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Chapter 2

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.

This figure is the 3-dimensional coordinate system. It has two intersecting planes drawn. The first is the x z-plane. The second is parallel to the y z-plane at the value of z = 3. They are perpendicular to each other.
2.17

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

This figure is the 3-dimensional coordinate system. It has a cylinder parallel to the y-axis and centered around the y-axis.
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).

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)(0,0,1)

2.55

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

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

2.57

This surface is a cylinder with radius 6.6.

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: (−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.

Answers may vary

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.

Answers may vary

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)

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 11 centered on the line y=1,z=1y=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 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 , 7 62 , 3 62 ⟩ ⟨ 2 62 , 7 62 , 3 62 ⟩

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.

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·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.

uu and vv are orthogonal; vv and ww 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.

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×v=〈6,−4,2〉;u×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 −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,

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, 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.

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〉,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.

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

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

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.

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.

−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,

This figure is the graph of an upside down parabola with its highest point at the origin of a rectangular coordinate system.
333.

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

This figure is the graph of an ellipse centered at the origin of a rectangular coordinate system.
335.

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

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

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 x22+y292=1x22+y292=1 in planes z=±22z=±22

359.

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

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 (x2+z2−1)3−x2z3=0.(x2+z2−1)3−x2z3=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 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,

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 −x2+y2−z2=1,−x2+y2−z2=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 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,

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.
377.

Plane of equation x=2,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 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,

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

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,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=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}.

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 ° ) ( 4000 , − 77 ° , 51 ° )

419.

43.17°W,43.17°W, 22.91°S22.91°S

421.

a. ρ2=0,ρ2=0, ρ+R2−r2−2Rsinφ=0;ρ+R2−r2−2Rsinφ=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〉;〈24,−5〉; b. 85;85; c. Can’t cross 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.

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=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