Calculus Volume 3

# Chapter 2

### Checkpoint

2.3

Vectors $a,a,$ $b,b,$ and $ee$ are equivalent.

2.4

$〈 3 , 7 〉 〈 3 , 7 〉$

2.5

a. $‖a‖=52,‖a‖=52,$ b. $b=〈−4,−3〉,b=〈−4,−3〉,$ c. $3a−4b=〈37,15〉3a−4b=〈37,15〉$

2.7

$v = 〈 −5 , 5 3 〉 v = 〈 −5 , 5 3 〉$

2.8

$〈 − 45 85 , − 10 85 〉 〈 − 45 85 , − 10 85 〉$

2.9

$a=16i−11j,a=16i−11j,$ $b=−22i−22jb=−22i−22j$

2.10

Approximately $516516$ mph

2.12

$5 2 5 2$

2.13

$z = −4 z = −4$

2.14

$( x + 2 ) 2 + ( y − 4 ) 2 + ( z + 5 ) 2 = 52 ( x + 2 ) 2 + ( y − 4 ) 2 + ( z + 5 ) 2 = 52$

2.15

$x 2 + ( y − 2 ) 2 + ( z + 2 ) 2 = 14 x 2 + ( y − 2 ) 2 + ( z + 2 ) 2 = 14$

2.16

The set of points forms the two planes $y=−2y=−2$ and $z=3.z=3.$

2.17

A cylinder of radius 4 centered on the line with $x=0andz=2.x=0andz=2.$

2.18

$S T → = 〈 −1 , −9 , 1 〉 = − i − 9 j + k S T → = 〈 −1 , −9 , 1 〉 = − i − 9 j + k$

2.19

$〈 1 3 10 , − 5 3 10 , 8 3 10 〉 〈 1 3 10 , − 5 3 10 , 8 3 10 〉$

2.20

$v = 〈 16 2 , 12 2 , 20 2 〉 v = 〈 16 2 , 12 2 , 20 2 〉$

2.21

7

2.22

a. $(r·p)q=〈12,−12,12〉;(r·p)q=〈12,−12,12〉;$ b. $‖p‖2=53‖p‖2=53$

2.23

$θ≈0.22θ≈0.22$ rad

2.24

$x = 5 x = 5$

2.25

a. $α≈1.04α≈1.04$ rad; b. $β≈2.58β≈2.58$ rad; c. $γ≈1.40γ≈1.40$ rad

2.26

Sales = $15,685.50; profit =$14,073.15

2.27

$v=p+q,v=p+q,$ where $p=185i+95jp=185i+95j$ and $q=75i−145jq=75i−145j$

2.28

21 knots

2.29

150 ft-lb

2.30

$i − 9 j + 2 k i − 9 j + 2 k$

2.31

Up (the positive z-direction)

2.32

$− i − i$

2.33

$− k − k$

2.34

$16 16$

2.35

$40 40$

2.36

$8 i − 35 j + 2 k 8 i − 35 j + 2 k$

2.37

$〈 −3 194 , −13 194 , 4 194 〉 〈 −3 194 , −13 194 , 4 194 〉$

2.38

$6 13 6 13$

2.39

$17 17$

2.40

$88$ units3

2.41

No, the triple scalar product is $−4≠0,−4≠0,$ so the three vectors form the adjacent edges of a parallelepiped. They are not coplanar.

2.42

$2020$ N

2.43

Possible set of parametric equations: $x=1+4t,y=−3+t,z=2+6t;x=1+4t,y=−3+t,z=2+6t;$

related set of symmetric equations: $x−14=y+3=z−26x−14=y+3=z−26$

2.44

$x = −1 − 7 t , y = 3 − t , z = 6 − 2 t , 0 ≤ t ≤ 1 x = −1 − 7 t , y = 3 − t , z = 6 − 2 t , 0 ≤ t ≤ 1$

2.45

$10 7 10 7$

2.46

These lines are skew because their direction vectors are not parallel and there is no point $(x,y,z)(x,y,z)$ that lies on both lines.

2.47

$−2(x−1)+(y+1)+3(z−1)=0−2(x−1)+(y+1)+3(z−1)=0$ or $−2x+y+3z=0−2x+y+3z=0$

2.48

$15 21 15 21$

2.49

$x = t , y = 7 − 3 t , z = 4 − 2 t x = t , y = 7 − 3 t , z = 4 − 2 t$

2.50

$1.441.44$ rad

2.51

$9 30 9 30$

2.53

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 $x232+y222=1,x232+y222=1,$ the trace in the xz-plane is hyperbola $x232−z252=1,x232−z252=1,$ and the trace in the yz-plane is hyperbola $y222−z252=1y222−z252=1$ (see the following figure).

2.54

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

2.55

The rectangular coordinates of the point are $(532,52,4).(532,52,4).$

2.56

$( 8 2 , 3 π 4 , −7 ) ( 8 2 , 3 π 4 , −7 )$

2.57

This surface is a cylinder with radius $6.6.$

2.58

Cartesian: $(−32,−12,3),(−32,−12,3),$ cylindrical: $(1,−5π6,3)(1,−5π6,3)$

2.59

a. This is the set of all points $1313$ units from the origin. This set forms a sphere with radius $13.13.$ b. This set of points forms a half plane. The angle between the half plane and the positive x-axis is $θ=2π3.θ=2π3.$ c. Let $PP$ be a point on this surface. The position vector of this point forms an angle of $φ=π4φ=π4$ 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

$( 4000 , 151 ° , 124 ° ) ( 4000 , 151 ° , 124 ° )$

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. $PQ→=〈2,2〉;PQ→=〈2,2〉;$ b. $PQ→=2i+2jPQ→=2i+2j$

3 .

a. $QP→=〈−2,−2〉;QP→=〈−2,−2〉;$ b. $QP→=−2i−2jQP→=−2i−2j$

5 .

a. $PQ→+PR→=〈0,6〉;PQ→+PR→=〈0,6〉;$ b. $PQ→+PR→=6jPQ→+PR→=6j$

7 .

a. $2PQ→−2PR→=〈8,−4〉;2PQ→−2PR→=〈8,−4〉;$ b. $2PQ→−2PR→=8i−4j2PQ→−2PR→=8i−4j$

9 .

a. $〈12,12〉;〈12,12〉;$ b. $12i+12j12i+12j$

11 .

$〈 3 5 , 4 5 〉 〈 3 5 , 4 5 〉$

13 .

$Q ( 0 , 2 ) Q ( 0 , 2 )$

15 .

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

17 .

$15 15$

19 .

$λ = −3 λ = −3$

21 .

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

23 .

25 .

$v = 〈 21 5 , 28 5 〉 v = 〈 21 5 , 28 5 〉$

27 .

$v = 〈 21 34 34 , − 35 34 34 〉 v = 〈 21 34 34 , − 35 34 34 〉$

29 .

$u = 〈 3 , 1 〉 u = 〈 3 , 1 〉$

31 .

$u = 〈 0 , 5 〉 u = 〈 0 , 5 〉$

33 .

$u = 〈 −5 3 , 5 〉 u = 〈 −5 3 , 5 〉$

35 .

$θ = 7 π 4 θ = 7 π 4$

37 .

39 .

a. $z0=f(x0)+f′(x0);z0=f(x0)+f′(x0);$ b. $u=11+[f′(x0)]2〈1,f′(x0)〉u=11+[f′(x0)]2〈1,f′(x0)〉$

43 .

$D ( 6 , 1 ) D ( 6 , 1 )$

45 .

$〈 60.62 , 35 〉 〈 60.62 , 35 〉$

47 .

The horizontal and vertical components are $750750$ ft/sec and $1299.041299.04$ ft/sec, respectively.

49 .

The magnitude of resultant force is $94.7194.71$ lb; the direction angle is $13.42°.13.42°.$

51 .

The magnitude of the third vector is $60.0360.03$ N; the direction angle is $259.38°.259.38°.$

53 .

The new ground speed of the airplane is $572.19572.19$ mph; the new direction is $N41.82E.N41.82E.$

55 .

$‖T1‖=30.13lb,‖T1‖=30.13lb,$ $‖T2‖=38.35lb‖T2‖=38.35lb$

57 .

$‖v1‖=750‖v1‖=750$ lb, $‖v2‖=1299‖v2‖=1299$ lb

59 .

The two horizontal and vertical components of the force of tension are $2828$ lb and $4242$ lb, respectively.

### Section 2.2 Exercises

61 .

a. $(2,0,5),(2,0,0),(2,3,0),(0,3,0),(0,3,5),(0,0,5);(2,0,5),(2,0,0),(2,3,0),(0,3,0),(0,3,5),(0,0,5);$ b. $3838$

63 .

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

65 .

A cylinder of radius $11$ centered on the line $y=1,z=1y=1,z=1$

67 .

$z = 1 z = 1$

69 .

$z = −2 z = −2$

71 .

$( x + 1 ) 2 + ( y − 7 ) 2 + ( z − 4 ) 2 = 16 ( x + 1 ) 2 + ( y − 7 ) 2 + ( z − 4 ) 2 = 16$

73 .

$( x + 3 ) 2 + ( y − 3.5 ) 2 + ( z − 8 ) 2 = 29 4 ( x + 3 ) 2 + ( y − 3.5 ) 2 + ( z − 8 ) 2 = 29 4$

75 .

Center $C(0,0,2)C(0,0,2)$ and radius $11$

77 .

a. $PQ→=〈−4,−1,2〉;PQ→=〈−4,−1,2〉;$ b. $PQ→=−4i−j+2kPQ→=−4i−j+2k$

79 .

a. $PQ→=〈6,−24,24〉;PQ→=〈6,−24,24〉;$ b. $PQ→=6i−24j+24kPQ→=6i−24j+24k$

81 .

$Q ( 5 , 2 , 8 ) Q ( 5 , 2 , 8 )$

83 .

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

85 .

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

87 .

$‖u−v‖=38,‖u−v‖=38,$ $‖−2u‖=229‖−2u‖=229$

89 .

$‖u−v‖=2,‖u−v‖=2,$ $‖−2u‖=213‖−2u‖=213$

91 .

$a = 3 5 i − 4 5 j a = 3 5 i − 4 5 j$

93 .

$〈 2 62 i − 7 62 j + 3 62 k 〉 〈 2 62 i − 7 62 j + 3 62 k 〉$

95 .

$〈 − 2 6 , 1 6 , 1 6 〉 〈 − 2 6 , 1 6 , 1 6 〉$

97 .

Equivalent vectors

99 .

$u = 〈 70 59 , − 10 59 , 30 59 〉 u = 〈 70 59 , − 10 59 , 30 59 〉$

101 .

$u = 〈 − 4 5 sin t , − 4 5 cos t , − 2 5 〉 u = 〈 − 4 5 sin t , − 4 5 cos t , − 2 5 〉$

103 .

$〈 5 154 , 15 154 , − 60 154 〉 〈 5 154 , 15 154 , − 60 154 〉$

105 .

$α=−7,α=−7,$ $β=−15β=−15$

111 .

a. $F=〈30,40,0〉;F=〈30,40,0〉;$ b. $53°53°$

113 .

$D = 10 k D = 10 k$

115 .

$F 4 = 〈 −20 , −7 , −3 〉 F 4 = 〈 −20 , −7 , −3 〉$

117 .

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

119 .

a. $F=−294kF=−294k$ N; b. $F1=〈−4933,49,−98〉,F1=〈−4933,49,−98〉,$ $F2=〈−4933,−49,−98〉,F2=〈−4933,−49,−98〉,$ and $F3=〈9833,0,−98〉F3=〈9833,0,−98〉$ (each component is expressed in newtons)

121 .

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

b.

### Section 2.3 Exercises

123 .

6

125 .

0

127 .

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

129 .

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

131 .

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

133 .

a. $θ=π4θ=π4$ rad; b. $θθ$ is acute.

135 .

$θ = π 2 θ = π 2$

137 .

$θ = π 3 θ = π 3$

139 .

$θ=2θ=2$ rad

141 .

Orthogonal

143 .

Not orthogonal

145 .

$a=〈−4α3,α〉,a=〈−4α3,α〉,$ where $α≠0α≠0$ is a real number

147 .

$u=−αi+αj+βk,u=−αi+αj+βk,$ where $αα$ and $ββ$ are real numbers such that $α2+β2≠0α2+β2≠0$

149 .

$α = −6 α = −6$

151 .

a. $OP→=4i+5j,OP→=4i+5j,$ $OQ→=5i−7j;OQ→=5i−7j;$ b. $105.8°105.8°$

153 .

$68.33 ° 68.33 °$

155 .

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

161 .

a. $cosα=23,cosβ=23,cosα=23,cosβ=23,$ and $cosγ=13;cosγ=13;$ b. $α=48°,α=48°,$ $β=48°,β=48°,$ and $γ=71°γ=71°$

163 .

a. $cosα=−130,cosβ=530,cosα=−130,cosβ=530,$ and $cosγ=230;cosγ=230;$ b. $α=101°,α=101°,$ $β=24°,β=24°,$ and $γ=69°γ=69°$

167 .

a. $w=〈8029,3229〉;w=〈8029,3229〉;$ b. $compuv=1629compuv=1629$

169 .

a. $w=〈2413,0,1613〉;w=〈2413,0,1613〉;$ b. $compuv=813compuv=813$

171 .

a. $w=〈2425,−1825〉;w=〈2425,−1825〉;$ b. $q=〈5125,6825〉,q=〈5125,6825〉,$ $v=w+q=〈2425,−1825〉+〈5125,6825〉v=w+q=〈2425,−1825〉+〈5125,6825〉$

173 .

a. $22;22;$ b. $109.47°109.47°$

175 .

$17 N · m 17 N · m$

177 .

1175 $ft·lbft·lb$

179 .

W = 43301.27 $ft-lbft-lb$

181 .

a. $‖F1+F2‖=52.9‖F1+F2‖=52.9$ lb; b. The direction angles are $α=74.5°,α=74.5°,$ $β=36.7°,β=36.7°,$ and $γ=57.7°.γ=57.7°.$

### Section 2.4 Exercises

183 .

a. $u×v=〈0,0,4〉;u×v=〈0,0,4〉;$
b.

185 .

a. $u×v=〈6,−4,2〉;u×v=〈6,−4,2〉;$
b.

187 .

$−2 j − 4 k −2 j − 4 k$

189 .

$w = − 1 3 6 i − 7 3 6 j − 2 3 6 k w = − 1 3 6 i − 7 3 6 j − 2 3 6 k$

191 .

$w = − 4 21 i − 2 21 j − 1 21 k w = − 4 21 i − 2 21 j − 1 21 k$

193 .

$α = 10 α = 10$

197 .

$−3 i + 11 j + 2 k −3 i + 11 j + 2 k$

199 .

$w = 〈 −1 , e t , − e − t 〉 w = 〈 −1 , e t , − e − t 〉$

201 .

$−26 i + 17 j + 9 k −26 i + 17 j + 9 k$

203 .

$72 ° 72 °$

209 .

$7 7$

211 .

a. $56;56;$ b. $562;562;$ c. $56595659$

213 .

a. $2;2;$ b. $22$

215 .

$v·(u×w)=−1,v·(u×w)=−1,$ $w·(u×v)=1w·(u×v)=1$

217 .

$a=〈1,2,3〉,a=〈1,2,3〉,$ $b=〈0,2,5〉,b=〈0,2,5〉,$ $c=〈8,9,2〉;c=〈8,9,2〉;$ $a·(b×c)=−9a·(b×c)=−9$

219 .

a. $α=1;α=1;$ b. $h=1,h=1,$

225 .

Yes, $AD→=αAB→+βAC→,AD→=αAB→+βAC→,$ where $α=−1α=−1$ and $β=1.β=1.$

227 .

$− k − k$

229 .

$〈 0 , ± 4 5 , ∓ 2 5 〉 〈 0 , ± 4 5 , ∓ 2 5 〉$

233 .

$w=〈w3−1,w3+1,w3〉,w=〈w3−1,w3+1,w3〉,$ where $w3w3$ is any real number

235 .

8.66 ft-lb

237 .

559 N

239 .

$F = 4.8 × 10 −15 k N F = 4.8 × 10 −15 k N$

241 .

a. $B(t)=〈2sint5,−2cost5,15〉;B(t)=〈2sint5,−2cost5,15〉;$
b.

### Section 2.5 Exercises

243 .

a. $r=〈−3,5,9〉+t〈7,−12,−7〉,r=〈−3,5,9〉+t〈7,−12,−7〉,$ $t∈ℝ;t∈ℝ;$ b. $x=−3+7t,y=5−12t,z=9−7t,x=−3+7t,y=5−12t,z=9−7t,$ $t∈ℝ;t∈ℝ;$ c. $x+37=y−5−12=z−9−7;x+37=y−5−12=z−9−7;$ d. $x=−3+7t,y=5−12t,z=9−7t,x=−3+7t,y=5−12t,z=9−7t,$ $t∈[0,1]t∈[0,1]$

245 .

a. $r=〈−1,0,5〉+t〈5,0,−2〉,r=〈−1,0,5〉+t〈5,0,−2〉,$ $t∈ℝ;t∈ℝ;$ b. $x=−1+5t,y=0,z=5−2t,x=−1+5t,y=0,z=5−2t,$ $t∈ℝ;t∈ℝ;$ c. $x+15=z−5−2,y=0;x+15=z−5−2,y=0;$ d. $x=−1+5t,y=0,z=5−2t,x=−1+5t,y=0,z=5−2t,$ $t∈[0,1]t∈[0,1]$

247 .

a. $x=1+t,y=−2+2t,z=3+3t,x=1+t,y=−2+2t,z=3+3t,$ $t∈ℝ;t∈ℝ;$ b. $x−11=y+22=z−33;x−11=y+22=z−33;$ c. $(0,−4,0)(0,−4,0)$

249 .

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

251 .

a. $P(1,3,5),P(1,3,5),$ $v=〈1,1,4〉;v=〈1,1,4〉;$ b. $33$

253 .

$2 2 3 2 2 3$

255 .

a. Parallel; b. $2323$

259 .

$( −12 , 6 , −4 ) ( −12 , 6 , −4 )$

261 .

The lines are skew.

263 .

The lines are equal.

265 .

a. $x=1+t,y=1−t,z=1+2t,x=1+t,y=1−t,z=1+2t,$ $t∈ℝ;t∈ℝ;$ b. For instance, the line passing through $AA$ with direction vector $j:x=1,z=1;j:x=1,z=1;$ c. For instance, the line passing through $AA$ and point $(2,0,0)(2,0,0)$ that belongs to $LL$ is a line that intersects; $L:x−1−1=y−1=z−1L:x−1−1=y−1=z−1$

267 .

a. $3x−2y+4z=0;3x−2y+4z=0;$ b. $3x−2y+4z=03x−2y+4z=0$

269 .

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

271 .

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

273 .

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

275 .

$( 3 , 0 , 0 ) ( 3 , 0 , 0 )$

277 .

$x=−2+2t,y=1−3t,z=3+t,x=−2+2t,y=1−3t,z=3+t,$ $t∈ℝt∈ℝ$

281 .

a. $−2y+3z−1=0;−2y+3z−1=0;$ b. $〈0,−2,3〉·〈x−1,y−1,z−1〉=0;〈0,−2,3〉·〈x−1,y−1,z−1〉=0;$ c. $x=0,y=−2t,z=3t,x=0,y=−2t,z=3t,$ $t∈ℝt∈ℝ$

283 .

a. Answers may vary; b. $x−11=z−6−1,y=4x−11=z−6−1,y=4$

285 .

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

287 .

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

289 .

$16 14 16 14$

291 .

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

293 .

a. The planes are parallel.

295 .

$1 6 1 6$

297 .

a. $1829;1829;$ b. $P(−5129,13029,6229)P(−5129,13029,6229)$

299 .

$4 x − 3 y = 0 4 x − 3 y = 0$

301 .

a. $v(1)=〈cos1,−sin1,2〉;v(1)=〈cos1,−sin1,2〉;$ b. $(cos1)(x−sin1)−(sin1)(y−cos1)+2(z−2)=0;(cos1)(x−sin1)−(sin1)(y−cos1)+2(z−2)=0;$
c.

### Section 2.6 Exercises

303 .

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

305 .

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

307 .

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

309 .

a. Cylinder; b. The x-axis

311 .

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

313 .

b.

315 .

d.

317 .

a.

319 .

$−x29+y214+z214=1,−x29+y214+z214=1,$ hyperboloid of one sheet with the x-axis as its axis of symmetry

321 .

$−x2103+y22−z210=1,−x2103+y22−z210=1,$ hyperboloid of two sheets with the y-axis as its axis of symmetry

323 .

$y=−z25+x25,y=−z25+x25,$ hyperbolic paraboloid with the y-axis as its axis of symmetry

325 .

$x215+y23+z25=1,x215+y23+z25=1,$ ellipsoid

327 .

$x240+y28−z25=0,x240+y28−z25=0,$ elliptic cone with the z-axis as its axis of symmetry

329 .

$x=y22+z23,x=y22+z23,$ elliptic paraboloid with the x-axis as its axis of symmetry

331 .

Parabola $y=−x24,y=−x24,$

333 .

Ellipse $y24+z2100=1,y24+z2100=1,$

335 .

Ellipse $y24+z2100=1,y24+z2100=1,$

337 .

a. Ellipsoid; b. The third equation; c. $x2100+y2400+z2225=1x2100+y2400+z2225=1$

339 .

a. $(x+3)216+(z−2)28=1;(x+3)216+(z−2)28=1;$ b. Cylinder centered at $(−3,2)(−3,2)$ with rulings parallel to the y-axis

341 .

a. $(x−3)24+(y−2)2−(z+2)2=1;(x−3)24+(y−2)2−(z+2)2=1;$ b. Hyperboloid of one sheet centered at $(3,2,−2),(3,2,−2),$ with the z-axis as its axis of symmetry

343 .

a. $(x+3)2+y24−z23=0;(x+3)2+y24−z23=0;$ b. Elliptic cone centered at $(−3,0,0),(−3,0,0),$ with the z-axis as its axis of symmetry

345 .

$x 2 4 + y 2 16 + z 2 = 1 x 2 4 + y 2 16 + z 2 = 1$

347 .

$(1,−1,0)(1,−1,0)$ and $(133,4,53)(133,4,53)$

349 .

$x2+z2+4y=0,x2+z2+4y=0,$ elliptic paraboloid

351 .

$( 0 , 0 , 100 ) ( 0 , 0 , 100 )$

355 .

a. $x=2−z22,y=±z24−z2,x=2−z22,y=±z24−z2,$ where $z∈[−2,2];z∈[−2,2];$
b.

357 .

two ellipses of equations $x22+y292=1x22+y292=1$ in planes $z=±22z=±22$

359 .

a. $x239632+y239632+z239502=1;x239632+y239632+z239502=1;$
b.

;
c. The intersection curve is the ellipse of equation $x239632+y239632=(2950)(4950)39502,x239632+y239632=(2950)(4950)39502,$ and the intersection is an ellipse.; d. The intersection curve is the ellipse of equation $2y239632+z239502=1.2y239632+z239502=1.$

361 .

a.

b. The intersection curve is $(x2+z2−1)3−x2z3=0.(x2+z2−1)3−x2z3=0.$

### Section 2.7 Exercises

363 .

$( 2 3 , 2 , 3 ) ( 2 3 , 2 , 3 )$

365 .

$( −2 3 , −2 , 3 ) ( −2 3 , −2 , 3 )$

367 .

$( 2 , π 3 , 2 ) ( 2 , π 3 , 2 )$

369 .

$( 3 2 , − π 4 , 7 ) ( 3 2 , − π 4 , 7 )$

371 .

A cylinder of equation $x2+y2=16,x2+y2=16,$ with its center at the origin and rulings parallel to the z-axis,

373 .

Hyperboloid of two sheets of equation $−x2+y2−z2=1,−x2+y2−z2=1,$ with the y-axis as the axis of symmetry,

375 .

Cylinder of equation $x2−2x+y2=0,x2−2x+y2=0,$ with a center at $(1,0,0)(1,0,0)$ and radius $1,1,$ with rulings parallel to the z-axis,

377 .

Plane of equation $x=2,x=2,$

379 .

$z = 3 z = 3$

381 .

$r 2 + z 2 = 9 r 2 + z 2 = 9$

383 .

$r = 16 cos θ , r = 0 r = 16 cos θ , r = 0$

385 .

$( 0 , 0 , −3 ) ( 0 , 0 , −3 )$

387 .

$( 6 , −6 , 6 2 ) ( 6 , −6 , 6 2 )$

389 .

$( 4 , 0 , 90 ° ) ( 4 , 0 , 90 ° )$

391 .

$( 3 , 90 ° , 90 ° ) ( 3 , 90 ° , 90 ° )$

393 .

Sphere of equation $x2+y2+z2=9x2+y2+z2=9$ centered at the origin with radius $3,3,$

395 .

Sphere of equation $x2+y2+(z−1)2=1x2+y2+(z−1)2=1$ centered at $(0,0,1)(0,0,1)$ with radius $1,1,$

397 .

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

399 .

$φ=π3φ=π3$ or $φ=2π3;φ=2π3;$ Elliptic cone

401 .

$ρcosφ=6;ρcosφ=6;$ Plane at $z=6z=6$

403 .

$( 10 , π 4 , 0.3218 ) ( 10 , π 4 , 0.3218 )$

405 .

$( 3 2 , π 2 , π 4 ) ( 3 2 , π 2 , π 4 )$

407 .

$( 2 , − π 4 , 0 ) ( 2 , − π 4 , 0 )$

409 .

$( 8 , π 3 , 0 ) ( 8 , π 3 , 0 )$

411 .

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

413 .

Cylindrical system, ${ ( r , θ , z ) | r 2 + z 2 ≤ 9 , r ≥ 0 , π 2 ≤ θ ≤ 3 π 2 , ( r ≥ 3 cos ⁡ θ , − π 2 ≤ θ ≤ π 2 ) } { ( r , θ , z ) | r 2 + z 2 ≤ 9 , r ≥ 0 , π 2 ≤ θ ≤ 3 π 2 , ( r ≥ 3 cos ⁡ θ , − π 2 ≤ θ ≤ π 2 ) }$

415 .

The region is described by the set of points ${(r,θ,z)|0≤r≤1,0≤θ≤2π,r2≤z≤r}.{(r,θ,z)|0≤r≤1,0≤θ≤2π,r2≤z≤r}.$

417 .

$( 4000 , − 77 ° , 51 ° ) ( 4000 , − 77 ° , 51 ° )$

419 .

$43.17°W,43.17°W,$ $22.91°S22.91°S$

421 .

a. $ρ=0,ρ=0,$ $ρ+R2−r2−2Rsinφ=0;ρ+R2−r2−2Rsinφ=0;$
c.

### Review Exercises

423 .

True

425 .

False

427 .

a. $〈24,−5〉;〈24,−5〉;$ b. $85;85;$ c. Can’t dot a vector with a scalar; d. $−29−29$

429 .

$a = ± 2 a = ± 2$

431 .

$〈 1 14 , − 2 14 , − 3 14 〉 〈 1 14 , − 2 14 , − 3 14 〉$

433 .

$27 27$

435 .

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

437 .

$− x + 3 y + 8 z = 43 − x + 3 y + 8 z = 43$

439 .

$x=kx=k$ trace: $k2=y2+z2k2=y2+z2$ is a circle, $y=ky=k$ trace: $x2−z2=k2x2−z2=k2$ is a hyperbola (or a pair of lines if $k=0),k=0),$ $z=kz=k$ trace: $x2−y2=k2x2−y2=k2$ is a hyperbola (or a pair of lines if $k=0).k=0).$ The surface is a cone.

441 .

Cylindrical: $z=r2−1,z=r2−1,$ spherical: $cosφ=ρsin2φ−1ρcosφ=ρsin2φ−1ρ$

443 .

$x2−2x+y2+z2=1,x2−2x+y2+z2=1,$ sphere

445 .

331 N, and 244 N

447 .

$15 J 15 J$

449 .

More, $59.0959.09$ J

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