Chapter 2
> 来源: OpenStax《Calculus Volume 3》| 原页: https://openstax.org/books/calculus-volume-3/pages/chapter-2
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Chapter 2
Calculus Volume 3Chapter 2
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Chapter 2
Checkpoint
2.1
2.2
2.3
Vectors $\mathbf{\text{a}},$ $\mathbf{\text{b}},$ and $\mathbf{\text{e}}$ are equivalent.
2.4
$\left\langle {3,7} \right\rangle$
2.5
a\. $\left\| \mathbf{\text{a}} \right\| = 5\sqrt{2},$ b. $\mathbf{\text{b}} = \left\langle {-4,-3} \right\rangle,$ c. $3\mathbf{\text{a}} - 4\mathbf{\text{b}} = \left\langle {37,15} \right\rangle$
2.7
$\mathbf{\text{v}} = \left\langle {-5,5\sqrt{3}} \right\rangle$
2.8
$\left\langle {- \frac{45}{\sqrt{85}}, - \frac{10}{\sqrt{85}}} \right\rangle$
2.9
$\mathbf{\text{a}} = 16\mathbf{\text{i}} - 11\mathbf{\text{j}},$ $\mathbf{\text{b}} = - \frac{\sqrt{2}}{2}\mathbf{\text{i}} - \frac{\sqrt{2}}{2}\mathbf{\text{j}}$
2.10
Approximately $516$ mph
2.11
2.12
$5\sqrt{2}$
2.13
$z = -4$
2.14
$\left( {x + 2} \right)^{2} + \left( {y - 4} \right)^{2} + \left( {z + 5} \right)^{2} = 52$
2.15
$x^{2} + \left( {y - 2} \right)^{2} + \left( {z + 2} \right)^{2} = 14$
2.16
The set of points forms the two planes $y = -2$ and $z = 3.$
2.17
A cylinder of radius 4 centered on the line with $x = 0\ \text{and}\ z = 2.$
2.18
$\overset{\rightarrow}{ST} = \left\langle {-1,-9,1} \right\rangle = \text{−}\mathbf{\text{i}} - 9\mathbf{\text{j}} + \mathbf{\text{k}}$
2.19
$\left\langle {\frac{1}{3\sqrt{10}}, - \frac{5}{3\sqrt{10}},\frac{8}{3\sqrt{10}}} \right\rangle$
2.20
$\mathbf{\text{v}} = \left\langle {16\sqrt{2},12\sqrt{2},20\sqrt{2}} \right\rangle$
2.21
7
2.22
a\. $\left( {\mathbf{\text{r}} \cdot \mathbf{\text{p}}} \right)\mathbf{\text{q}} = \left\langle {12,-12,12} \right\rangle;$ b. $\left\| \mathbf{\text{p}} \right\|^{2} = 53$
2.23
$\theta \approx 0.22$ rad
2.24
$x = 5$
2.25
a\. $\alpha \approx 1.04$ rad; b. $\beta \approx 2.58$ rad; c. $\gamma \approx 1.40$ rad
2.26
Sales = \$15,685.50; profit = \$14,073.15
2.27
$\mathbf{\text{v}} = \mathbf{\text{p}} + \mathbf{\text{q}},$ where $\mathbf{\text{p}} = \frac{18}{5}\mathbf{\text{i}} + \frac{9}{5}\mathbf{\text{j}}$ and $\mathbf{\text{q}} = \frac{7}{5}\mathbf{\text{i}} - \frac{14}{5}\mathbf{\text{j}}$
2.28
21 knots
2.29
150 ft-lb
2.30
$\mathbf{\text{i}} - 9\mathbf{\text{j}} + 2\mathbf{\text{k}}$
2.31
Up (the positive *z*-direction)
2.32
$\text{−}\mathbf{\text{i}}$
2.33
$\text{−}\mathbf{\text{k}}$
2.34
$16$
2.35
$40$
2.36
$8\mathbf{\text{i}} - 35\mathbf{\text{j}} + 2\mathbf{\text{k}}$
2.37
$\left\langle {\frac{-3}{\sqrt{194}},\frac{-13}{\sqrt{194}},\frac{4}{\sqrt{194}}} \right\rangle$
2.38
$6\sqrt{13}$
2.39
$17$
2.40
$8$ units3
2.41
No, the triple scalar product is $-4 \neq 0,$ so the three vectors form the adjacent edges of a parallelepiped. They are not coplanar.
2.42
$20$ N
2.43
Possible set of parametric equations: $x = 1 + 4t,y = -3 + t,z = 2 + 6t;$
related set of symmetric equations: $\frac{x - 1}{4} = y + 3 = \frac{z - 2}{6}$
2.44
$x = -1 - 7t,y = 3 - t,z = 6 - 2t,0 \leq t \leq 1$
2.45
$\sqrt{\frac{10}{7}}$
2.46
These lines are skew because their direction vectors are not parallel and there is no point $\left( {x,y,z} \right)$ that lies on both lines.
2.47
$-2\left( {x - 1} \right) + \left( {y + 1} \right) + 3\left( {z - 1} \right) = 0$ or $-2x + y + 3z = 0$
2.48
$\frac{15}{\sqrt{21}}$
2.49
$x = t,y = 7 - 3t,z = 4 - 2t$
2.50
$1.44$ rad
2.51
$\frac{9}{\sqrt{30}}$
2.52
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 $\frac{x^{2}}{3^{2}} + \frac{y^{2}}{2^{2}} = 1,$ 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).
2.54
Hyperboloid of one sheet, centered at $\left( {0,0,1} \right)$
2.55
The rectangular coordinates of the point are $\left( {\frac{5\sqrt{3}}{2},\frac{5}{2},4} \right).$
2.56
$\left( {8\sqrt{2},\frac{3\pi}{4},-7} \right)$
2.57
This surface is a cylinder with radius $6.$
2.58
Cartesian: $\left( {- \frac{\sqrt{3}}{2}, - \frac{1}{2},\sqrt{3}} \right),$ cylindrical: $\left( {1, - \frac{5\pi}{6},\sqrt{3}} \right)$
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 $\theta = \frac{2\pi}{3}.$ c. Let $P$ be a point on this surface. The position vector of this point forms an angle of $\varphi = \frac{\pi}{4}$ 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
$\left( {4000,151\text{°},124\text{°}} \right)$
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\. $\overset{\rightarrow}{PQ} = \left\langle {2,2} \right\rangle;$ b. $\overset{\rightarrow}{PQ} = 2\textbf{i} + 2\textbf{j}$
3.
a\. $\overset{\rightarrow}{QP} = \left\langle {-2,-2} \right\rangle;$ b. $\overset{\rightarrow}{QP} = -2\textbf{i} - 2\textbf{j}$
5.
a\. $\overset{\rightarrow}{PQ} + \overset{\rightarrow}{PR} = \left\langle {0,6} \right\rangle;$ b. $\overset{\rightarrow}{PQ} + \overset{\rightarrow}{PR} = 6\textbf{j}$
7.
a\. $2\overset{\rightarrow}{PQ} - 2\overset{\rightarrow}{PR} = \left\langle {8,-4} \right\rangle;$ b. $2\overset{\rightarrow}{PQ} - 2\overset{\rightarrow}{PR} = 8\textbf{i} - 4\textbf{j}$
9.
a\. $\left\langle {\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}} \right\rangle;$ b. $\frac{1}{\sqrt{2}}\textbf{i} + \frac{1}{\sqrt{2}}\textbf{j}$
11.
$\left\langle {\frac{3}{5},\frac{4}{5}} \right\rangle$
13.
$Q(0,2)$
15.
a\. $\textbf{a} + \textbf{b} = 3\textbf{i} + 4\textbf{j},$ $\textbf{a} + \textbf{b} = \left\langle {3,4} \right\rangle;$ b. $\textbf{a} - \textbf{b} = \textbf{i} - 2\textbf{j},$ $\textbf{a} - \textbf{b} = \left\langle {1,-2} \right\rangle;$ c. Answers will vary; d. $2\textbf{a} = 4\textbf{i} + 2\textbf{j},$ $2\textbf{a} = \left\langle {4,2} \right\rangle,$ $\text{−}\textbf{b} = \text{−}\textbf{i} - 3\textbf{j},$ $\text{−}\textbf{b} = \left\langle {-1,-3} \right\rangle,$ $2\textbf{a} - \textbf{b} = 3\textbf{i} - \textbf{j},$ $2\textbf{a} - \textbf{b} = \left\langle {3,-1} \right\rangle$
17.
$15$
19.
$\lambda = -3$
21.
a\. $\textbf{a}(0) = \left\langle {1,0} \right\rangle,$ $\textbf{a}(\pi) = \left\langle {-1,0} \right\rangle;$ b. Answers may vary; c. Answers may vary
23.
Answers may vary
25.
$\textbf{v} = \left\langle {\frac{21}{5},\frac{28}{5}} \right\rangle$
27.
$\textbf{v} = \left\langle {\frac{21\sqrt{34}}{34}, - \frac{35\sqrt{34}}{34}} \right\rangle$
29.
$\textbf{u} = \left\langle {\sqrt{3},1} \right\rangle$
31.
$\textbf{u} = \left\langle {0,5} \right\rangle$
33.
$\textbf{u} = \left\langle {-5\sqrt{3},5} \right\rangle$
35.
$\theta = \frac{7\pi}{4}$
37.
Answers may vary
39.
a\. $z_{0} = f(x_{0}) + f^{\prime}(x_{0});$ b. $\textbf{u} = \frac{1}{\sqrt{1 + \left\lbrack {f^{\prime}(x_{0})} \right\rbrack^{2}}}\left\langle {1,f^{\prime}(x_{0})} \right\rangle$
43.
$D(6,1)$
45.
$\left\langle {60.62,35} \right\rangle$
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 $13.42\text{°}.$
51.
The magnitude of the third vector is $60.03$ N; the direction angle is $259.38\text{°}.$
53.
The new ground speed of the airplane is $572.19$ mph; the new direction is $\text{N}41.82\text{E}.$
55.
$\left\| \mathbf{\text{T}}_{1} \right\| = 30.13\ \text{lb},$ $\left\| \mathbf{\text{T}}_{2} \right\| = 38.35\ \text{lb}$
57.
$\left\| \textbf{v}_{1} \right\| = 750$ lb, $\left\| \textbf{v}_{2} \right\| = 1299$ 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\. $\left( {2,0,5} \right),\left( {2,0,0} \right),\left( {2,3,0} \right),\left( {0,3,0} \right),\left( {0,3,5} \right),\left( {0,0,5} \right);$ 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)
65.
A cylinder of radius $1$ centered on the line $y = 1,z = 1$
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 $C\left( {0,0,2} \right)$ and radius $1$
77.
a\. $\overset{\rightarrow}{PQ} = \left\langle {-4,-1,2} \right\rangle;$ b. $\overset{\rightarrow}{PQ} = -4\mathbf{\text{i}} - \mathbf{\text{j}} + 2\mathbf{\text{k}}$
79.
a\. $\overset{\rightarrow}{PQ} = \left\langle {6,-24,24} \right\rangle;$ b. $\overset{\rightarrow}{PQ} = 6\mathbf{\text{i}} - 24\textbf{j} + 24\mathbf{\text{k}}$
81.
$Q(5,2,8)$
83.
$\mathbf{\text{a}} + \textbf{b} = \left\langle {-6,4,-3} \right\rangle,$ $4\mathbf{\text{a}} = \left\langle {-4,-8,16} \right\rangle,$ $-5\mathbf{\text{a}} + 3\mathbf{\text{b}} = \left\langle {-10,28,-41} \right\rangle$
85.
$\mathbf{\text{a}} + \textbf{b} = \left\langle {-1,0,-1} \right\rangle,$ $4\mathbf{\text{a}} = \left\langle {0,0,-4} \right\rangle,$ $-5\mathbf{\text{a}} + 3\mathbf{\text{b}} = \left\langle {-3,0,5} \right\rangle$
87.
$\left\| {\mathbf{\text{u}} - \mathbf{\text{v}}} \right\| = \sqrt{38},$ $\left\| {-2\mathbf{\text{u}}} \right\| = 2\sqrt{29}$
89.
$\left\| {\mathbf{\text{u}} - \mathbf{\text{v}}} \right\| = 2,$ $\left\| {-2\mathbf{\text{u}}} \right\| = 2\sqrt{13}$
91.
$\mathbf{\text{a}} = \frac{3}{5}\mathbf{\text{i}} - \frac{4}{5}\mathbf{\text{j}}$
93.
$\left\langle \frac{2}{\sqrt{62}},\frac{7}{\sqrt{62}},\frac{3}{\sqrt{62}} \right\rangle$
95.
$\left\langle {- \frac{2}{\sqrt{6}},\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}}} \right\rangle$
97.
Equivalent vectors
99.
$\mathbf{\text{u}} = \left\langle {\frac{70}{\sqrt{59}}, - \frac{10}{\sqrt{59}},\frac{30}{\sqrt{59}}} \right\rangle$
101.
$\mathbf{\text{u}} = \left\langle {- \frac{4}{\sqrt{5}}\text{sin}\ t, - \frac{4}{\sqrt{5}}\text{cos}\ t, - \frac{2}{\sqrt{5}}} \right\rangle$
103.
$\left\langle {\frac{5}{\sqrt{154}},\frac{15}{\sqrt{154}}, - \frac{60}{\sqrt{154}}} \right\rangle$
105.
$\alpha = \text{−}\sqrt{7},$ $\beta = \text{−}\sqrt{15}$
111.
a\. $\mathbf{\text{F}} = \left\langle {30,40,0} \right\rangle;$ b. $53\text{°}$
113.
$\textbf{D} = 10\mathbf{\text{k}}$
115.
$\mathbf{\text{F}}_{4} = \left\langle {-20,-7,-3} \right\rangle$
117.
a\. $\mathbf{\text{F}} = -19.6\mathbf{\text{k}},$ $\left\| \textbf{F} \right\| = 19.6$ N; b. $\textbf{T} = 19.6\mathbf{\text{k}},$ $\left\| \textbf{T} \right\| = 19.6$ N
119.
a\. $\mathbf{\text{F}} = -294\mathbf{\text{k}}$ N; b. $\mathbf{\text{F}}_{1} = \left\langle {- \frac{49\sqrt{3}}{3},49,-98} \right\rangle,$ $\mathbf{\text{F}}_{2} = \left\langle {- \frac{49\sqrt{3}}{3},-49,-98} \right\rangle,$ and $\mathbf{\text{F}}_{3} = \left\langle {\frac{98\sqrt{3}}{3},0,-98} \right\rangle$ (each component is expressed in newtons)
121.
a\. $\mathbf{\text{v}}(1) = \left\langle {-0.84,0.54,2} \right\rangle$ (each component is expressed in centimeters per second); $\left\| {\mathbf{\text{v}}(1)} \right\| = 2.24$ (expressed in centimeters per second); $\mathbf{\text{a}}(1) = \left\langle {-0.54,-0.84,0} \right\rangle$ (each component expressed in centimeters per second squared);
b.
Section 2.3 Exercises
123.
6
125.
0
127.
$\left( {\mathbf{\text{a}} \cdot \textbf{b}} \right)\textbf{c} = \left\langle {-11,-11,11} \right\rangle;$ $\left( {\mathbf{\text{a}} \cdot \textbf{c}} \right)\textbf{b} = \left\langle {-20,-35,5} \right\rangle$
129.
$\left( {\mathbf{\text{a}} \cdot \textbf{b}} \right)\textbf{c} = \left\langle {1,0,-2} \right\rangle;$ $\left( {\mathbf{\text{a}} \cdot \textbf{c}} \right)\textbf{b} = \left\langle {1,0,-1} \right\rangle$
131.
a\. $\theta = 2.82$ rad; b. $\theta$ is not acute.
133.
a\. $\theta = \frac{\pi}{4}$ rad; b. $\theta$ is acute.
135.
$\theta = \frac{\pi}{2}$
137.
$\theta = \frac{\pi}{3}$
139.
$\theta = 2$ rad
141.
Orthogonal
143.
Not orthogonal
145.
$\mathbf{\text{a}} = \left\langle {- \frac{4\alpha}{3},\alpha} \right\rangle,$ where $\alpha \neq 0$ is a real number
147.
$\mathbf{\text{u}} = \text{−}\alpha\mathbf{\text{i}} + \alpha\textbf{j} + \beta\mathbf{\text{k}},$ where $\alpha$ and $\beta$ are real numbers such that $\alpha^{2} + \beta^{2} \neq 0$
149.
$\alpha = -6$
151.
a\. $\overset{\rightarrow}{OP} = 4\mathbf{\text{i}} + 5\mathbf{\text{j}},$ $\overset{\rightarrow}{OQ} = 5\mathbf{\text{i}} - 7\textbf{j};$ b. $105.8\text{°}$
153.
$68.33\text{°}$
155.
$\textbf{u}$ and $\textbf{v}$ are orthogonal; $\textbf{v}$ and $\textbf{w}$ are orthogonal.
161.
a\. $\text{cos}\ \alpha = \frac{2}{3},\text{cos}\ \beta = \frac{2}{3},$ and $\text{cos}\ \gamma = \frac{1}{3};$ b. $\alpha = 48\text{°},$ $\beta = 48\text{°},$ and $\gamma = 71\text{°}$
163.
a\. $\text{cos}\ \alpha = - \frac{1}{\sqrt{30}},\text{cos}\ \beta = \frac{5}{\sqrt{30}},$ and $\text{cos}\ \gamma = \frac{2}{\sqrt{30}};$ b. $\alpha = 101\text{°},$ $\beta = 24\text{°},$ and $\gamma = 69\text{°}$
167.
a\. $\mathbf{\text{w}} = \left\langle {\frac{80}{29},\frac{32}{29}} \right\rangle;$ b. $\text{comp}_{\text{u}}\textbf{v} = \frac{16}{\sqrt{29}}$
169.
a\. $\mathbf{\text{w}} = \left\langle {\frac{24}{13},0,\frac{16}{13}} \right\rangle;$ b. $\text{comp}_{\text{u}}\textbf{v} = \frac{8}{\sqrt{13}}$
171.
a\. $\mathbf{\text{w}} = \left\langle {\frac{24}{25}, - \frac{18}{25}} \right\rangle;$ b. $\textbf{q} = \left\langle {\frac{51}{25},\frac{68}{25}} \right\rangle,$ $\mathbf{\text{v}} = \textbf{w} + \textbf{q} = \left\langle {\frac{24}{25}, - \frac{18}{25}} \right\rangle + \left\langle {\frac{51}{25},\frac{68}{25}} \right\rangle$
173.
a\. $2\sqrt{2};$ b. $109.47\text{°}$
175.
$17\text{N} \cdot \text{m}$
177.
1175 $\text{ft} \cdot \text{lb}$
179.
W = 43301.27 $\text{ft-lb}$
181.
a\. $\left\| {\mathbf{\text{F}}_{1} + \textbf{F}_{2}} \right\| = 52.9$ lb; b. The direction angles are $\alpha = 74.5\text{°},$ $\beta = 36.7\text{°},$ and $\gamma = 57.7\text{°}.$
Section 2.4 Exercises
183.
a\. $\mathbf{\text{u}}\ \times \ \mathbf{\text{v}} = \left\langle {0,0,4} \right\rangle;$
b.
185.
a\. $\mathbf{\text{u}}\ \times \ \mathbf{\text{v}} = \left\langle {6,-4,2} \right\rangle;$
b.
187.
$-2\mathbf{\text{j}} - 4\mathbf{\text{k}}$
189.
$\mathbf{\text{w}} = - \frac{1}{3\sqrt{6}}\mathbf{\text{i}} - \frac{7}{3\sqrt{6}}\mathbf{\text{j}} - \frac{2}{3\sqrt{6}}\mathbf{\text{k}}$
191.
$\mathbf{\text{w}} = - \frac{4}{\sqrt{21}}\mathbf{\text{i}} - \frac{2}{\sqrt{21}}\mathbf{\text{j}} - \frac{1}{\sqrt{21}}\mathbf{\text{k}}$
193.
$\alpha = 10$
197.
$-3\mathbf{\text{i}} + 11\mathbf{\text{j}} + 2\mathbf{\text{k}}$
199.
$\mathbf{\text{w}} = \left\langle {-1,e^{t},\text{−}e^{\text{−}t}} \right\rangle$
201.
$-26\mathbf{\text{i}} + 17\mathbf{\text{j}} + 9\mathbf{\text{k}}$
203.
$72\text{°}$
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.
$\mathbf{\text{v}} \cdot (\mathbf{\text{u}}\ \times \ \text{w}) = -1,$ $\mathbf{\text{w}} \cdot (\mathbf{\text{u}}\ \times \ \mathbf{\text{v}}) = 1$
217.
$\mathbf{\text{a}} = \left\langle {1,2,3} \right\rangle,$ $\textbf{b} = \left\langle {0,2,5} \right\rangle,$ $\textbf{c} = \left\langle {8,9,2} \right\rangle;$ $\mathbf{\text{a}} \cdot (\text{b}\ \times \ \text{c}) = -9$
219.
a\. $\alpha = 1;$ b. $h = 1,$
225.
Yes, $\overset{\rightarrow}{AD} = \alpha\overset{\rightarrow}{AB} + \beta\overset{\rightarrow}{AC},$ where $\alpha = -1$ and $\beta = 1.$
227.
$\text{−}\mathbf{\text{k}}$
229.
$\left\langle {0,\text{±}4\sqrt{5}, \mp 2\sqrt{5}} \right\rangle$
233.
$\mathbf{\text{w}} = \left\langle {w_{3} - 1,w_{3} + 1,w_{3}} \right\rangle,$ where $w_{3}$ is any real number
235.
8.66 ft-lb
237.
559 N
239.
$\mathbf{\text{F}} = 4.8\ \times \ 10^{-15}\mathbf{\text{k}}\ \textbf{N}$
241.
a\. $\textbf{B}(t) = \left\langle {\frac{2\ \text{sin}\ t}{\sqrt{5}}, - \frac{2\ \text{cos}\ t}{\sqrt{5}},\frac{1}{\sqrt{5}}} \right\rangle;$
b.
Section 2.5 Exercises
243.
a\. $\mathbf{\text{r}} = \left\langle {-3,5,9} \right\rangle + t\left\langle {7,-12,-7} \right\rangle,$ $t \in \mathbb{R}\text{;}$ b. $x = -3 + 7t,y = 5 - 12t,z = 9 - 7t,$ $t \in \mathbb{R}\text{;}$ c. $\frac{x + 3}{7} = \frac{y - 5}{-12} = \frac{z - 9}{-7};$ d. $x = -3 + 7t,y = 5 - 12t,z = 9 - 7t,$ $t \in \lbrack 0,1\rbrack$
245.
a\. $\mathbf{\text{r}} = \left\langle {-1,0,5} \right\rangle + t\left\langle {5,0,-2} \right\rangle,$ $t \in \mathbb{R};$ b. $x = -1 + 5t,y = 0,z = 5 - 2t,$ $t \in \mathbb{R};$ c. $\frac{x + 1}{5} = \frac{z - 5}{-2},y = 0;$ d. $x = -1 + 5t,y = 0,z = 5 - 2t,$ $t \in \lbrack 0,1\rbrack$
247.
a\. $x = 1 + t,y = -2 + 2t,z = 3 + 3t,$ $t \in \mathbb{R}\text{;}$ b. $\frac{x - 1}{1} = \frac{y + 2}{2} = \frac{z - 3}{3};$ c. $(0,-4,0)$
249.
a\. $x = 3 + t,y = 1,z = 5,$ $t \in \mathbb{R};$ b. $y = 1,z = 5;$ c. The line does not intersect the *xy*-plane.
251.
a\. $P(1,3,5),$ $v = \left\langle {1,1,4} \right\rangle;$ b. $\sqrt{3}$
253.
$\frac{2\sqrt{2}}{\sqrt{3}}$
255.
a\. Parallel; b. $\frac{\sqrt{2}}{\sqrt{3}}$
259.
$\left( {-12,6,-4} \right)$
261.
The lines are skew.
263.
The lines are equal.
265.
a\. $x = 1 + t,y = 1 - t,z = 1 + 2t,$ $t \in \mathbb{R};$ b. For instance, the line passing through $A$ with direction vector $\textbf{j}:x = 1,z = 1;$ 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\. $3x - 2y + 4z = 0;$ b. $3x - 2y + 4z = 0$
269.
a\. $\left( {x - 1} \right) + 2\left( {y - 2} \right) + 3\left( {z - 3} \right) = 0;$ b. $x + 2y + 3z - 14 = 0$
271.
a\. $\textbf{n} = 4\mathbf{\text{i}} + 5\textbf{j} + 10\textbf{k};$ b. $\left( {5,0,0} \right),$ $\left( {0,4,0} \right),$ and $\left( {0,0,2} \right);$
c.
273.
a\. $\mathbf{\text{n}} = 3\mathbf{\text{i}} - 2\mathbf{\text{j}} + 4\mathbf{\text{k}};$ b. $\left( {0,0,0} \right);$
c.
275.
$\left( {3,0,0} \right)$
277.
$x = -2 + 2t,y = 1 - 3t,z = 3 + t,$ $t \in \mathbb{R}$
281.
a\. $-2y + 3z - 1 = 0;$ b. $\left\langle {0,-2,3} \right\rangle \cdot \left\langle {x - 1,y - 1,z - 1} \right\rangle = 0;$ c. $x = 0,y = -2t,z = 3t,$ $t \in \mathbb{R}$
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.
$2x - 5y - 3z + 15 = 0$
287.
The line intersects the plane at point $P\left( {-3,4,0} \right).$
289.
$\frac{16}{\sqrt{14}}$
291.
a\. The planes are neither parallel nor orthogonal; b. $62\text{°}$
293.
a\. The planes are parallel.
295.
$\frac{1}{\sqrt{6}}$
297.
a\. $\frac{18}{\sqrt{29}};$ b. $P\left( {- \frac{51}{29},\frac{130}{29},\frac{62}{29}} \right)$
299.
$4x - 3y = 0$
301.
a\. $\mathbf{\text{v}}(1) = \left\langle {\text{cos}\ 1,\text{−}\text{sin}\ 1,2} \right\rangle;$ b. $\left( {\text{cos}\ 1} \right)(x - \text{sin}\ 1) - \left( {\text{sin}\ 1} \right)(y - \text{cos}\ 1) + 2(z - 2) = 0;$
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.
$- \frac{x^{2}}{9} + \frac{y^{2}}{\frac{1}{4}} + \frac{z^{2}}{\frac{1}{4}} = 1,$ 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 $y = - \frac{x^{2}}{4},$
333.
Ellipse $\frac{y^{2}}{4} + \frac{z^{2}}{100} = 1,$
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\. $\frac{\left( {x + 3} \right)^{2}}{16} + \frac{\left( {z - 2} \right)^{2}}{8} = 1;$ b. Cylinder centered at $\left( {-3,2} \right)$ with rulings parallel to the *y*-axis
341.
a\. $\frac{\left( {x - 3} \right)^{2}}{4} + \left( {y - 2} \right)^{2} - \left( {z + 2} \right)^{2} = 1;$ b. Hyperboloid of one sheet centered at $\left( {3,2,-2} \right),$ with the *z*-axis as its axis of symmetry
343.
a\. $\left( {x + 3} \right)^{2} + \frac{y^{2}}{4} - \frac{z^{2}}{3} = 0;$ b. Elliptic cone centered at $\left( {-3,0,0} \right),$ with the *z*-axis as its axis of symmetry
345.
$\frac{x^{2}}{4} + \frac{y^{2}}{16} + z^{2} = 1$
347.
$\left( {1,-1,0} \right)$ and $\left( {\frac{13}{3},4,\frac{5}{3}} \right)$
349.
$x^{2} + z^{2} + 4y = 0,$ elliptic paraboloid
351.
$\left( {0,0,100} \right)$
355.
a\. $x = 2 - \frac{z^{2}}{2},y = \pm \frac{z}{2}\sqrt{4 - z^{2}},$ where $z \in \left\lbrack {-2,2} \right\rbrack;$
b.
357.
two ellipses of equations $\frac{x^{2}}{2} + \frac{y^{2}}{\frac{9}{2}} = 1$ in planes $z = \text{±}2\sqrt{2}$
359.
a\. $\frac{x^{2}}{3963^{2}} + \frac{y^{2}}{3963^{2}} + \frac{z^{2}}{3950^{2}} = 1;$
b.
;
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 $\frac{2y^{2}}{3963^{2}} + \frac{z^{2}}{3950^{2}} = 1.$
361.
a.
b. The intersection curve is $\left( {x^{2} + z^{2} - 1} \right)^{3} - x^{2}z^{3} = 0.$
Section 2.7 Exercises
363.
$\left( {2\sqrt{3},2,3} \right)$
365.
$\left( {-2\sqrt{3},-2,3} \right)$
367.
$\left( {2,\frac{\pi}{3},2} \right)$
369.
$\left( {3\sqrt{2}, - \frac{\pi}{4},7} \right)$
371.
A cylinder of equation $x^{2} + y^{2} = 16,$ with its center at the origin and rulings parallel to the *z*-axis,
373.
Hyperboloid of two sheets of equation $\text{−}x^{2} + y^{2} - z^{2} = 1,$ with the *y*-axis as the axis of symmetry,
375.
Cylinder of equation $x^{2} - 2x + y^{2} = 0,$ with a center at $\left( {1,0,0} \right)$ and radius $1,$ with rulings parallel to the *z*-axis,
377.
Plane of equation $x = 2,$
379.
$z = 3$
381.
$r^{2} + z^{2} = 9$
383.
$r = 16\ \text{cos}\ \theta,r = 0$
385.
$\left( {0,0,-3} \right)$
387.
$\left( {6,-6,6\sqrt{2}} \right)$
389.
$\left( {4,0,90\text{°}} \right)$
391.
$\left( {3,90\text{°},90\text{°}} \right)$
393.
Sphere of equation $x^{2} + y^{2} + z^{2} = 9$ centered at the origin with radius $3,$
395.
Sphere of equation $x^{2} + y^{2} + \left( {z - 1} \right)^{2} = 1$ centered at $\left( {0,0,1} \right)$ with radius $1,$
397.
The *xy*-plane of equation $z = 0,$
399.
$\varphi = \frac{\pi}{3}$ or $\varphi = \frac{2\pi}{3};$ Elliptic cone
401.
$\rho\ \text{cos}\ \varphi = 6;$ Plane at $z = 6$
403.
$\left( {\sqrt{10},\frac{\pi}{4},0.3218} \right)$
405.
$\left( {3\sqrt{2},\frac{\pi}{2},\frac{\pi}{4}} \right)$
407.
$\left( {2, - \frac{\pi}{4},0} \right)$
409.
$\left( {8,\frac{\pi}{3},0} \right)$
411.
Cartesian system, $\left\{ \left( {x,y,z} \right) \middle| 0 \leq x \leq a,0 \leq y \leq a,0 \leq z \leq a \right\}$
413.
Cylindrical system, $$\left\{ \begin{array}{l}
{(r,\theta,z)|r^{2} + z^{2} \leq 9,r \geq 0,\frac{\pi}{2} \leq \theta \leq \frac{3\pi}{2}} \\
{,(r \geq 3\cos\theta, - \frac{\pi}{2} \leq \theta \leq \frac{\pi}{2})}
\end{array} \right\}$$
415.
The region is described by the set of points $\left\{ \left( {r,\theta,z} \right) \middle| 0 \leq r \leq 1,0 \leq \theta \leq 2\pi,r^{2} \leq z \leq r \right\}.$
417.
$\left( {4000,\text{−}77\text{°},51\text{°}} \right)$
419.
$43.17\text{°}\text{W},$ $22.91\text{°}\text{S}$
421.
a\. $\rho^{2} = 0,$ $\rho + R^{2} - r^{2} - 2R\ \text{sin}\ \varphi = 0;$
c.
Review Exercises
423.
True
425.
False
427.
a\. $\left\langle {24,-5} \right\rangle;$ b. $\sqrt{85};$ c. Can’t cross a vector with a scalar; d. $-29$
429.
$a = \text{±}2$
431.
$\left\langle {\frac{1}{\sqrt{14}}, - \frac{2}{\sqrt{14}}, - \frac{3}{\sqrt{14}}} \right\rangle$
433.
$27$
435.
$x = 1 - 3t,y = 3 + 3t,z = 5 - 8t,\mathbf{\text{r}}(t) = \left( {1 - 3t} \right)\mathbf{\text{i}} + 3\left( {1 + t} \right)\mathbf{\text{j}} + \left( {5 - 8t} \right)\mathbf{\text{k}}$
437.
$\text{−}x + 3y + 8z = 43$
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.
441.
Cylindrical: $z = r^{2} - 1,$ spherical: $\text{cos}\ \varphi = \rho\ \text{sin}^{2}\varphi - \frac{1}{\rho}$
443.
$x^{2} - 2x + y^{2} + z^{2} = 1,$ sphere
445.
331 N, and 244 N
447.
$15\ \text{J}$
449.
More, $59.09$ J
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