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  1. Preface
  2. Unit 1. Mechanics
    1. 1 Units and Measurement
      1. Introduction
      2. 1.1 The Scope and Scale of Physics
      3. 1.2 Units and Standards
      4. 1.3 Unit Conversion
      5. 1.4 Dimensional Analysis
      6. 1.5 Estimates and Fermi Calculations
      7. 1.6 Significant Figures
      8. 1.7 Solving Problems in Physics
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 2 Vectors
      1. Introduction
      2. 2.1 Scalars and Vectors
      3. 2.2 Coordinate Systems and Components of a Vector
      4. 2.3 Algebra of Vectors
      5. 2.4 Products of Vectors
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 3 Motion Along a Straight Line
      1. Introduction
      2. 3.1 Position, Displacement, and Average Velocity
      3. 3.2 Instantaneous Velocity and Speed
      4. 3.3 Average and Instantaneous Acceleration
      5. 3.4 Motion with Constant Acceleration
      6. 3.5 Free Fall
      7. 3.6 Finding Velocity and Displacement from Acceleration
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 4 Motion in Two and Three Dimensions
      1. Introduction
      2. 4.1 Displacement and Velocity Vectors
      3. 4.2 Acceleration Vector
      4. 4.3 Projectile Motion
      5. 4.4 Uniform Circular Motion
      6. 4.5 Relative Motion in One and Two Dimensions
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    5. 5 Newton's Laws of Motion
      1. Introduction
      2. 5.1 Forces
      3. 5.2 Newton's First Law
      4. 5.3 Newton's Second Law
      5. 5.4 Mass and Weight
      6. 5.5 Newton’s Third Law
      7. 5.6 Common Forces
      8. 5.7 Drawing Free-Body Diagrams
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    6. 6 Applications of Newton's Laws
      1. Introduction
      2. 6.1 Solving Problems with Newton’s Laws
      3. 6.2 Friction
      4. 6.3 Centripetal Force
      5. 6.4 Drag Force and Terminal Speed
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    7. 7 Work and Kinetic Energy
      1. Introduction
      2. 7.1 Work
      3. 7.2 Kinetic Energy
      4. 7.3 Work-Energy Theorem
      5. 7.4 Power
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    8. 8 Potential Energy and Conservation of Energy
      1. Introduction
      2. 8.1 Potential Energy of a System
      3. 8.2 Conservative and Non-Conservative Forces
      4. 8.3 Conservation of Energy
      5. 8.4 Potential Energy Diagrams and Stability
      6. 8.5 Sources of Energy
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    9. 9 Linear Momentum and Collisions
      1. Introduction
      2. 9.1 Linear Momentum
      3. 9.2 Impulse and Collisions
      4. 9.3 Conservation of Linear Momentum
      5. 9.4 Types of Collisions
      6. 9.5 Collisions in Multiple Dimensions
      7. 9.6 Center of Mass
      8. 9.7 Rocket Propulsion
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    10. 10 Fixed-Axis Rotation
      1. Introduction
      2. 10.1 Rotational Variables
      3. 10.2 Rotation with Constant Angular Acceleration
      4. 10.3 Relating Angular and Translational Quantities
      5. 10.4 Moment of Inertia and Rotational Kinetic Energy
      6. 10.5 Calculating Moments of Inertia
      7. 10.6 Torque
      8. 10.7 Newton’s Second Law for Rotation
      9. 10.8 Work and Power for Rotational Motion
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    11. 11 Angular Momentum
      1. Introduction
      2. 11.1 Rolling Motion
      3. 11.2 Angular Momentum
      4. 11.3 Conservation of Angular Momentum
      5. 11.4 Precession of a Gyroscope
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    12. 12 Static Equilibrium and Elasticity
      1. Introduction
      2. 12.1 Conditions for Static Equilibrium
      3. 12.2 Examples of Static Equilibrium
      4. 12.3 Stress, Strain, and Elastic Modulus
      5. 12.4 Elasticity and Plasticity
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    13. 13 Gravitation
      1. Introduction
      2. 13.1 Newton's Law of Universal Gravitation
      3. 13.2 Gravitation Near Earth's Surface
      4. 13.3 Gravitational Potential Energy and Total Energy
      5. 13.4 Satellite Orbits and Energy
      6. 13.5 Kepler's Laws of Planetary Motion
      7. 13.6 Tidal Forces
      8. 13.7 Einstein's Theory of Gravity
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    14. 14 Fluid Mechanics
      1. Introduction
      2. 14.1 Fluids, Density, and Pressure
      3. 14.2 Measuring Pressure
      4. 14.3 Pascal's Principle and Hydraulics
      5. 14.4 Archimedes’ Principle and Buoyancy
      6. 14.5 Fluid Dynamics
      7. 14.6 Bernoulli’s Equation
      8. 14.7 Viscosity and Turbulence
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  3. Unit 2. Waves and Acoustics
    1. 15 Oscillations
      1. Introduction
      2. 15.1 Simple Harmonic Motion
      3. 15.2 Energy in Simple Harmonic Motion
      4. 15.3 Comparing Simple Harmonic Motion and Circular Motion
      5. 15.4 Pendulums
      6. 15.5 Damped Oscillations
      7. 15.6 Forced Oscillations
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 16 Waves
      1. Introduction
      2. 16.1 Traveling Waves
      3. 16.2 Mathematics of Waves
      4. 16.3 Wave Speed on a Stretched String
      5. 16.4 Energy and Power of a Wave
      6. 16.5 Interference of Waves
      7. 16.6 Standing Waves and Resonance
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 17 Sound
      1. Introduction
      2. 17.1 Sound Waves
      3. 17.2 Speed of Sound
      4. 17.3 Sound Intensity
      5. 17.4 Normal Modes of a Standing Sound Wave
      6. 17.5 Sources of Musical Sound
      7. 17.6 Beats
      8. 17.7 The Doppler Effect
      9. 17.8 Shock Waves
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  4. A | Units
  5. B | Conversion Factors
  6. C | Fundamental Constants
  7. D | Astronomical Data
  8. E | Mathematical Formulas
  9. F | Chemistry
  10. G | The Greek Alphabet
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
    14. Chapter 14
    15. Chapter 15
    16. Chapter 16
    17. Chapter 17
  12. Index

Problems

2.1 Scalars and Vectors

25.

A scuba diver makes a slow descent into the depths of the ocean. His vertical position with respect to a boat on the surface changes several times. He makes the first stop 9.0 m from the boat but has a problem with equalizing the pressure, so he ascends 3.0 m and then continues descending for another 12.0 m to the second stop. From there, he ascends 4 m and then descends for 18.0 m, ascends again for 7 m and descends again for 24.0 m, where he makes a stop, waiting for his buddy. Assuming the positive direction up to the surface, express his net vertical displacement vector in terms of the unit vector. What is his distance to the boat?

26.

In a tug-of-war game on one campus, 15 students pull on a rope at both ends in an effort to displace the central knot to one side or the other. Two students pull with force 196 N each to the right, four students pull with force 98 N each to the left, five students pull with force 62 N each to the left, three students pull with force 150 N each to the right, and one student pulls with force 250 N to the left. Assuming the positive direction to the right, express the net pull on the knot in terms of the unit vector. How big is the net pull on the knot? In what direction?

27.

Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point and what is the compass direction of a line connecting your starting point to your final position? Use a graphical method.

28.

For the vectors given in the following figure, use a graphical method to find the following resultants: (a) A+BA+B, (b) C+BC+B, (c) D+FD+F, (d) ABAB, (e) DFDF, (f) A+2FA+2F, (g)C2D+3FC2D+3F; and (h) A4D+2FA4D+2F.

The x y coordinate system is shown, with positive x to the right and positive y up. Vector A has magnitude 10.0 and makes an angle of 30 degrees above the positive x direction. Vector B has magnitude 5.0 and makes an angle of 53 degrees above the positive x direction. Vector C has magnitude 12.0 and makes an angle of 60 degrees below the positive x direction. Vector D has magnitude 20.0 and makes an angle of 37 degrees above the negative x direction. Vector F has magnitude 20.0 and makes an angle of 30 degrees below the negative x direction.
29.

A delivery man starts at the post office, drives 40 km north, then 20 km west, then 60 km northeast, and finally 50 km north to stop for lunch. Use a graphical method to find his net displacement vector.

30.

An adventurous dog strays from home, runs three blocks east, two blocks north, one block east, one block north, and two blocks west. Assuming that each block is about 100 m, how far from home and in what direction is the dog? Use a graphical method.

31.

In an attempt to escape a desert island, a castaway builds a raft and sets out to sea. The wind shifts a great deal during the day and he is blown along the following directions: 2.50 km and 45.0°45.0° north of west, then 4.70 km and 60.0°60.0° south of east, then 1.30 km and 25.0°25.0° south of west, then 5.10 km straight east, then 1.70 km and 5.00°5.00° east of north, then 7.20 km and 55.0°55.0° south of west, and finally 2.80 km and 10.0°10.0° north of east. Use a graphical method to find the castaway’s final position relative to the island.

32.

A small plane flies 40.0 km in a direction 60°60° north of east and then flies 30.0 km in a direction 15°15° north of east. Use a graphical method to find the total distance the plane covers from the starting point and the direction of the path to the final position.

33.

A trapper walks a 5.0-km straight-line distance from his cabin to the lake, as shown in the following figure. Use a graphical method (the parallelogram rule) to determine the trapper’s displacement directly to the east and displacement directly to the north that sum up to his resultant displacement vector. If the trapper walked only in directions east and north, zigzagging his way to the lake, how many kilometers would he have to walk to get to the lake?

North is up, east is to the right. A house and lake are shown. The x y coordiante system is also shown, with the origin near the house, the positive x direction to the right nad the positive y direction up. The vector from the house to the lake is shown as a straight red arrow, labeled as vector S, magnitude S=5.0 kilometers, and at an angle of 40 degrees above the posiitve x direction. Two meandering paths, path 1 and path 2, from the house to the lake are shown as dashed line.
34.

A surveyor measures the distance across a river that flows straight north by the following method. Starting directly across from a tree on the opposite bank, the surveyor walks 100 m along the river to establish a baseline. She then sights across to the tree and reads that the angle from the baseline to the tree is 35°35°. How wide is the river?

35.

A pedestrian walks 6.0 km east and then 13.0 km north. Use a graphical method to find the pedestrian’s resultant displacement and geographic direction.

36.

The magnitudes of two displacement vectors are A = 20 m and B = 6 m. What are the largest and the smallest values of the magnitude of the resultant R=A+B?R=A+B?

2.2 Coordinate Systems and Components of a Vector

37.

Assuming the +x-axis is horizontal and points to the right, resolve the vectors given in the following figure to their scalar components and express them in vector component form.

38.

Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point? What is your displacement vector? What is the direction of your displacement? Assume the +x-axis is to the east.

39.

You drive 7.50 km in a straight line in a direction 15°15° east of north. (a) Find the distances you would have to drive straight east and then straight north to arrive at the same point. (b) Show that you still arrive at the same point if the east and north legs are reversed in order. Assume the +x-axis is to the east.

40.

A sledge is being pulled by two horses on a flat terrain. The net force on the sledge can be expressed in the Cartesian coordinate system as vector F=(−2980.0i^+8200.0j^)NF=(−2980.0i^+8200.0j^)N, where i^i^ and j^j^ denote directions to the east and north, respectively. Find the magnitude and direction of the pull.

41.

A trapper walks a 5.0-km straight-line distance from her cabin to the lake, as shown in the following figure. Determine the east and north components of her displacement vector. How many more kilometers would she have to walk if she walked along the component displacements? What is her displacement vector?

The vector from the cabin to the lake is vector S, magnitude 5.0 kilometers and pointing 40 degrees north of east. Two additional meandering paths are shown and labeled path 1 and path 2.
42.

The polar coordinates of a point are 4π/34π/3 and 5.50 m. What are its Cartesian coordinates?

43.

Two points in a plane have polar coordinates P1(2.500m,π/6)P1(2.500m,π/6) and P2(3.800m,2π/3)P2(3.800m,2π/3). Determine their Cartesian coordinates and the distance between them in the Cartesian coordinate system. Round the distance to a nearest centimeter.

44.

A chameleon is resting quietly on a lanai screen, waiting for an insect to come by. Assume the origin of a Cartesian coordinate system at the lower left-hand corner of the screen and the horizontal direction to the right as the +x-direction. If its coordinates are (2.000 m, 1.000 m), (a) how far is it from the corner of the screen? (b) What is its location in polar coordinates?

45.

Two points in the Cartesian plane are A(2.00 m, −4.00 m) and B(−3.00 m, 3.00 m). Find the distance between them and their polar coordinates.

46.

A fly enters through an open window and zooms around the room. In a Cartesian coordinate system with three axes along three edges of the room, the fly changes its position from point b(4.0 m, 1.5 m, 2.5 m) to point e(1.0 m, 4.5 m, 0.5 m). Find the scalar components of the fly’s displacement vector and express its displacement vector in vector component form. What is its magnitude?

2.3 Algebra of Vectors

47.

For vectors B=i^4j^B=i^4j^ and A=−3i^2j^A=−3i^2j^, calculate (a) A+BA+B and its magnitude and direction angle, and (b) ABAB and its magnitude and direction angle.

48.

A particle undergoes three consecutive displacements given by vectors D1=(3.0i^4.0j^2.0k^)mmD1=(3.0i^4.0j^2.0k^)mm, D2=(1.0i^7.0j^+4.0k^)mmD2=(1.0i^7.0j^+4.0k^)mm, and D3=(−7.0i^+4.0j^+1.0k^)mmD3=(−7.0i^+4.0j^+1.0k^)mm. (a) Find the resultant displacement vector of the particle. (b) What is the magnitude of the resultant displacement? (c) If all displacements were along one line, how far would the particle travel?

49.

Given two displacement vectors A=(3.00i^4.00j^+4.00k^)mA=(3.00i^4.00j^+4.00k^)m and B=(2.00i^+3.00j^7.00k^)mB=(2.00i^+3.00j^7.00k^)m, find the displacements and their magnitudes for (a) C=A+BC=A+B and (b) D=2ABD=2AB.

50.

A small plane flies 40.0km40.0km in a direction 60°60° north of east and then flies 30.0km30.0km in a direction 15°15° north of east. Use the analytical method to find the total distance the plane covers from the starting point, and the geographic direction of its displacement vector. What is its displacement vector?

51.

In an attempt to escape a desert island, a castaway builds a raft and sets out to sea. The wind shifts a great deal during the day, and she is blown along the following straight lines: 2.50 km and 45.0°45.0° north of west, then 4.70 km and 60.0°60.0° south of east, then 1.30 km and 25.0°25.0° south of west, then 5.10 km due east, then 1.70 km and 5.00°5.00° east of north, then 7.20 km and 55.0°55.0° south of west, and finally 2.80 km and 10.0°10.0° north of east. Use the analytical method to find the resultant vector of all her displacement vectors. What is its magnitude and direction?

52.

Assuming the +x-axis is horizontal to the right for the vectors given in the following figure, use the analytical method to find the following resultants: (a) A+B,A+B, (b) C+BC+B, (c) D+FD+F, (d) ABAB, (e) DFDF, (f) A+2FA+2F, (g) C2D+3FC2D+3F, and (h) A4D+2FA4D+2F.

The x y coordinate system has positive x to the right and positive y up. Vector A has magnitude 10.0 and points 30 degrees counterclockwise from the positive x direction. Vector B has magnitude 5.0 and points 53 degrees counterclockwise from the positive x direction. Vector C has magnitude 12.0 and points 60 degrees clockwise from the positive x direction. Vector D has magnitude 20.0 and points 37 degrees clockwise from the negative x direction. Vector F has magnitude 20.0 and points 30 degrees counterclockwise from the negative x direction.
Figure 2.33
53.

Given the vectors in the preceding figure, find vector RR that solves equations (a) D+R=FD+R=F and (b) C2D+5R=3FC2D+5R=3F. Assume the +x-axis is horizontal to the right.

54.

A delivery man starts at the post office, drives 40 km north, then 20 km west, then 60 km northeast, and finally 50 km north to stop for lunch. Use the analytical method to determine the following: (a) Find his net displacement vector. (b) How far is the restaurant from the post office? (c) If he returns directly from the restaurant to the post office, what is his displacement vector on the return trip? (d) What is his compass heading on the return trip? Assume the +x-axis is to the east.

55.

An adventurous dog strays from home, runs three blocks east, two blocks north, and one block east, one block north, and two blocks west. Assuming that each block is about a 100 yd, use the analytical method to find the dog’s net displacement vector, its magnitude, and its direction. Assume the +x-axis is to the east. How would your answer be affected if each block was about 100 m?

56.

If D=(6.00i^8.00j^)mD=(6.00i^8.00j^)m, B=(−8.00i^+3.00j^)mB=(−8.00i^+3.00j^)m, and A=(26.0i^+19.0j^)mA=(26.0i^+19.0j^)m, find the unknown constants a and b such that aD+bB+A=0aD+bB+A=0.

57.

Given the displacement vector D=(3i^4j^)m,D=(3i^4j^)m, find the displacement vector RR so that D+R=−4Dj^D+R=−4Dj^.

58.

Find the unit vector of direction for the following vector quantities: (a) Force F=(3.0i^2.0j^)NF=(3.0i^2.0j^)N, (b) displacement D=(−3.0i^4.0j^)mD=(−3.0i^4.0j^)m, and (c) velocity v=(−5.00i^+4.00j^)m/sv=(−5.00i^+4.00j^)m/s.

59.

At one point in space, the direction of the electric field vector is given in the Cartesian system by the unit vector E^=1/5i^2/5j^E^=1/5i^2/5j^. If the magnitude of the electric field vector is E = 400.0 V/m, what are the scalar components ExEx, EyEy, and EzEz of the electric field vector EE at this point? What is the direction angle θEθE of the electric field vector at this point?

60.

A barge is pulled by the two tugboats shown in the following figure. One tugboat pulls on the barge with a force of magnitude 4000 units of force at 15°15° above the line AB (see the figure and the other tugboat pulls on the barge with a force of magnitude 5000 units of force at 12°12° below the line AB. Resolve the pulling forces to their scalar components and find the components of the resultant force pulling on the barge. What is the magnitude of the resultant pull? What is its direction relative to the line AB?

The situation in the problem is illustrated as viewed from above. Line A B is vertical on the page, with A at the top and B at the bottom. Two tugboats above the barge are pulling it. The one on the right with 5000 units at an angle of 12 degrees counterclockwise from the line A B and the one on the right with 4000 units at an angle of 15 degrees.
Figure 2.34
61.

In the control tower at a regional airport, an air traffic controller monitors two aircraft as their positions change with respect to the control tower. One plane is a cargo carrier Boeing 747 and the other plane is a Douglas DC-3. The Boeing is at an altitude of 2500 m, climbing at 10°10° above the horizontal, and moving 30°30° north of west. The DC-3 is at an altitude of 3000 m, climbing at 5°5° above the horizontal, and cruising directly west. (a) Find the position vectors of the planes relative to the control tower. (b) What is the distance between the planes at the moment the air traffic controller makes a note about their positions?

2.4 Products of Vectors

62.

Assuming the +x-axis is horizontal to the right for the vectors in the following figure, find the following scalar products: (a) A·CA·C, (b) A·FA·F, (c) D·CD·C, (d) A·(F+2C)A·(F+2C), (e) i^·Bi^·B, (f) j^·Bj^·B, (g) (3i^j^)·B(3i^j^)·B, and (h) B^·BB^·B.

The x y coordinate system has positive x to the right and positive y up. Vector A has magnitude 10.0 and points 30 degrees counterclockwise from the positive x direction. Vector B has magnitude 5.0 and points 53 degrees counterclockwise from the positive x direction. Vector C has magnitude 12.0 and points 60 degrees clockwise from the positive x direction. Vector D has magnitude 20.0 and points 37 degrees clockwise from the negative x direction. Vector F has magnitude 20.0 and points 30 degrees counterclockwise from the negative x direction.
63.

Assuming the +x-axis is horizontal to the right for the vectors in the preceding figure, find (a) the component of vector AA along vector CC, (b) the component of vector CC along vector AA, (c) the component of vector i^i^ along vector FF, and (d) the component of vector FF along vector i^i^.

64.

Find the angle between vectors for (a) D=(−3.0i^4.0j^)mD=(−3.0i^4.0j^)m and A=(−3.0i^+4.0j^)mA=(−3.0i^+4.0j^)m and (b) D=(2.0i^4.0j^+k^)mD=(2.0i^4.0j^+k^)m and B=(−2.0i^+3.0j^+2.0k^)mB=(−2.0i^+3.0j^+2.0k^)m.

65.

Find the angles that vector D=(2.0i^4.0j^+k^)mD=(2.0i^4.0j^+k^)m makes with the x-, y-, and z- axes.

66.

Show that the force vector D=(2.0i^4.0j^+k^)ND=(2.0i^4.0j^+k^)N is orthogonal to the force vector G=(3.0i^+4.0j^+10.0k^)NG=(3.0i^+4.0j^+10.0k^)N.

67.

Assuming the +x-axis is horizontal to the right for the vectors in the previous figure, find the following vector products: (a) A×CA×C, (b) A×FA×F, (c) D×CD×C, (d) A×(F+2C)A×(F+2C), (e) i^×Bi^×B, (f) j^×Bj^×B, (g) (3i^j^)×B(3i^j^)×B, and (h) B^×BB^×B.

68.

Find the cross product A×CA×C for (a) A=2.0i^4.0j^+k^A=2.0i^4.0j^+k^ and C=3.0i^+4.0j^+10.0k^C=3.0i^+4.0j^+10.0k^, (b) A=3.0i^+4.0j^+10.0k^A=3.0i^+4.0j^+10.0k^ and C=2.0i^4.0j^+k^C=2.0i^4.0j^+k^, (c) A=−3.0i^4.0j^A=−3.0i^4.0j^ and C=−3.0i^+4.0j^C=−3.0i^+4.0j^, and (d) C=−2.0i^+3.0j^+2.0k^C=−2.0i^+3.0j^+2.0k^ and A=−9.0j^A=−9.0j^.

69.

For the vectors in the earlier figure, find (a) (A×F)·D(A×F)·D, (b) (A×F)·(D×B)(A×F)·(D×B), and (c) (A·F)(D×B)(A·F)(D×B).

70.

(a) If A×F=B×FA×F=B×F, can we conclude A=BA=B? (b) If A·F=B·FA·F=B·F, can we conclude A=BA=B? (c) If FA=BFFA=BF, can we conclude A=BA=B? Why or why not?

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