Checkpoint
Vectors a,a, b,b, and ee are equivalent.
〈 3 , 7 〉 〈 3 , 7 〉
a. ‖a‖=52,‖a‖=52, b. b=〈−4,−3〉,b=〈−4,−3〉, c. 3a−4b=〈37,15〉3a−4b=〈37,15〉
v = 〈 −5 , 5 3 〉 v = 〈 −5 , 5 3 〉
〈 − 45 85 , − 10 85 〉 〈 − 45 85 , − 10 85 〉
a=16i−11j,a=16i−11j, b=−22i−22jb=−22i−22j
Approximately 516516 mph
5 2 5 2
z = −4 z = −4
( x + 2 ) 2 + ( y − 4 ) 2 + ( z + 5 ) 2 = 52 ( x + 2 ) 2 + ( y − 4 ) 2 + ( z + 5 ) 2 = 52
x 2 + ( y − 2 ) 2 + ( z + 2 ) 2 = 14 x 2 + ( y − 2 ) 2 + ( z + 2 ) 2 = 14
The set of points forms the two planes y=−2y=−2 and z=3.z=3.
A cylinder of radius 4 centered on the line with x=0andz=2.x=0andz=2.
S T → = 〈 −1 , −9 , 1 〉 = − i − 9 j + k S T → = 〈 −1 , −9 , 1 〉 = − i − 9 j + k
〈 1 3 10 , − 5 3 10 , 8 3 10 〉 〈 1 3 10 , − 5 3 10 , 8 3 10 〉
v = 〈 16 2 , 12 2 , 20 2 〉 v = 〈 16 2 , 12 2 , 20 2 〉
7
a. (r·p)q=〈12,−12,12〉;(r·p)q=〈12,−12,12〉; b. ‖p‖2=53‖p‖2=53
θ≈0.22θ≈0.22 rad
x = 5 x = 5
a. α≈1.04α≈1.04 rad; b. β≈2.58β≈2.58 rad; c. γ≈1.40γ≈1.40 rad
Sales = $15,685.50; profit = $14,073.15
v=p+q,v=p+q, where p=185i+95jp=185i+95j and q=75i−145jq=75i−145j
21 knots
150 ft-lb
i − 9 j + 2 k i − 9 j + 2 k
Up (the positive z-direction)
− i − i
− k − k
16 16
40 40
8 i − 35 j + 2 k 8 i − 35 j + 2 k
〈 −3 194 , −13 194 , 4 194 〉 〈 −3 194 , −13 194 , 4 194 〉
6 13 6 13
17 17
88 units3
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.
2020 N
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
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
10 7 10 7
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(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
15 21 15 21
x = t , y = 7 − 3 t , z = 4 − 2 t x = t , y = 7 − 3 t , z = 4 − 2 t
1.441.44 rad
9 30 9 30
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).
Hyperboloid of one sheet, centered at (0,0,1)(0,0,1)
The rectangular coordinates of the point are (532,52,4).(532,52,4).
( 8 2 , 3 π 4 , −7 ) ( 8 2 , 3 π 4 , −7 )
This surface is a cylinder with radius 6.6.
Cartesian: (−32,−12,3),(−32,−12,3), cylindrical: (1,−5π6,3)(1,−5π6,3)
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.
( 4000 , 151 ° , 124 ° ) ( 4000 , 151 ° , 124 ° )
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
a. PQ→=〈2,2〉;PQ→=〈2,2〉; b. PQ→=2i+2jPQ→=2i+2j
a. QP→=〈−2,−2〉;QP→=〈−2,−2〉; b. QP→=−2i−2jQP→=−2i−2j
a. PQ→+PR→=〈0,6〉;PQ→+PR→=〈0,6〉; b. PQ→+PR→=6jPQ→+PR→=6j
a. 2PQ→−2PR→=〈8,−4〉;2PQ→−2PR→=〈8,−4〉; b. 2PQ→−2PR→=8i−4j2PQ→−2PR→=8i−4j
a. 〈12,12〉;〈12,12〉; b. 12i+12j12i+12j
〈 3 5 , 4 5 〉 〈 3 5 , 4 5 〉
Q ( 0 , 2 ) Q ( 0 , 2 )
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〉
15 15
λ = −3 λ = −3
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
Answers may vary
v = 〈 21 5 , 28 5 〉 v = 〈 21 5 , 28 5 〉
v = 〈 21 34 34 , − 35 34 34 〉 v = 〈 21 34 34 , − 35 34 34 〉
u = 〈 3 , 1 〉 u = 〈 3 , 1 〉
u = 〈 0 , 5 〉 u = 〈 0 , 5 〉
u = 〈 −5 3 , 5 〉 u = 〈 −5 3 , 5 〉
θ = 7 π 4 θ = 7 π 4
Answers may vary
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)〉
D ( 6 , 1 ) D ( 6 , 1 )
〈 60.62 , 35 〉 〈 60.62 , 35 〉
The horizontal and vertical components are 750750 ft/sec and 1299.041299.04 ft/sec, respectively.
The magnitude of resultant force is 94.7194.71 lb; the direction angle is 13.42°.13.42°.
The magnitude of the third vector is 60.0360.03 N; the direction angle is 259.38°.259.38°.
The new ground speed of the airplane is 572.19572.19 mph; the new direction is N41.82E.N41.82E.
‖T1‖=30.13lb,‖T1‖=30.13lb, ‖T2‖=38.35lb‖T2‖=38.35lb
‖v1‖=750‖v1‖=750 lb, ‖v2‖=1299‖v2‖=1299 lb
The two horizontal and vertical components of the force of tension are 2828 lb and 4242 lb, respectively.
Section 2.2 Exercises
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
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)
A cylinder of radius 11 centered on the line y=1,z=1y=1,z=1
z = 1 z = 1
z = −2 z = −2
( x + 1 ) 2 + ( y − 7 ) 2 + ( z − 4 ) 2 = 16 ( x + 1 ) 2 + ( y − 7 ) 2 + ( z − 4 ) 2 = 16
( x + 3 ) 2 + ( y − 3.5 ) 2 + ( z − 8 ) 2 = 29 4 ( x + 3 ) 2 + ( y − 3.5 ) 2 + ( z − 8 ) 2 = 29 4
Center C(0,0,2)C(0,0,2) and radius 11
a. PQ→=〈−4,−1,2〉;PQ→=〈−4,−1,2〉; b. PQ→=−4i−j+2kPQ→=−4i−j+2k
a. PQ→=〈6,−24,24〉;PQ→=〈6,−24,24〉; b. PQ→=6i−24j+24kPQ→=6i−24j+24k
Q ( 5 , 2 , 8 ) Q ( 5 , 2 , 8 )
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〉
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〉
‖u−v‖=38,‖u−v‖=38, ‖−2u‖=229‖−2u‖=229
‖u−v‖=2,‖u−v‖=2, ‖−2u‖=213‖−2u‖=213
a = 3 5 i − 4 5 j a = 3 5 i − 4 5 j
⟨ 2 62 , 7 62 , 3 62 ⟩ ⟨ 2 62 , 7 62 , 3 62 ⟩
〈 − 2 6 , 1 6 , 1 6 〉 〈 − 2 6 , 1 6 , 1 6 〉
Equivalent vectors
u = 〈 70 59 , − 10 59 , 30 59 〉 u = 〈 70 59 , − 10 59 , 30 59 〉
u = 〈 − 4 5 sin t , − 4 5 cos t , − 2 5 〉 u = 〈 − 4 5 sin t , − 4 5 cos t , − 2 5 〉
〈 5 154 , 15 154 , − 60 154 〉 〈 5 154 , 15 154 , − 60 154 〉
α=−7,α=−7, β=−15β=−15
a. F=〈30,40,0〉;F=〈30,40,0〉; b. 53°53°
D = 10 k D = 10 k
F 4 = 〈 −20 , −7 , −3 〉 F 4 = 〈 −20 , −7 , −3 〉
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
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)
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
6
0
(a·b)c=〈−11,−11,11〉;(a·b)c=〈−11,−11,11〉; (a·c)b=〈−20,−35,5〉(a·c)b=〈−20,−35,5〉
(a·b)c=〈1,0,−2〉;(a·b)c=〈1,0,−2〉; (a·c)b=〈1,0,−1〉(a·c)b=〈1,0,−1〉
a. θ=2.82θ=2.82 rad; b. θθ is not acute.
a. θ=π4θ=π4 rad; b. θθ is acute.
θ = π 2 θ = π 2
θ = π 3 θ = π 3
θ=2θ=2 rad
Orthogonal
Not orthogonal
a=〈−4α3,α〉,a=〈−4α3,α〉, where α≠0α≠0 is a real number
u=−αi+αj+βk,u=−αi+αj+βk, where αα and ββ are real numbers such that α2+β2≠0α2+β2≠0
α = −6 α = −6
a. OP→=4i+5j,OP→=4i+5j, OQ→=5i−7j;OQ→=5i−7j; b. 105.8°105.8°
68.33 ° 68.33 °
uu and vv are orthogonal; vv and ww are orthogonal.
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°
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°
a. w=〈8029,3229〉;w=〈8029,3229〉; b. compuv=1629compuv=1629
a. w=〈2413,0,1613〉;w=〈2413,0,1613〉; b. compuv=813compuv=813
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〉
a. 22;22; b. 109.47°109.47°
17 N · m 17 N · m
1175 ਫੁੱਟ-ਪੌਂਡ
ਡਬਲਯੂ = 43301.27 ਫੁੱਟ-ਪੌਂਡ
a. ‖F1+F2‖=52.9 ਪੌਂਡ; b. ਦਿਸ਼ਾ ਕੋਣ α=74.5°, β=36.7°, ਅਤੇ γ=57.7° ਹਨ।
ਭਾਗ 2.4 ਅਭਿਆਸ
a. u×v=〈0,0,4〉; b.
a. u×v=〈6,−4,2〉; b.
−2 ਜੇ − 4 ਕੇ
ਡਬਲਯੂ = − 136 ਆਈ − 736 ਜੇ − 236 ਕੇ
ਡਬਲਯੂ = − 421 ਆਈ − 221 ਜੇ − 121 ਕੇ
α = 10
−3 ਆਈ + 11 ਜੇ + 2 ਕੇ
ਡਬਲਯੂ = 〈−1, e^t, − e^−t〉
−26 ਆਈ + 17 ਜੇ + 9 ਕੇ
72°
7
a. 56; b. 562; c. 5659
a. 2; b. 22
v·(u×w)=−1, w·(u×v)=1
a=〈1,2,3〉, b=〈0,2,5〉, c=〈8,9,2〉; a·(b×c)=−9
a. α=1; b. h=1,
ਹਾਂ, AD→=αAB→+βAC→, ਜਿੱਥੇ α=−1 ਅਤੇ β=1.
− ਕੇ
〈0, ± 4/5, ∓ 2/5〉
w=〈w3−1, w3+1, w3〉, ਜਿੱਥੇ w3 ਕੋਈ ਵੀ ਅਸਲ ਸੰਖਿਆ ਹੈ
8.66 ft-lb
559 N
F = 4.8 × 10 −15 k N F = 4.8 × 10 −15 k N
a. B(t)=〈2sint5,−2cost5,15〉;B(t)=〈2sint5,−2cost5,15〉; b.
Section 2.5 Exercises
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]
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]
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)
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.
a. P(1,3,5),P(1,3,5), v=〈1,1,4〉;v=〈1,1,4〉; b. 33
2 2 3 2 2 3
a. Parallel; b. 2323
( −12 , 6 , −4 ) ( −12 , 6 , −4 )
The lines are skew.
The lines are equal.
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
a. 3x−2y+4z=0;3x−2y+4z=0; b. 3x−2y+4z=03x−2y+4z=0
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
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.
a. n=3i−2j+4k;n=3i−2j+4k; b. (0,0,0);(0,0,0); c.
( 3 , 0 , 0 ) ( 3 , 0 , 0 )
x=−2+2t,y=1−3t,z=3+t,x=−2+2t,y=1−3t,z=3+t, t∈ℝt∈ℝ
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.
a. Answers may vary; b. x−11=z−6−1,y=4x−11=z−6−1,y=4
2 x − 5 y − 3 z + 15 = 0 2 x − 5 y − 3 z + 15 = 0
The line intersects the plane at point P(−3,4,0).P(−3,4,0).
16 14 16 14
a. The planes are neither parallel nor orthogonal; b. 62°62°
a. The planes are parallel.
1 6 1 6
a. 1829;1829; b. P(−5129,13029,6229)P(−5129,13029,6229)
4 x − 3 y = 0 4 x − 3 y = 0
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
The surface is a cylinder with the rulings parallel to the y-axis.
The surface is a cylinder with rulings parallel to the y-axis.
The surface is a cylinder with rulings parallel to the x-axis.
a. Cylinder; b. The x-axis
a. Hyperboloid of two sheets; b. The x-axis
b.
d.
a.
−x29+y214+z214=1,−x29+y214+z214=1, hyperboloid of one sheet with the x-axis as its axis of symmetry
−x2103+y22−z210=1,−x2103+y22−z210=1, hyperboloid of two sheets with the y-axis as its axis of symmetry
y=−z25+x25,y=−z25+x25, hyperbolic paraboloid with the y-axis as its axis of symmetry
x215+y23+z25=1,x215+y23+z25=1, ellipsoid
x240+y28−z25=0,x240+y28−z25=0, elliptic cone with the z-axis as its axis of symmetry
x=y22+z23,x=y22+z23, elliptic paraboloid with the x-axis as its axis of symmetry
Parabola y=−x24,y=−x24,
Ellipse y24+z2100=1,y24+z2100=1,
Ellipse y24+z2100=1,y24+z2100=1,
a. Ellipsoid; b. The third equation; c. x2100+y2400+z2225=1x2100+y2400+z2225=1
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
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
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
x 2 4 + y 2 16 + z 2 = 1 x 2 4 + y 2 16 + z 2 = 1
(1,−1,0)(1,−1,0) and (133,4,53)(133,4,53)
x2+z2+4y=0,x2+z2+4y=0, elliptic paraboloid
( 0 , 0 , 100 ) ( 0 , 0 , 100 )
a. x=2−z22,y=±z24−z2,x=2−z22,y=±z24−z2, where z∈[−2,2];z∈[−2,2]; b.
two ellipses of equations x22+y292=1x22+y292=1 in planes z=±22z=±22
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.
a.
b. The intersection curve is (x2+z2−1)3−x2z3=0.(x2+z2−1)3−x2z3=0.
Section 2.7 Exercises
( 2 3 , 2 , 3 ) ( 2 3 , 2 , 3 )
( −2 3 , −2 , 3 ) ( −2 3 , −2 , 3 )
( 2 , π 3 , 2 ) ( 2 , π 3 , 2 )
( 3 2 , − π 4 , 7 ) ( 3 2 , − π 4 , 7 )
A cylinder of equation x2+y2=16,x2+y2=16, with its center at the origin and rulings parallel to the z-axis,
Hyperboloid of two sheets of equation −x2+y2−z2=1,−x2+y2−z2=1, with the y-axis as the axis of symmetry,
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,
Plane of equation x=2,x=2,
z = 3 z = 3
r 2 + z 2 = 9 r 2 + z 2 = 9
r = 16 cos θ , r = 0 r = 16 cos θ , r = 0
( 0 , 0 , −3 ) ( 0 , 0 , −3 )
( 6 , −6 , 6 2 ) ( 6 , −6 , 6 2 )
( 4 , 0 , 90 ° ) ( 4 , 0 , 90 ° )
( 3 , 90 ° , 90 ° ) ( 3 , 90 ° , 90 ° )
Sphere of equation x2+y2+z2=9x2+y2+z2=9 centered at the origin with radius 3,3,
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,
The xy-plane of equation z=0,z=0,
φ=π3φ=π3 or φ=2π3;φ=2π3; Elliptic cone
ρcosφ=6;ρcosφ=6; Plane at z=6z=6
( 10 , π 4 , 0.3218 ) ( 10 , π 4 , 0.3218 )
( 3 2 , π 2 , π 4 ) ( 3 2 , π 2 , π 4 )
( 2 , − π 4 , 0 ) ( 2 , − π 4 , 0 )
( 8 , π 3 , 0 ) ( 8 , π 3 , 0 )
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}
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 ) }
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}.
( 4000 , − 77 ° , 51 ° ) ( 4000 , − 77 ° , 51 ° )
43.17°W,43.17°W, 22.91°S22.91°S
a. ρ2=0,ρ2=0, ρ+R2−r2−2Rsinφ=0;ρ+R2−r2−2Rsinφ=0; c.
Review Exercises
True
False
a. 〈24,−5〉;〈24,−5〉; b. 85;85; c. Can’t cross a vector with a scalar; d. −29−29
a = ± 2 a = ± 2
〈 1 14 , − 2 14 , − 3 14 〉 〈 1 14 , − 2 14 , − 3 14 〉
27 27
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
− x + 3 y + 8 z = 43 − x + 3 y + 8 z = 43
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.
Cylindrical: z=r2−1,z=r2−1, spherical: cosφ=ρsin2φ−1ρcosφ=ρsin2φ−1ρ
x2−2x+y2+z2=1,x2−2x+y2+z2=1, sphere
331 N, and 244 N
15 J 15 J
More, 59.0959.09 J