms a sphere with radius $13 .$ b. This set of points forms a half plane. The angle between the half plane and the positive x-axis is $θ = \frac{2 π}{3} .$ c. Let $P$ be a point on this surface. The position vector of this point forms an angle of $φ = \frac{π}{4}$ with the positive z-axis, which means that points closer to the origin are closer to the axis. These points form a half-cone.

2.60

$(4000, 151 °, 124 °)$

2.61

Spherical coordinates with the origin located at the center of the earth, the z-axis aligned with the North Pole, and the x-axis aligned with the prime meridian

Section 2.1 Exercises

1.

a. $\vec{P Q} = 〈 2, 2 〉;$ b. $\vec{P Q} = 2 i + 2 j$

3.

a. $\vec{Q P} = 〈 −2, −2 〉;$ b. $\vec{Q P} = −2 i - 2 j$

5.

a. $\vec{P Q} + \vec{P R} = 〈 0, 6 〉;$ b. $\vec{P Q} + \vec{P R} = 6 j$

7.

a. $2 \vec{P Q} - 2 \vec{P R} = 〈 8, −4 〉;$ b. $2 \vec{P Q} - 2 \vec{P R} = 8 i - 4 j$

9.

a. $〈 \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} 〉;$ b. $\frac{1}{\sqrt{2}} i + \frac{1}{\sqrt{2}} j$

11.

$〈 \frac{3}{5}, \frac{4}{5} 〉$

13.

$Q (0, 2)$

15.

a. $a + b = 3 i + 4 j,$ $a + b = 〈 3, 4 〉;$ b. $a - b = i - 2 j,$ $a - b = 〈 1, −2 〉;$ c. Answers will vary; d. $2 a = 4 i + 2 j,$ $2 a = 〈 4, 2 〉,$ $− b = − i - 3 j,$ $− b = 〈 −1, −3 〉,$ $2 a - b = 3 i - j,$ $2 a - b = 〈 3, −1 〉$

17.

$15$

19.

$λ = −3$

21.

a. $a (0) = 〈 1, 0 〉,$ $a (π) = 〈 −1, 0 〉;$ b. Answers may vary; c. Answers may vary

23.

Answers may vary

25.

$v = 〈 \frac{21}{5}, \frac{28}{5} 〉$

27.

$v = 〈 \frac{21 \sqrt{34}}{34}, - \frac{35 \sqrt{34}}{34} 〉$

29.

$u = 〈 \sqrt{3}, 1 〉$

31.

$u = 〈 0, 5 〉$

33.

$u = 〈 −5 \sqrt{3}, 5 〉$

35.

$θ = \frac{7 π}{4}$

37.

Answers may vary

39.

a. $z_{0} = f (x_{0}) + f^{'} (x_{0});$ b. $u = \frac{1}{\sqrt{1 + {[f^{'} (x_{0})]}^{2}}} 〈 1, f^{'} (x_{0}) 〉$

43.

$D (6, 1)$

45.

$〈 60.62, 35 〉$

47.

The horizontal and vertical components are $750$ ft/sec and $1299.04$ ft/sec, respectively.

49.

The magnitude of resultant force is $94.71$ lb; the direction angle is $13.42 ° .$

51.

The magnitude of the third vector is $60.03$ N; the direction angle is $259.38 ° .$

53.

The new ground speed of the airplane is $572.19$ mph; the new direction is $N 41.82 E .$

55.

$‖ T_{1} ‖ = 30.13 lb,$ $‖ T_{2} ‖ = 38.35 lb$

57.

$‖ v_{1} ‖ = 750$ lb, $‖ v_{2} ‖ = 1299$ lb

59.

The two horizontal and vertical components of the force of tension are $28$ lb and $42$ lb, respectively.

Section 2.2 Exercises

61.

a. $(2, 0, 5), (2, 0, 0), (2, 3, 0), (0, 3, 0), (0, 3, 5), (0, 0, 5);$ b. $\sqrt{38}$

63.

A union of two planes: $y = 5$ (a plane parallel to the xz-plane) and $z = 6$ (a plane parallel to the xy-plane)

This figure is the first octant of the 3-dimensional coordinate system. It has two planes drawn. The first plane is parallel to the x y-plane and is at z = 6. The second plane is parallel to the x z-plane and is at y = 5. The planes are perpendicular.

65.

A cylinder of radius $1$ centered on the line $y = 1, z = 1$

This figure is the first octant of the 3-dimensional coordinate system. It has a cylinder drawn. The axis of the cylinder is parallel to the x-axis.

67.

$z = 1$

69.

$z = −2$

71.

${(x + 1)}^{2} + {(y - 7)}^{2} + {(z - 4)}^{2} = 16$

73.

${(x + 3)}^{2} + {(y - 3.5)}^{2} + {(z - 8)}^{2} = \frac{29}{4}$

75.

Center $C (0, 0, 2)$ and radius $1$

77.

a. $\vec{P Q} = 〈 −4, −1, 2 〉;$ b. $\vec{P Q} = −4 i - j + 2 k$

79.

a. $\vec{P Q} = 〈 6, −24, 24 〉;$ b. $\vec{P Q} = 6 i - 24 j + 24 k$

81.

$Q (5, 2, 8)$

83.

$a + b = 〈 −6, 4, −3 〉,$ $4 a = 〈 −4, −8, 16 〉,$ $−5 a + 3 b = 〈 −10, 28, −41 〉$

85.

$a + b = 〈 −1, 0, −1 〉,$ $4 a = 〈 0, 0, −4 〉,$ $−5 a + 3 b = 〈 −3, 0, 5 〉$

87.

$‖ u - v ‖ = \sqrt{38},$ $‖ −2 u ‖ = 2 \sqrt{29}$

89.

$‖ u - v ‖ = 2,$ $‖ −2 u ‖ = 2 \sqrt{13}$

91.

$a = \frac{3}{5} i - \frac{4}{5} j$

93.

$⟨ \frac{2}{\sqrt{62}}, \frac{7}{\sqrt{62}}, \frac{3}{\sqrt{62}} ⟩$

95.

$〈 - \frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}} 〉$

97.

Equivalent vectors

99.

$u = 〈 \frac{70}{\sqrt{59}}, - \frac{10}{\sqrt{59}}, \frac{30}{\sqrt{59}} 〉$

101.

$u = 〈 - \frac{4}{\sqrt{5}} sin t, - \frac{4}{\sqrt{5}} cos t, - \frac{2}{\sqrt{5}} 〉$

103.

$〈 \frac{5}{\sqrt{154}}, \frac{15}{\sqrt{154}}, - \frac{60}{\sqrt{154}} 〉$

105.

$α = − \sqrt{7},$ $β = − \sqrt{15}$

111.

a. $F = 〈 30, 40, 0 〉;$ b. $53 °$

113.

$D = 10 k$

115.

$F_{4} = 〈 −20, −7, −3 〉$

117.

a. $F = −19.6 k,$ $‖ F ‖ = 19.6$ N; b. $T = 19.6 k,$ $‖ T ‖ = 19.6$ N

119.

a. $F = −294 k$ N; b. $F_{1} = 〈 - \frac{49 \sqrt{3}}{3}, 49, −98 〉,$ $F_{2} = 〈 - \frac{49 \sqrt{3}}{3}, −49, −98 〉,$ and $F_{3} = 〈 \frac{98 \sqrt{3}}{3}, 0, −98 〉$ (each component is expressed in newtons)

121.

a. $v (1) = 〈 −0.84, 0.54, 2 〉$ (each component is expressed in centimeters per second); $‖ v (1) ‖ = 2.24$ (expressed in centimeters per second); $a (1) = 〈 −0.54, −0.84, 0 〉$ (each component expressed in centimeters per second squared);

b.

This figure is of the 3-dimensional coordinate system above the xy-plane. It has a spiral drawn resembling a spring. The spiral is around the z-axis. The spiral starts on the x-axis at x = 1.

Section 2.3 Exercises

123.

6

125.

0

127.

$(a \cdot b) c = 〈 −11, −11, 11 〉;$ $(a \cdot c) b = 〈 −20, −35, 5 〉$

129.

$(a \cdot b) c = 〈 1, 0, −2 〉;$ $(a \cdot c) b = 〈 1, 0, −1 〉$

131.

a. $θ = 2.82$ rad; b. $θ$ is not acute.

133.

a. $θ = \frac{π}{4}$ rad; b. $θ$ is acute.

135.

$θ = \frac{π}{2}$

137.

$θ = \frac{π}{3}$

139.

$θ = 2$ rad

141.

Orthogonal

143.

Not orthogonal

145.

$a = 〈 - \frac{4 α}{3}, α 〉,$ where $α \neq 0$ is a real number

147.

$u = − α i + α j + β k,$ where $α$ and $β$ are real numbers such that $α^{2} + β^{2} \neq 0$

149.

$α = −6$

151.

a. $\vec{O P} = 4 i + 5 j,$ $\vec{O Q} = 5 i - 7 j;$ b. $105.8 °$

153.

$68.33 °$

155.

$u$ and $v$ are orthogonal; $v$ and $w$ are orthogonal.

161.

a. $cos α = \frac{2}{3}, cos β = \frac{2}{3},$ and $cos γ = \frac{1}{3};$ b. $α = 48 °,$ $β = 48 °,$ and $γ = 71 °$

163.

a. $cos α = - \frac{1}{\sqrt{30}}, cos β = \frac{5}{\sqrt{30}},$ and $cos γ = \frac{2}{\sqrt{30}};$ b. $α = 101 °,$ $β = 24 °,$ and $γ = 69 °$

167.

a. $w = 〈 \frac{80}{29}, \frac{32}{29} 〉;$ b. ${comp}_{u} v = \frac{16}{\sqrt{29}}$

169.

a. $w = 〈 \frac{24}{13}, 0, \frac{16}{13} 〉;$ b. ${comp}_{u} v = \frac{8}{\sqrt{13}}$

171.

a. $w = 〈 \frac{24}{25}, - \frac{18}{25} 〉;$ b. $q = 〈 \frac{51}{25}, \frac{68}{25} 〉,$ $v = w + q = 〈 \frac{24}{25}, - \frac{18}{25} 〉 + 〈 \frac{51}{25}, \frac{68}{25} 〉$

173.

a. $2 \sqrt{2};$ b. $109.47 °$

175.

$17 N \cdot m$

177.

1175 $ft \cdot lb$

179.

W = 43301.27 $ft-lb$

181.

a. $‖ F_{1} + F_{2} ‖ = 52.9$ lb; b. The direction angles are $α = 74.5 °,$ $β = 36.7 °,$ and $γ = 57.7 ° .$

Section 2.4 Exercises

183.

a. $u \times v = 〈 0, 0, 4 〉;$
b.

This figure is the first octant of the 3-dimensional coordinate system. On the x-axis there is a vector labeled “u.” In the x y-plane there is a vector labeled “v.” On the z-axis there is the vector labeled “u cross v.”

185.

a. $u \times v = 〈 6, −4, 2 〉;$
b.

This figure is the first octant of the 3-dimensional coordinate system and shows three vectors. The first vector is labeled u and has components <2, 3, 0>. The second vector is labeled v and has components <0, 1, 2>.” The third vector is labeled u cross v and has components <6, -4, 2>.”

187.

$−2 j - 4 k$

189.

$w = - \frac{1}{3 \sqrt{6}} i - \frac{7}{3 \sqrt{6}} j - \frac{2}{3 \sqrt{6}} k$

191.

$w = - \frac{4}{\sqrt{21}} i - \frac{2}{\sqrt{21}} j - \frac{1}{\sqrt{21}} k$

193.

$α = 10$

197.

$−3 i + 11 j + 2 k$

199.

$w = 〈 −1, e^{t}, − e^{− t} 〉$

201.

$−26 i + 17 j + 9 k$

203.

$72 °$

209.

$7$

211.

a. $5 \sqrt{6};$ b. $\frac{5 \sqrt{6}}{2};$ c. $\frac{5 \sqrt{6}}{\sqrt{59}}$

213.

a. $2;$ b. $2$

215.

$v \cdot (u \times w) = −1,$ $w \cdot (u \times v) = 1$

217.

$a = 〈 1, 2, 3 〉,$ $b = 〈 0, 2, 5 〉,$ $c = 〈 8, 9, 2 〉;$ $a \cdot (b \times c) = −9$

219.

a. $α = 1;$ b. $h = 1,$

This figure is the first octant of the 3-dimensional coordinate system. There is a parallelepided drawn. From the origin there are three vectors to vertices on the parallelepiped. They are vectors to the points A (2, 1, 0); B (1, 2, 0); and C (0, 1, alpha).

225.

Yes, $\vec{A D} = α \vec{A B} + β \vec{A C},$ where $α = −1$ and $β = 1 .$

227.

$− k$

229.

$〈 0, ± 4 \sqrt{5}, \mp 2 \sqrt{5} 〉$

233.

$w = 〈 w_{3} - 1, w_{3} + 1, w_{3} 〉,$ where $w_{3}$ is any real number

235.

8.66 ft-lb

237.

559 N

239.

$F = 4.8 \times 10^{−15} k N$

241.

a. $B (t) = 〈 \frac{2 sin t}{\sqrt{5}}, - \frac{2 cos t}{\sqrt{5}}, \frac{1}{\sqrt{5}} 〉;$
b.

This figure is the first octant of the 3-dimensional coordinate system. There is a curve sketched that is increasing. On the curve is a point labeled “P.” At P there is a tangent vector to the curve labeled “v(1).” Also from P there is a vector towards the inside of the curve labeled “a(1).” Finally, there is a vector from P labeled “B(1)” pointing towards the z-axis.

Section 2.5 Exercises

243.

a. $r = 〈 −3, 5, 9 〉 + t 〈 7, −12, −7 〉,$ $t \in ℝ;$ b. $x = −3 + 7 t, y = 5 - 12 t, z = 9 - 7 t,$ $t \in ℝ;$ c. $\frac{x + 3}{7} = \frac{y - 5}{−12} = \frac{z - 9}{−7};$ d. $x = −3 + 7 t, y = 5 - 12 t, z = 9 - 7 t,$ $t \in [0, 1]$

245.

a. $r = 〈 −1, 0, 5 〉 + t 〈 5, 0, −2 〉,$ $t \in ℝ;$ b. $x = −1 + 5 t, y = 0, z = 5 - 2 t,$ $t \in ℝ;$ c. $\frac{x + 1}{5} = \frac{z - 5}{−2}, y = 0;$ d. $x = −1 + 5 t, y = 0, z = 5 - 2 t,$ $t \in [0, 1]$

247.

a. $x = 1 + t, y = −2 + 2 t, z = 3 + 3 t,$ $t \in ℝ;$ b. $\frac{x - 1}{1} = \frac{y + 2}{2} = \frac{z - 3}{3};$ c. $(0, −4, 0)$

249.

a. $x = 3 + t, y = 1, z = 5,$ $t \in ℝ;$ b. $y = 1, z = 5;$ c. The line does not intersect the xy-plane.

251.

a. $P (1, 3, 5),$ $v = 〈 1, 1, 4 〉;$ b. $\sqrt{3}$

253.

$\frac{2 \sqrt{2}}{\sqrt{3}}$

255.

a. Parallel; b. $\frac{\sqrt{2}}{\sqrt{3}}$

259.

$(−12, 6, −4)$

261.

The lines are skew.

263.

The lines are equal.

265.

a. $x = 1 + t, y = 1 - t, z = 1 + 2 t,$ $t \in ℝ;$ b. For instance, the line passing through $A$ with direction vector $j : x = 1, z = 1;$ c. For instance, the line passing through $A$ and point $(2, 0, 0)$ that belongs to $L$ is a line that intersects; $L : \frac{x - 1}{−1} = y - 1 = z - 1$

267.

a. $3 x - 2 y + 4 z = 0;$ b. $3 x - 2 y + 4 z = 0$

269.

a. $(x - 1) + 2 (y - 2) + 3 (z - 3) = 0;$ b. $x + 2 y + 3 z - 14 = 0$

271.

a. $n = 4 i + 5 j + 10 k;$ b. $(5, 0, 0),$ $(0, 4, 0),$ and $(0, 0, 2);$
c.

This figure is the first octant of the 3-dimensional coordinate system. It has a triangle drawn with vertices on the x, y, and z axes.

273.

a. $n = 3 i - 2 j + 4 k;$ b. $(0, 0, 0);$
c.

This figure is the 3-dimensional coordinate system represented in a box. It has a tilted parallelogram inside the box representing a plane.

275.

$(3, 0, 0)$

277.

$x = −2 + 2 t, y = 1 - 3 t, z = 3 + t,$ $t \in ℝ$

281.

a. $−2 y + 3 z - 1 = 0;$ b. $〈 0, −2, 3 〉 \cdot 〈 x - 1, y - 1, z - 1 〉 = 0;$ c. $x = 0, y = −2 t, z = 3 t,$ $t \in ℝ$

Answers may vary by a sign, depending on how the vector cross multiplication is performed.

283.

a. Answers may vary; b. $\frac{x - 1}{1} = \frac{z - 6}{−1}, y = 4$

285.

$2 x - 5 y - 3 z + 15 = 0$

287.

The line intersects the plane at point $P (−3, 4, 0) .$

289.

$\frac{16}{\sqrt{14}}$

291.

a. The planes are neither parallel nor orthogonal; b. $62 °$

293.

a. The planes are parallel.

295.

$\frac{1}{\sqrt{6}}$

297.

a. $\frac{18}{\sqrt{29}};$ b. $P (- \frac{51}{29}, \frac{130}{29}, \frac{62}{29})$

299.

$4 x - 3 y = 0$

301.

a. $v (1) = 〈 cos 1, − sin 1, 2 〉;$ b. $(cos 1) (x - sin 1) - (sin 1) (y - cos 1) + 2 (z - 2) = 0;$
c.

This figure is the first octant of the 3-dimensional coordinate system. It has a parallelogram grid drawn representing a plane. There is a curve from y = 1 increasing. The curve intersects the plane. At the point the curve intersects the plane, there is a vector labeled “v(1).” It is upward parallel to the z-axis.

Section 2.6 Exercises

303.

The surface is a cylinder with the rulings parallel to the y-axis.

This figure is a circular cylinder inside of a box. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

305.

The surface is a cylinder with rulings parallel to the y-axis.

This figure is a surface inside of a box. Its cross section parallel to the x z plane would be a cosine curve. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

307.

The surface is a cylinder with rulings parallel to the x-axis.

This figure is a surface inside of a box. Its cross section parallel to the y z plane would be an upside down parabola. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

309.

a. Cylinder; b. The x-axis

311.

a. Hyperboloid of two sheets; b. The x-axis

313.

b.

315.

d.

317.

a.

319.

$- \frac{x^{2}}{9} + \frac{y^{2}}{\frac{1}{4}} + \frac{z^{2}}{\frac{1}{4}} = 1,$ hyperboloid of one sheet with the x-axis as its axis of symmetry

321.

$- \frac{x^{2}}{\frac{10}{3}} + \frac{y^{2}}{2} - \frac{z^{2}}{10} = 1,$ hyperboloid of two sheets with the y-axis as its axis of symmetry

323.

$y = - \frac{z^{2}}{5} + \frac{x^{2}}{5},$ hyperbolic paraboloid with the y-axis as its axis of symmetry

325.

$\frac{x^{2}}{15} + \frac{y^{2}}{3} + \frac{z^{2}}{5} = 1,$ ellipsoid

327.

$\frac{x^{2}}{40} + \frac{y^{2}}{8} - \frac{z^{2}}{5} = 0,$ elliptic cone with the z-axis as its axis of symmetry

329.

$x = \frac{y^{2}}{2} + \frac{z^{2}}{3},$ elliptic paraboloid with the x-axis as its axis of symmetry

331.

Parabola $y = - \frac{x^{2}}{4},$

This figure is the graph of an upside down parabola with its highest point at the origin of a rectangular coordinate system.

333.

Ellipse $\frac{y^{2}}{4} + \frac{z^{2}}{100} = 1,$

This figure is the graph of an ellipse centered at the origin of a rectangular coordinate system.

335.

Ellipse $\frac{y^{2}}{4} + \frac{z^{2}}{100} = 1,$

337.

a. Ellipsoid; b. The third equation; c. $\frac{x^{2}}{100} + \frac{y^{2}}{400} + \frac{z^{2}}{225} = 1$

339.

a. $\frac{{(x + 3)}^{2}}{16} + \frac{{(z - 2)}^{2}}{8} = 1;$ b. Cylinder centered at $(−3, 2)$ with rulings parallel to the y-axis

341.

a. $\frac{{(x - 3)}^{2}}{4} + {(y - 2)}^{2} - {(z + 2)}^{2} = 1;$ b. Hyperboloid of one sheet centered at $(3, 2, −2),$ with the z-axis as its axis of symmetry

343.

a. ${(x + 3)}^{2} + \frac{y^{2}}{4} - \frac{z^{2}}{3} = 0;$ b. Elliptic cone centered at $(−3, 0, 0),$ with the z-axis as its axis of symmetry

345.

$\frac{x^{2}}{4} + \frac{y^{2}}{16} + z^{2} = 1$

347.

$(1, −1, 0)$ and $(\frac{13}{3}, 4, \frac{5}{3})$

349.

$x^{2} + z^{2} + 4 y = 0,$ elliptic paraboloid

351.

$(0, 0, 100)$

355.

a. $x = 2 - \frac{z^{2}}{2}, y = \pm \frac{z}{2} \sqrt{4 - z^{2}},$ where $z \in [−2, 2];$
b.

This figure is a surface inside of a box. It is a sphere with a figure eight curve on the side of the sphere. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

357.

This figure is a surface inside of a box. It is a solid oval with an elliptical cylinder vertically intersecting. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

two ellipses of equations $\frac{x^{2}}{2} + \frac{y^{2}}{\frac{9}{2}} = 1$ in planes $z = ± 2 \sqrt{2}$

359.

a. $\frac{x^{2}}{3963^{2}} + \frac{y^{2}}{3963^{2}} + \frac{z^{2}}{3950^{2}} = 1;$
b.

This figure is a surface inside of a box. It is a sphere. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

;
c. The intersection curve is the ellipse of equation $\frac{x^{2}}{3963^{2}} + \frac{y^{2}}{3963^{2}} = \frac{(2950) (4950)}{3950^{2}},$ and the intersection is an ellipse.; d. The intersection curve is the ellipse of equation $\frac{2 y^{2}}{3963^{2}} + \frac{z^{2}}{3950^{2}} = 1 .$

361.

a.

This figure is a surface inside of a box. It is a heart. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

b. The intersection curve is ${(x^{2} + z^{2} - 1)}^{3} - x^{2} z^{3} = 0 .$

This figure is a curve on a rectangular coordinate system. It is the shape of a heart centered about the y-axis.

Section 2.7 Exercises

363.

$(2 \sqrt{3}, 2, 3)$

365.

$(−2 \sqrt{3}, −2, 3)$

367.

$(2, \frac{π}{3}, 2)$

369.

$(3 \sqrt{2}, - \frac{π}{4}, 7)$

371.

A cylinder of equation $x^{2} + y^{2} = 16,$ with its center at the origin and rulings parallel to the z-axis,

This figure is a right circular cylinder, vertical. It is inside of a box. The edges of the box represent the x, y, and z axes.

373.

Hyperboloid of two sheets of equation $− x^{2} + y^{2} - z^{2} = 1,$ with the y-axis as the axis of symmetry,

This figure is a elliptic cone surface that is horizontal. It is inside of a box. The edges of the box represent the x, y, and z axes.

375.

Cylinder of equation $x^{2} - 2 x + y^{2} = 0,$ with a center at $(1, 0, 0)$ and radius $1,$ with rulings parallel to the z-axis,

377.

Plane of equation $x = 2,$

This figure is a vertical parallelogram where x = 2 and parallel to the y z-plane. It is inside of a box. The edges of the box represent the x, y, and z axes.

379.

$z = 3$

381.

$r^{2} + z^{2} = 9$

383.

$r = 16 cos θ, r = 0$

385.

$(0, 0, −3)$

387.

$(6, −6, 6 \sqrt{2})$

389.

$(4, 0, 90 °)$

391.

$(3, 90 °, 90 °)$

393.

Sphere of equation $x^{2} + y^{2} + z^{2} = 9$ centered at the origin with radius $3,$

This figure is a sphere. It is inside of a box. The edges of the box represent the x, y, and z axes.

395.

Sphere of equation $x^{2} + y^{2} + {(z - 1)}^{2} = 1$ centered at $(0, 0, 1)$ with radius $1,$

This figure is a sphere of radius 1 centered in a box. The center of the sphere is the point (0, 0, 1).

397.

The xy-plane of equation $z = 0,$

This figure is a parallelogram representing a plane. It is parallel to the x y-plane at z = 0. It is inside of a box. The edges of the box represent the x, y, and z axes.

399.

$φ = \frac{π}{3}$ or $φ = \frac{2 π}{3};$ Elliptic cone

401.

$ρ cos φ = 6;$ Plane at $z = 6$

403.

$(\sqrt{10}, \frac{π}{4}, 0.3218)$

405.

$(3 \sqrt{2}, \frac{π}{2}, \frac{π}{4})$

407.

$(2, - \frac{π}{4}, 0)$

409.

$(8, \frac{π}{3}, 0)$

411.

Cartesian system, ${(x, y, z) | 0 \leq x \leq a, 0 \leq y \leq a, 0 \leq z \leq a}$

413.

Cylindrical system, ${\begin{cases} (r, θ, z) | r^{2} + z^{2} \leq 9, r \geq 0, \frac{π}{2} \leq θ \leq \frac{3 π}{2} \\ , (r \geq 3 \cos θ, - \frac{π}{2} \leq θ \leq \frac{π}{2}) \end{cases}}$

415.

The region is described by the set of points ${(r, θ, z) | 0 \leq r \leq 1, 0 \leq θ \leq 2 π, r^{2} \leq z \leq r} .$

This figure is a paraboloid, vertical. It is inside of a box. The edges of the box represent the x, y, and z axes.

417.

$(4000, − 77 °, 51 °)$

419.

$43.17 ° W,$ $22.91 ° S$

421.

a. $ρ^{2} = 0,$ $ρ + R^{2} - r^{2} - 2 R sin φ = 0;$
c.

This figure is a torus. It is inside of a box. The edges of the box represent the x, y, and z axes.

Review Exercises

423.

True

425.

False

427.

a. $〈 24, −5 〉;$ b. $\sqrt{85};$ c. Can’t cross a vector with a scalar; d. $−29$

429.

$a = ± 2$

431.

$〈 \frac{1}{\sqrt{14}}, - \frac{2}{\sqrt{14}}, - \frac{3}{\sqrt{14}} 〉$

433.

$27$

435.

$x = 1 - 3 t, y = 3 + 3 t, z = 5 - 8 t, r (t) = (1 - 3 t) i + 3 (1 + t) j + (5 - 8 t) k$

437.

$− x + 3 y + 8 z = 43$

439.

$x = k$ trace: $k^{2} = y^{2} + z^{2}$ is a circle, $y = k$ trace: $x^{2} - z^{2} = k^{2}$ is a hyperbola (or a pair of lines if $k = 0),$ $z = k$ trace: $x^{2} - y^{2} = k^{2}$ is a hyperbola (or a pair of lines if $k = 0) .$ The surface is a cone.

This figure is an elliptical cone on its side. It is inside of a box. The edges of the box represent the x, y, and z axes.

441.

Cylindrical: $z = r^{2} - 1,$ spherical: $cos φ = ρ {sin}^{2} φ - \frac{1}{ρ}$

443.

$x^{2} - 2 x + y^{2} + z^{2} = 1,$ sphere

445.

331 N, and 244 N

447.

$15 J$

449.

More, $59.09$ J