Vector \(\vec{A} = a \hat{\imath} - b \hat{\kmath}\) and vector \(\vec{B} = -c \hat{\jmath} + d \hat{\kmath}\). (a) In terms of the positive scalar quantities a, b, c, and d, what are \(\vec{A} \cdot \vec{B}\) and \(\vec{A} \times \vec{B}\)? (b) If c = 0, what is the magnitude of \(\vec{A} \cdot \vec{B}\) and what are the magnitude and direction of \(\vec{A} \times \vec{B}\) ? Does your result for the direction for \(\vec{A} \times \vec{B}\) agree with the result you get if you sketch \(\vec{A}\) and \(\vec{B}\) in the xz-plane and apply the right-hand rule? The scalar product can be described as the magnitude of \(\vec{B}\) times the component of \(\vec{A}\) that is parallel to \(\vec{B}\). Does this agree with your result? The magnitude of the vector product can be described as the magnitude of \(\vec{B}\) times the component of \(\vec{A}\) that is perpendicular to \(\vec{B}\). Does this agree with your result?

How many correct experiments do we need to disprove a theory? How many do we need to prove a theory? Explain.

Suppose you are asked to compute the tangent of 5.00 meters. Is this possible? Why or why not?

What is your height in centimeters? What is your weight in newtons?

The U.S. National Institute of Standards and Technology (NIST) maintains several accurate copies of the international standard kilogram. Even after careful cleaning, these national standard kilograms are gaining mass at an average rate of about \(1 \ \mu \mathrm{g} > \mathrm{y (y = year)}\) when compared every 10 years or so to the standard international kilogram. Does this apparent increase have any importance? Explain.

What physical phenomena (other than a pendulum or cesium clock) could you use to define a time standard?

Describe how you could measure the thickness of a sheet of paper with an ordinary ruler.

The quantity \(\pi = 3.14159 \ldots\) is a number with no dimensions, since it is a ratio of two lengths. Describe two or three other geometrical or physical quantities that are dimensionless.

What are the units of volume? Suppose another student tells you that a cylinder of radius r and height h has volume given by \(\pi r^3 h\). Explain why this cannot be right.

Three archers each fire four arrows at a target. Joe’s four arrows hit points that are spread around in a region that goes 10 cm above, 10 cm below, 10 cm to the left, and 10 cm to the right of the center of the target. All four of Moe’s arrows hit within 1 cm of a point 20 cm from the center, and Flo’s four arrows hit within 1 cm of the center. The contest judge says that one of the archers is precise but not accurate, another archer is accurate but not precise, and the third archer is both accurate and precise. Which description applies to which archer? Explain.

Is the vector \((\hat{\imath}+\hat{\jmath}+\hat{k})\) a unit vector? Is the vector \((3.0 \hat{\imath}-2.0 \hat{\jmath})\) a unit vector? Justify your answers.

A circular racetrack has a radius of 500 m. What is the displacement of a bicyclist when she travels around the track from the north side to the south side? When she makes one complete circle around the track? Explain.

Can you find two vectors with different lengths that have a vector sum of zero? What length restrictions are required for three vectors to have a vector sum of zero? Explain.

The “direction of time” is said to proceed from past to future. Does this mean that time is a vector quantity? Explain.

Air traffic controllers give instructions called “vectors” to tell airline pilots in which direction they are to fly. If these are the only instructions given, is the name “vector” used correctly? Why or why not?

Can you find a vector quantity that has a magnitude of zero but components that are not zero? Explain. Can the magnitude of a vector be less than the magnitude of any of its components? Explain.

(a) Does it make sense to say that a vector is negative? Why? (b) Does it make sense to say that one vector is the negative of another? Why? Does your answer here contradict what you said in part (a)?

If \(\vec{C}=\vec{A}+\vec{B}\), what must be true about the directions and magnitudes of \(\vec{A}\) and \(\vec{B}\) if C = A + B ? What must be true about the directions and magnitudes of \(\vec{A}\) and \(\vec{B}\) if C = 0?

If \(\vec{A}\) and \(\vec{B}\) are nonzero vectors, is it possible for \(both \ \vec{A} \cdot \vec{B}\) and \(\vec{A} \times \vec{B}\) to be zero? Explain.

What does \(\vec{A} \cdot \vec{A}\), the scalar product of a vector with itself, give? What about \(\vec{A} \times \vec{A}\), the vector product of a vector with itself?

Let \(\vec{A}\) represent any nonzero vector. Why is \(\vec{A}/A\) a unit vector, and what is its direction? If \(\theta\) is the angle that \(\vec{A}\) makes with the +x-axis, explain why \((\vec{A}/A) \cdot \hat{\imath}\) is called the direction cosine for that axis.

Figure 1.7 shows the result of an unacceptable error in the stopping position of a train. If a train travels 890 km from Berlin to Paris and then overshoots the end of the track by 10.0 m, what is the percent error in the total distance covered? Is it correct to write the total distance covered by the train as 890,010 m? Explain.

Which of the following are legitimate mathematical operations: (a) \(\vec{A} \cdot (\vec{B}- \vec{C})\); (b) \((\vec{A} - \vec{B}) \times \vec{C}\); (c) \(\vec{A} \cdot (\vec{B} \times \vec{C})\); (d) \(\vec{A} \times )\vec{B} \times \vec{C})\); (e) \(\vec{A} \times (\vec{B} \cdot \vec{C})\)? In each case, give the reason for your answer.

Consider the vector products \(\vec{A} \times (\vec{B} \times \vec{C})\) and \((\vec{A} \times \vec{B}) \times \vec{C}\). Give an example that illustrates the general rule that these two vector products do not have the same magnitude or direction. Can you choose vectors \(\vec{A}, \vec{B}\), and \(\vec{C}\) such that these two vector products are equal? If so, give an example.

Show that, no matter what \(\vec{A}\) and \(\vec{B}\) are, \(\vec{A} \cdot (\vec{A} \times \vec{B}) = 0\). (Hint: Do not look for an elaborate mathematical proof. Consider the definition of the direction of the cross product.)

(a) If \(\vec{A} \cdot \vec{B}= 0\), does it necessarily follow that A = 0 or B = 0? Explain. (b) If \(\vec{A} \times \vec{B} = \mathbf{0}\), does it necessarily follow that A = 0 or B = 0 ? Explain.

If \(\vec{A}= \mathbf{0}\) for a vector in the xy-plane, does it follow that \(A_x = -A_y\) ? What can you say about \(A_x\) and \(A_y\)?

White Dwarfs and Neutron Stars. Recall that density is mass divided by volume, and consult Appendix B as needed. (a) Calculate the average density of the earth in \(\mathrm{g/cm}^3\), assuming our planet is a perfect sphere. (b) In about 5 billion years, at the end of its lifetime, our sun will end up as a white dwarf that has about the same mass as it does now but is reduced to about 15,000 km in diameter. What will be its density at that stage? (c) A neutron star is the remnant of certain supernovae (explosions of giant stars). Typically, neutron stars are about 20 km in diameter and have about the same mass as our sun. What is a typical neutron star density in \(\mathrm{g/cm}^3\)?

The Hydrogen Maser. A maser is a laser-type device that produces electromagnetic waves with frequencies in the microwave and radio-wave bands of the electromagnetic spectrum. You can use the radio waves generated by a hydrogen maser as a standard of frequency. The frequency of these waves is 1,420,405,751.786 hertz. (A hertz is another name for one cycle per second.) A clock controlled by a hydrogen maser is off by only 1 s in 100,000 years. For the following questions, use only three significant figures. (The large number of significant figures given for the frequency simply illustrates the remarkable accuracy to which it has been measured.) (a) What is the time for one cycle of the radio wave? (b) How many cycles occur in 1 h? (c) How many cycles would have occurred during the age of the earth, which is estimated to be \(4.6 \times 10^9\) years? (d) By how many seconds would a hydrogen maser clock be off after a time interval equal to the age of the earth?

An Earthlike Planet. In January 2006 astronomers reported the discovery of a planet, comparable in size to the earth, orbiting another star and having a mass about 5.5 times the earth’s mass. It is believed to consist of a mixture of rock and ice, similar to Neptune. If this planet has the same density as Neptune \((1.76 \ \mathrm{g/cm}^3)\), what is its radius expressed (a) in kilometers and (b) as a multiple of earth’s radius? Consult Appendix F for astronomical data.

A rectangular piece of aluminum is \(7.60 \pm 0.01 \ \mathrm{cm long}\) and \(1.90 \pm 0.01 \ \mathrm{cm wide}\). (a) Find the area of the rectangle and the uncertainty in the area. (b) Verify that the fractional uncertainty in the area is equal to the sum of the fractional uncertainties in the length and in the width. (This is a general result.)

BIO Estimate the number of atoms in your body. (Hint: Based on what you know about biology and chemistry, what are the most common types of atom in your body? What is the mass of each type of atom? Appendix D gives the atomic masses of different elements, measured in atomic mass units; you can find the value of an atomic mass unit, or 1 u, in Appendix E.)

BIO Biological tissues are typically made up of 98% water. Given that the density of water is \(1.0 \times 10^3 \ \mathrm{kg/m}^3\), estimate the mass of (a) the heart of an adult human; (b) a cell with a diameter of \(0.5 \ \mu \mathrm{m}\); (c) a honeybee.

Vector \(\vec{A} = 3.0 \hat{\imath} - 4.0 \hat{\kmath}\). (a) Construct a unit vector that is parallel to \(\vec{A}\) . (b) Construct a unit vector that is antiparallel to \(\vec{A}\) . (c) Construct two unit vectors that are perpendicular to \(\vec{A}\) and that have no y-component.

Three horizontal ropes pull on a large stone stuck in the ground, producing the vector forces \(\vec{A}, \vec{B}\), and \(\vec{C}\) shown in Fig. P1.56. Find the magnitude and direction of a fourth force on the stone that will make the vector sum of the four forces zero.

As noted in Exercise 1.23, a spelunker is surveying a cave. She follows a passage 180 m straight west, then 210 m in a direction \(45^{\circ}\) east of south, and then 280 m at \(30^{\circ}\) east of north. After a fourth displacement, she finds herself back where she started. Use the method of components to determine the magnitude and direction of the fourth displacement. Draw the vector-addition diagram and show that it is in qualitative agreement with your numerical solution.

Emergency Landing. A plane leaves the airport in Galisteo and flies 170 km at \(68.0^{\circ}\) east of north; then it changes direction to fly 230 km at \(36.0^{\circ}\) south of east, after which it makes an immediate emergency landing in a pasture. When the airport sends out a rescue crew, in which direction and how far should this crew fly to go directly to this plane?

A charged object with electric charge q produces an electric field. The SI unit for electric field is N/C, where N is the SI unit for force and C is the SI unit for charge. If at point P there are electric fields from two or more charged objects, then the resultant field is the vector sum of the fields from each object. At point P the electric field \(\vec{E}_1\) from charge \(q_1\) is 450 N/C in the +y-direction, and the electric field \(\vec{E}_2\) from charge \(q_2\) is 600 N/C in the direction \(36.9^{\circ}\) from the -y-axis toward the -x-axis. What are the magnitude and direction of the resultant field \(\vec{E}= \vec{E}_1 + \vec{E}_2\) at point P due to these two charges?

A sailor in a small sailboat encounters shifting winds. She sails 2.00 km east, next 3.50 km southeast, and then an additional distance in an unknown direction. Her final position is 5.80 km directly east of the starting point (Fig. P1.60). Find the magnitude and direction of the third leg of the journey. Draw the vector-addition diagram and show that it is in qualitative agreement with your numerical solution.

BIO Dislocated Shoulder. A patient with a dislocated shoulder is put into a traction apparatus as shown in Fig. P1.61. The pulls \(\vec{A}\) and \(\vec{B}\) have equal magnitudes and must combine to produce an outward traction force of 12.8 N on the patient’s arm. How large should these pulls be?

On a training flight, a student pilot flies from Lincoln, Nebraska, to Clarinda, Iowa, next to St. Joseph, Missouri, and then to Manhattan, Kansas (Fig. P1.62). The directions are shown relative to north: \(0^{\circ}\) is north, \(90^{\circ}\) is east, \(180^{\circ}\) is south, and \(270^{\circ}\) is west. Use the method of components to find (a) the distance she has to fly from Manhattan to get back to Lincoln, and (b) the direction (relative to north) she must fly to get there. Illustrate your solutions with a vector diagram.

You leave the airport in College Station and fly 23.0 km in a direction \(34.0^{\circ}\) south of east. You then fly 46.0 km due north. How far and in what direction must you then fly to reach a private landing strip that is 32.0 km due west of the College Station airport?

Getting Back. An explorer in Antarctica leaves his shelter during a whiteout. He takes 40 steps northeast, next 80 steps at \(60^{\circ}\) north of west, and then 50 steps due south. Assume all of his steps are equal in length. (a) Sketch, roughly to scale, the three vectors and their resultant. (b) Save the explorer from becoming hopelessly lost by giving him the displacement, calculated by using the method of components, that will return him to his shelter.

As a test of orienteering skills, your physics class holds a contest in a large, open field. Each contestant is told to travel 20.8 m due north from the starting point, then 38.0 m due east, and finally 18.0 m in the direction \(33.0^{\circ}\) west of south. After the specified displacements, a contestant will find a silver dollar hidden under a rock. The winner is the person who takes the shortest time to reach the location of the silver dollar. Remembering what you learned in class, you run on a straight line from the starting point to the hidden coin. How far and in what direction do you run?

You are standing on a street corner with your friend. You then travel 14.0 m due west across the street and into your apartment building. You travel in the elevator 22.0 m upward to your floor, walk 12.0 m north to the door of your apartment, and then walk 6.0 m due east to your balcony that overlooks the street. Your friend is standing where you left her. Now how far are you from your friend?

You are lost at night in a large, open field. Your GPS tells you that you are 122.0 m from your truck, in a direction \(58.0^{\circ}\) east of south. You walk 72.0 m due west along a ditch. How much farther, and in what direction, must you walk to reach your truck?

You live in a town where the streets are straight but are in a variety of directions. On Saturday you go from your apartment to the grocery store by driving 0.60 km due north and then 1.40 km in the direction \(60.0^{\circ}\) west of north. On Sunday you again travel from your apartment to the same store but this time by driving 0.80 km in the direction \(50.0^{\circ}\) north of west and then in a straight line to the store. (a) How far is the store from your apartment? (b) On which day do you travel the greater distance, and how much farther do you travel? Or, do you travel the same distance on each route to the store?

While following a treasure map, you start at an old oak tree. You first walk 825 m directly south, then turn and walk 1.25 km at \(30.0^{\circ}\) west of north, and finally walk 1.00 km at \(32.0^{\circ}\) north of east, where you find the treasure: a biography of Isaac Newton! (a) To return to the old oak tree, in what direction should you head and how far will you walk? Use components to solve this problem. (b) To see whether your calculation in part (a) is reasonable, compare it with a graphical solution drawn roughly to scale.

A fence post is 52.0 m from where you are standing, in a direction \(37.0^{\circ}\) north of east. A second fence post is due south from you. How far are you from the second post if the distance between the two posts is 68.0 m?

A dog in an open field runs 12.0 m east and then 28.0 m in a direction \(50.0^{\circ}\) west of north. In what direction and how far must the dog then run to end up 10.0 m south of her original starting point?

Ricardo and Jane are standing under a tree in the middle of a pasture. An argument ensues, and they walk away in different directions. Ricardo walks 26.0 m in a direction \(60.0^{\circ}\) west of north. Jane walks 16.0 m in a direction \(30.0^{\circ}\) south of west. They then stop and turn to face each other. (a) What is the distance between them? (b) In what direction should Ricardo walk to go directly toward Jane?

You are camping with Joe and Karl. Since all three of you like your privacy, you don’t pitch your tents close together. Joe’s tent is 21.0 m from yours, in the direction \(23.0^{\circ}\) south of east. Karl’s tent is 32.0 m from yours, in the direction \(37.0^{\circ}\) north of east. What is the distance between Karl’s tent and Joe’s tent?

Bond Angle in Methane. In the methane molecule, \(\mathrm{CH}_4\), each hydrogen atom is at a corner of a regular tetrahedron with the carbon atom at the center. In coordinates for which one of the C-H bonds is in the direction of \(\hat{\imath} + \hat{\jmath} + \hat{\kmath}\), an adjacent C-H bond is in the \(\hat{\imath} - \hat{\jmath} - \hat{\kmath}\) direction. Calculate the angle between these two bonds.

The work W done by a constant force \(\vec{F}\) on an object that undergoes displacement \(\vec{s}\) from point 1 to point 2 is \(W = \vec{F} \cdot \vec{s}\) . For F in newtons (N) and s in meters (m), W is in joules (J). If, during a displacement of the object, \(\vec{F}\) has constant direction \(60.0^{\circ}\) above the -x-axis and constant magnitude 5.00 N and if the displacement is 0.800 m in the +x-direction, what is the work done by the force \(\vec{F}\)?

Magnetic fields are produced by moving charges and exert forces on moving charges. When a particle with charge q is moving with velocity \(\vec{v}\) in a magnetic field \(\vec{B}\) , the force \(\vec{F}\) that the field exerts on the particle is given by \(\vec{F}=q \vec{v} \times \vec{B}\). The SI units are as follows: For charge it is the coulomb (C), for magnetic field it is tesla (T), for force it is newton (N), and for velocity it is m/s. If \(q = -8.00 \times 10^{-6} \ \mathrm{C}, \vec{v}\) is \(3.00 \times 10^4 \ \mathrm{m/s}\) in the +x-direction, and \(\vec{B}\) is 5.00 T in the -y-direction, what are the magnitude and direction of the force that the magnetic field exerts on the charged particle?

Vectors \(\vec{A}\) and \(\vec{B}\) have scalar product -6.00, and their vector product has magnitude +9.00. What is the angle between these two vectors?

Torque is a vector quantity that specifies the effectiveness of a force in causing the rotation of an object. The torque that a force \(\vec{F}\) exerts on a rigid object depends on the point where the force acts and on the location of the axis of rotation. If \(\vec{r}\) is the length vector from the axis to the point of application of the force, then the torque is \(\vec{r} \times \vec{F}\). If \(\vec{F}\) is 22.0 N in the -y-direction and if \(\vec{r}\) is in the xy-plane at an angle of \(36^{\circ}\) from the +y-axis toward the -x-axis and has magnitude 4.0 m, what are the magnitude and direction of the torque exerted by \(\vec{F}\)?

Vector \(\vec{A} = a \hat{\imath} - b \hat{\kmath}\) and vector \(\vec{B} = -c \hat{\jmath} + d \hat{\kmath}\). (a) In terms of the positive scalar quantities a, b, c, and d, what are \(\vec{A} \cdot \vec{B}\) and \(\vec{A} \times \vec{B}\)? (b) If c = 0, what is the magnitude of \(\vec{A} \cdot \vec{B}\) and what are the magnitude and direction of \(\vec{A} \times \vec{B}\) ? Does your result for the direction for \(\vec{A} \times \vec{B}\) agree with the result you get if you sketch \(\vec{A}\) and \(\vec{B}\) in the xz-plane and apply the right-hand rule? The scalar product can be described as the magnitude of \(\vec{B}\) times the component of \(\vec{A}\) that is parallel to \(\vec{B}\). Does this agree with your result? The magnitude of the vector product can be described as the magnitude of \(\vec{B}\) times the component of \(\vec{A}\) that is perpendicular to \(\vec{B}\). Does this agree with your result?

Vectors \(\vec{A}\) and \(\vec{B}\) are in the xy-plane. Vector \(\vec{A}\) is in the +x-direction, and the direction of vector \(\vec{B}\) is at an angle \(\theta\) from the +x-axis measured toward the +y-axis. (a) If \(\theta\) is in the range \(0^{\circ} \leq \theta \leq 180^{\circ}\), for what two values of \(\theta\) does the scalar product \(\vec{A} \cdot \vec{B}\) have its maximum magnitude? For each of these values of \(\theta\), what is the magnitude of the vector product \(\vec{A} \times \vec{B}\)? (b) If \(\theta\) is in the range \(0^{\circ} \leq \theta \leq 180^{\circ}\), for what value of \(\theta\) does the vector product \(\vec{A} \times \vec{B}\) have its maximum value? For this value of \(\theta\), what is the magnitude of the scalar product \(\vec{A} \cdot \vec{B}\)? (c) What is the angle \(\theta\) in the range \(0^{\circ} \leq \theta \leq 180^{\circ}\) for which \(\vec{A} \cdot \vec{B}\) is twice \(|\vec{A} \times \vec{B}|\)?

Vector \(\vec{A}\) has magnitude 12.0 m, and vector \(\vec{B}\) has magnitude 16.0 m. The scalar product \(\vec{A} \cdot \vec{B}\) is \(112.0 \ \mathrm{m}^2\). What is the magnitude of the vector product between these two vectors?

Vector \(\vec{A}\) has magnitude 5.00 m and lies in the xy-plane in a direction \(53.0^{\circ}\) from the +x-axis axis measured toward the +y-axis. Vector \(\vec{B}\) has magnitude 8.00 m and a direction you can adjust. (a) You want the vector product \(\vec{A} \times \vec{B}\) to have a positive z-component of the largest possible magnitude. What direction should you select for vector \(\vec{B}\) ? (b) What is the direction of \(\vec{B}\) for which \(\vec{A} \times \vec{B}\) has the most negative z-component? (c) What are the two directions of \(\vec{B}\) for which \(\vec{A} \times \vec{B}\) is zero?

The scalar product of vectors \(\vec{A}\) and \(\vec{B}\) is \(+48.0 \ \mathrm{~m}^2\). Vector \(\vec{A}\) has magnitude 9.00 m and direction \(28.0^{\circ}\) west of south. If vector \(\vec{B}\) has direction \(39.0^{\circ}\) south of east, what is the magnitude of \(\vec{B}\) ?

Obtain a unit vector perpendicular to the two vectors given in Exercise 1.39.

You are given vectors \(\vec{A}=5.0 \hat{\imath}-6.5 \hat{\jmath}\) and \(\vec{B}=3.5 \hat{\imath}-7.0 \hat{\jmath}\). A third vector, \(\vec{C}\), lies in the xy-plane. Vector \(\vec{C}\) is perpendicular to vector \(\vec{A}\), and the scalar product of \(\vec{C}\) with \(\vec{B}\) is 15.0. From this information, find the components of vector \(\vec{C}\).

Two vectors \(\vec{A}\) and \(\vec{B}\) have magnitudes A = 3.00 and B = 3.00. Their vector product is \(\vec{A} \times \vec{B}= -5.00 \hat{\kmath} + 2.00 \hat{\imath}\).. What is the angle between \(\vec{A}\) and \(\vec{B}\)?

DATA You are a team leader at a pharmaceutical company. Several technicians are preparing samples, and you want to compare the densities of the samples (density = mass/volume) by using the mass and volume values they have reported. Unfortunately, you did not specify what units to use. The technicians used a variety of units in reporting their values, as shown in the following table. List the sample IDs in order of increasing density of the sample.

DATA You are a mechanical engineer working for a manufacturing company. Two forces, \(\vec{F}_1\) and \(\vec{F}_2\), act on a component part of a piece of equipment. Your boss asked you to find the magnitude of the larger of these two forces. You can vary the angle between \(\vec{F}_1\) and \(\vec{F}_2\) from \(0^{\circ}\) to \(90^{\circ}\) while the magnitude of each force stays constant. And, you can measure the magnitude of the resultant force they produce (their vector sum), but you cannot directly measure the magnitude of each separate force. You measure the magnitude of the resultant force for four angles \(\theta\) between the directions of the two forces as follows: (a) What is the magnitude of the larger of the two forces? (b) When the equipment is used on the production line, the angle between the two forces is \(30.0^{\circ}\). What is the magnitude of the resultant force in this case?

DATA Navigating in the Solar System. The Mars Polar Lander spacecraft was launched on January 3, 1999. On December 3, 1999, the day Mars Polar Lander impacted the Martian surface at high velocity and probably disintegrated, the positions of the earth and Mars were given by these coordinates: With these coordinates, the sun is at the origin and the earth’s orbit is in the xy-plane. The earth passes through the +x-axis once a year on the autumnal equinox, the first day of autumn in the northern hemisphere (on or about September 22). One AU, or astronomical unit, is equal to \(1.496 \times 10^8 \ \mathrm{km}\), the average distance from the earth to the sun. (a) Draw the positions of the sun, the earth, and Mars on December 3, 1999. (b) Find these distances in AU on December 3, 1999: from (i) the sun to the earth; (ii) the sun to Mars; (iii) the earth to Mars. (c) As seen from the earth, what was the angle between the direction to the sun and the direction to Mars on December 3, 1999? (d) Explain whether Mars was visible from your current location at midnight on December 3, 1999. (When it is midnight, the sun is on the opposite side of the earth from you.)

Completed Pass. The football team at Enormous State University (ESU) uses vector displacements to record its plays, with the origin taken to be the position of the ball before the play starts. In a certain pass play, the receiver starts at \(+1.0 \hat{\imath} - 5.0 \hat{\jmath}\), where the units are yards, \(\hat{\imath}\) is to the right, and \(\hat{\jmath}\) is downfield. Subsequent displacements of the receiver are \(+9.0 \hat{\imath}\) (he is in motion before the snap), \(+11.0 \hat{\jmath}\) (breaks downfield), \(-6.0 \hat{\imath} + 4.0 \hat{\jmath}\) (zigs), and \(+12.0 \hat{\imath} + 18.0 \hat{\jmath}\) (zags). Meanwhile, the quarterback has dropped straight back to a position \(-7.0 \hat{\jmath}\). How far and in which direction must the quarterback throw the ball? (Like the coach, you’ll be well advised to diagram the situation before solving this numerically.)

Navigating in the Big Dipper. All of the stars of the Big Dipper (part of the constellation Ursa Major) may appear to be the same distance from the earth, but in fact they are very far from each other. Figure P1.91 shows the distances from the earth to each of these stars. The distances are given in light-years (ly), the distance that light travels in one year. One light-year equals \(9.461 \times 10^{15} \ \mathrm{m}\). (a) Alkaid and Merak are \(25.6^{\circ}\) apart in the earth’s sky. In a diagram, show the relative positions of Alkaid, Merak, and our sun. Find the distance in light-years from Alkaid to Merak. (b) To an inhabitant of a planet orbiting Merak, how many degrees apart in the sky would Alkaid and our sun be?

What is total volume of the gas-exchanging region of the lungs? (a) \(2000 \ \mu \mathrm{m}^3\); (b) \(2 \ \mathrm{m}^3\); (c) 2.0 L; (d) 120 L.

If we assume that alveoli are spherical, what is the diameter of a typical alveolus? (a) 0.20 mm; (b) 2 mm; (c) 20 mm; (d) 200 mm.

Individuals vary considerably in total lung volume. Figure P1.94 shows the results of measuring the total lung volume and average alveolar volume of six individuals. From these data, what can you infer about the relationship among alveolar size, total lung volume, and number of alveoli per individual? As the total volume of the lungs increases, Individuals vary considerably in total lung volume. Figure P1.94 shows the results of measuring the total lung volume and average alveolar volume of six individuals. From these data, what can you infer about the relationship among alveolar size, total lung volume, and number of alveoli per individual? As the total volume of the lungs increases, (a) the number and volume of individual alveoli increase; (b) the number of alveoli increases and the volume of individual alveoli decreases; (c) the volume of the individual alveoli remains constant and the number of alveoli increases; (d) both the number of alveoli and the volume of individual alveoli remain constant.