Skip to ContentGo to accessibility pageKeyboard shortcuts menu
University Physics Volume 1

# Chapter 2

Menu

### Check Your Understanding

2.1

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; $D→=−16mu^D→=−16mu^$

2.3

G = 28.2 cm, $θG=291°θG=291°$

2.4

$D→=(−5.0i^−3.0j^)cmD→=(−5.0i^−3.0j^)cm$; the fly moved 5.0 cm to the left and 3.0 cm down from its landing site.

2.5

5.83 cm, $211°211°$

2.6

$D → = ( −20 m ) i ^ D → = ( −20 m ) i ^$

2.7

35.2 m/s = 126.4 km/h

2.8

$G → = ( 10.25 i ^ − 26.22 j ^ ) cm G → = ( 10.25 i ^ − 26.22 j ^ ) cm$

2.9

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

2.10

$v^=0.8i^+0.6j^v^=0.8i^+0.6j^$, $36.87°36.87°$ north of east

2.11

$A→·B→=−57.3A→·B→=−57.3$, $F→·C→=27.8F→·C→=27.8$

2.13

$131.9 ° 131.9 °$

2.14

$W1=1.5JW1=1.5J$, $W2=0.3JW2=0.3J$

2.15

$A→×B→=−40.1k^A→×B→=−40.1k^$ or, equivalently, $|A→×B→|=40.1|A→×B→|=40.1$, and the direction is into the page; $C→×F→=+157.6k^C→×F→=+157.6k^$ or, equivalently, $|C→×F→|=157.6|C→×F→|=157.6$, and the direction is out of the page.

2.16

a. $−2k^−2k^$, b. 2, c. $153.4°153.4°$, d. $135°135°$

### 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. $C=A→·B→C=A→·B→$, b. $C→=A→×B→C→=A→×B→$ or $C→=A→−B→C→=A→−B→$, c. $C→=A→×B→C→=A→×B→$, d. $C→=AB→C→=AB→$, e. $C→+2A→=B→C→+2A→=B→$, f. $C→=A→×B→C→=A→×B→$, g. left side is a scalar and right side is a vector, h. $C→=2A→×B→C→=2A→×B→$, i. $C→=A→/BC→=A→/B$, j. $C→=A→/BC→=A→/B$

23.

They are orthogonal.

### Problems

25.

$h→=−49mu^h→=−49mu^$, 49 m

27.

30.8 m, $35.7°35.7°$ west of north

29.

134 km, $80°80°$

31.

7.34 km, $63.5°63.5°$ south of east

33.

3.8 km east, 3.2 km north, 7.0 km

35.

14.3 km, $65°65°$

37.

a. $A→=+8.66i^+5.00j^A→=+8.66i^+5.00j^$, b. $B→=+3.01i^+3.99j^B→=+3.01i^+3.99j^$, c. $C→=+6.00i^−10.39j^C→=+6.00i^−10.39j^$, d. $D→=−15.97i^+12.04j^D→=−15.97i^+12.04j^$, f. $F→=−17.32i^−10.00j^F→=−17.32i^−10.00j^$

39.

a. 1.94 km, 7.24 km; b. proof

41.

3.8 km east, 3.2 km north, 2.0 km, $D→=(3.8i^+3.2j^)kmD→=(3.8i^+3.2j^)km$

43.

$P1(2.165m,1.250m)P1(2.165m,1.250m)$, $P2(−1.900m,3.290m)P2(−1.900m,3.290m)$, 6.09 m

45.

8.60 m, $A(25m,0.647π)A(25m,0.647π)$, $B(32m,0.75π)B(32m,0.75π)$

47.

a. $A→+B→=−4i^−6j^A→+B→=−4i^−6j^$, $|A→+B→|=7.211,θ=236°|A→+B→|=7.211,θ=236°$; b. $A→−B→=–2i^+2j^A→−B→=–2i^+2j^$, $|A→−B→|=22,θ=135°|A→−B→|=22,θ=135°$

49.

a. $C→=(5.0i^−1.0j^−3.0k^)m,C=5.92mC→=(5.0i^−1.0j^−3.0k^)m,C=5.92m$;
b. $D→=(4.0i^−11.0j^+15.0k^)m,D=19.03mD→=(4.0i^−11.0j^+15.0k^)m,D=19.03m$

51.

$D→=(3.3i^−6.6j^)kmD→=(3.3i^−6.6j^)km$, $i^i^$ is to the east, 7.34 km, $−63.5°−63.5°$

53.

a. $R→=−1.35i^−22.04j^R→=−1.35i^−22.04j^$, b. $R→=−17.98i^+0.89j^R→=−17.98i^+0.89j^$

55.

$D→=(200i^+300j^)ydD→=(200i^+300j^)yd$, D = 360.5 yd, $56.3°56.3°$ 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.

$R → = −3 i ^ − 16 j ^ R → = −3 i ^ − 16 j ^$

59.

$E→=EE^E→=EE^$, $Ex=+178.9V/mEx=+178.9V/m$, $Ey=−357.8V/mEy=−357.8V/m$, $Ez=0.0V/mEz=0.0V/m$, $θE=−tan−1(2)θE=−tan−1(2)$

61.

a. $–34.290 R→B=(–12.278i^+7.089j^+2.500k^)km–34.290R→B=(–12.278i^+7.089j^+2.500k^)km$, $R→D=(−34.290i^+3.000k^)kmR→D=(−34.290i^+3.000k^)km$; b. $|R→B−R→D|=23.131km|R→B−R→D|=23.131km$

63.

a. 0, b. 0, c. –0.866, d. –17.32

65.

$θ i = 64.12 ° , θ j = 150.79 ° , θ k = 77.39 ° θ i = 64.12 ° , θ j = 150.79 ° , θ k = 77.39 °$

67.

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

69.

a. 0, b. 0, c. $+–20,000k^+–20,000k^$

### Additional Problems

71.

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

73.

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

75.

$d PM = 6.2 nmi = 11.4 km , d NP = 7.2 nmi = 13.3 km d PM = 6.2 nmi = 11.4 km , d NP = 7.2 nmi = 13.3 km$

77.

proof

79.

a. 10.00 m, b. $5πm5πm$, c. 0

81.

22.2 km/h, $35.8°35.8°$ south of west

83.

270 m, $4.2°4.2°$ north of west

85.

$B→=−4.0i^+3.0j^B→=−4.0i^+3.0j^$ or $B→=4.0i^−3.0j^B→=4.0i^−3.0j^$

87.

proof

### Challenge Problems

89.

$G H = 19 N / 17 ≈ 4.6 N G H = 19 N / 17 ≈ 4.6 N$

91.

proof

Do you know how you learn best?
Kinetic by OpenStax offers access to innovative study tools designed to help you maximize your learning potential.
Order a print copy

As an Amazon Associate we earn from qualifying purchases.

Citation/Attribution

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Attribution information
• If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
Access for free at https://openstax.org/books/university-physics-volume-1/pages/1-introduction
• If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
Access for free at https://openstax.org/books/university-physics-volume-1/pages/1-introduction
Citation information

© Feb 10, 2022 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.