Chapter 2

Check Your Understanding

a. not equal because they are orthogonal; b. not equal because they have different magnitudes; c. not equal because they have different magnitudes and directions; d. not equal because they are antiparallel; e. equal.

2.2

16 m; $\vec{D} = −16 m \hat{u}$

2.3

G = 28.2 cm, $θ_{G} = 291 °$

2.4

$\vec{D} = (−5.0 \hat{i} - 3.0 \hat{j}) cm$ ; the fly moved 5.0 cm to the left and 3.0 cm down from its landing site.

2.5

5.83 cm, $211 °$

2.6

$\vec{D} = (−20 m) \hat{i}$

2.7

35.2 m/s = 126.7 km/h

2.8

$\vec{G} = (10.25 \hat{i} - 26.22 \hat{j}) cm$

2.9

D = 55.7 N; direction $65.7 °$ north of east

2.10

$\hat{v} = 0.8 \hat{i} + 0.6 \hat{j}$ , $36.87 °$ north of east

2.11

$\vec{A} \cdot \vec{B} = −57.3$ , $\vec{F} \cdot \vec{C} = 27.8$

2.13

$131.9 °$

2.14

$W_{1} = 1.5 J$ , $W_{2} = 0.3 J$

2.15

$\vec{A} \times \vec{B} = −40.1 \hat{k}$ or, equivalently, $| \vec{A} \times \vec{B} | = 40.1$ , and the direction is into the page; $\vec{C} \times \vec{F} = + 157.6 \hat{k}$ or, equivalently, $| \vec{C} \times \vec{F} | = 157.6$ , and the direction is out of the page.

2.16

a. $−2 \hat{k}$ , b. 2, c. $153.4 °$ , d. $135 °$

Conceptual Questions

1.

scalar

3.

answers may vary

5.

parallel, sum of magnitudes, antiparallel, zero

7.

no, yes

9.

zero, yes

11.

no

13.

equal, equal, the same

15.

a unit vector of the x-axis

17.

They are equal.

19.

yes

21.

a. $C = \vec{A} \cdot \vec{B}$ , b. $\vec{C} = \vec{A} \times \vec{B}$ or $\vec{C} = \vec{A} - \vec{B}$ , c. $\vec{C} = \vec{A} \times \vec{B}$ , d. $\vec{C} = A \vec{B}$ , e. $\vec{C} + 2 \vec{A} = \vec{B}$ , f. $\vec{C} = \vec{A} \times \vec{B}$ , g. left side is a scalar and right side is a vector, h. $\vec{C} = 2 \vec{A} \times \vec{B}$ , i. $\vec{C} = \vec{A} / B$ , j. $\vec{C} = \vec{A} / B$

23.

They are orthogonal.

Problems

25.

$\vec{h} = −49 m \hat{u}$ , 49 m

27.

30.8 m, $35.7 °$ west of north (equivalently, $54.2 °$ north of west or $125.7 °$ from east)

29.

134 km, $80 °$

31.

7.34 km, $63.5 °$ south of east

33.

3.8 km east, 3.2 km north, 7.0 km

35.

14.3 km, $65 °$

37.

a. $\vec{A} = + 8.66 \hat{i} + 5.00 \hat{j}$ , b. $\vec{B} = + 3.01 \hat{i} + 3.99 \hat{j}$ , c. $\vec{C} = + 6.00 \hat{i} - 10.39 \hat{j}$ , d. $\vec{D} = −15.97 \hat{i} + 12.04 \hat{j}$ , f. $\vec{F} = −17.32 \hat{i} - 10.00 \hat{j}$

The figure shows an x-y axis in the upper left corner followed by 5 unique vectors drawn on the x-y plane. Vector A has a magnitude of 10.0 and points 30 degrees above the x-axis. Vector B has a magnitude of 5.0 and points 53 degrees above the x-axis. Vector C has a magnitude of 12.0 and points 60 degrees below the x-axis. Vector D has a magnitude of 20.0 and points 37 degrees above the negative x-axis. Vector F has a magnitude of 20.0 and points 30 degrees below the negative x-axis.

39.

a. 1.94 km, 7.24 km; b. proof

41.

3.8 km east, 3.2 km north, 2.0 km, $\vec{D} = (3.8 \hat{i} + 3.2 \hat{j}) km$

43.

$P_{1} (2.165 m, 1.250 m)$ , $P_{2} (−1.900 m, 3.290 m)$ , 4.55 m

45.

8.60 m, $A (2 \sqrt{5} m, 0.647 π)$ , $B (3 \sqrt{2} m, 0.75 π)$

47.

a. $\vec{A} + \vec{B} = −4 \hat{i} - 6 \hat{j}$ , $| \vec{A} + \vec{B} | = 7.211, θ = 236 °$ ; b. $\vec{A} - \vec{B} = –2 \hat{i} + 2 \hat{j}$ , $| \vec{A} - \vec{B} | = 2 \sqrt{2}, θ = 135 °$

49.

a. $\vec{C} = (5.0 \hat{i} - 1.0 \hat{j} - 3.0 \hat{k}) m, C = 5.92 m$ ;
b. $\vec{D} = (4.0 \hat{i} - 11.0 \hat{j} + 15.0 \hat{k}) m, D = 19.03 m$

51.

$\vec{D} = (3.3 \hat{i} - 6.6 \hat{j}) km$ , $\hat{i}$ is to the east, 7.34 km, $−63.5 °$

53.

a. $\vec{R} = −1.35 \hat{i} - 22.04 \hat{j}$ , b. $\vec{R} = −17.98 \hat{i} + 0.89 \hat{j}$

55.

$\vec{D} = (200 \hat{i} + 300 \hat{j}) yd$ , D = 360.5 yd, $56.3 °$ north of east; The numerical answers would stay the same but the physical unit would be meters. The physical meaning and distances would be about the same because 1 yd is comparable with 1 m.

57.

$\vec{R} = (−3 \hat{i} - 16 \hat{j}) m$

59.

$\vec{E} = E \hat{E}$ , $E_{x} = + 178.9 V / m$ , $E_{y} = −357.8 V / m$ , $E_{z} = 0.0 V / m$ , $θ_{E} = − {tan}^{−1} (2)$

61.

a. $–34.290 {\vec{R}}_{B} = (–12.278 \hat{i} + 7.089 \hat{j} + 2.500 \hat{k}) km$ , ${\vec{R}}_{D} = (−34.290 \hat{i} + 3.000 \hat{k}) km$ ; b. $| {\vec{R}}_{B} - {\vec{R}}_{D} | = 23.131 km$

63.

a. 0, b. 0, c. –0.866, d. –17.32

65.

$θ_{i} = 64.12 °, θ_{j} = 150.79 °, θ_{k} = 77.39 °$

67.

a. $−120 \hat{k}$ , b. $0 \hat{k}$ , c. $94 \hat{k}$ , d. $–240 \hat{k}$ , e. $4.0 \hat{k}$ , f. $−3.0 \hat{k}$ , g. $15 \hat{k}$ , h. 0

69.

a. 0, b. 0, c. $20,000 \hat{k}$

Additional Problems

71.

a. 18.4 km and 26.2 km, b. 31.5 km and 5.56 km

73.

a. $(r, π - φ)$ , b. $(2 r, φ + 2 π)$ , c. $(3 r, − φ)$

75.

$d_{PM} = 6.2 nmi = 11.4 km, d_{NP} = 7.2 nmi = 13.3 km$

77.

proof

79.

a. 10.00 m, b. $5 π m$ , c. 0

81.

22.2 km/h, $35.8 °$ south of west

83.

270 m, $4.2 °$ north of west

85.

$\vec{B} = −4.0 \hat{i} + 3.0 \hat{j}$ or $\vec{B} = 4.0 \hat{i} - 3.0 \hat{j}$

87.

proof

Challenge Problems

89.

$G_{H} = 19 N / \sqrt{17} \approx 4.6 N$

91.

proof