What does the scalar product of a vector with it-self,

Problem 1DQ How many correct experiments do we need to disprove a theory? How many do we need to prove a theory? Explain.

Problem 1E Starting with the definition 1 in. = 2.54 cm, find the number of (a) kilometers in 1.00 mile and (b) feet in 1.00 km.

Problem 2E According to the label on a bottle of salad dressing, the volume of the contents is 0.473 liter (L). Using only the conversions 1 L = 1000 cm3 and 1 in. = 2.54 cm, express this volume in cubic inches.

Problem 2DQ A guidebook describes the rate of climb of a mountain trail as 120 meters per kilometer. How can you express this as a number with no units?

Problem 3DQ Suppose you are asked to compute the tangent of 5.00 meters. Is this possible? Why or why not?

Problem 3E How many nanoseconds does it take light to travel 1.00 ft in vacuum? (This result is a useful quantity to remember.)

Problem 4DQ A highway contractor stated that in building a bridge deck he poured 250 yards of concrete. What do you think he meant?

Problem 4E The density of gold is 19.3 g/cm3. What is this value in kilograms per cubic meter?

Problem 102CP The vector called the ?position vector points from the origin (0, 0, 0) to an arbitrary point in space with coordinates (?x?, ?y?, z?). Use what you know about vectors to prove the following: All points (?x?, ?y?, ?z?) that satisfy the equation ?Ax + ?By + ?Cz = 0, where ?A?, ?B?, and ?C are constants, lie in a plane that passes through the origin and that is perpendicular to the vector . Sketch this vector and the plane.

Problem 5DQ What is your height in centimeters? What is your weight in newtons?

Problem 5E The most powerful engine available for the classic 1963 Chevrolet Corvette Sting Ray developed 360 horsepower and had a displacement of 327 cubic inches. Express this displacement in liters (L) by using only the conversions 1 L = 1000 cm3 and 1 in. = 2.54 cm.

Problem 6DQ 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 µg/y (y = year) when compared every 10 years or so to the standard international kilogram. Does this apparent increase have any importance? Explain.

Problem 6E A square field measuring 100.0 m by 100.0 m has an area of 1.00 hectare. An acre has an area of 43,600 ft2. If a lot has an area of 12.0 acres, what is its area in hectares?

A power supply has a fixed output voltage of \(12.0 \mathrm{~V}\), but you need \(V_{\mathrm{T}}=3.0 \mathrm{~V}\) for an experiment. (a) Using the voltage divider shown in Fig. 19-66, what should \(R_2\) be if \(R_1\) is \(10.0 \Omega\) ? (b) What will the terminal voltage \(V_{\mathrm{T}}\) be if you connect a load to the \(3.0-\mathrm{V}\) terminal, assuming the load has a resistance of \(7.0 \Omega\) ?

Problem 7E How many years older will you be 1.00 gigasecond from now? (Assume a 365-day year.)

Problem 8DQ Describe how you could measure the thickness of a sheet of paper with an ordinary ruler.

Problem 8E While driving in an exotic foreign land, you see a speed limit sign that reads 180,000 furlongs per fortnight. How many miles per hour is this? (One furlong is mile, and a fortnight is 14 days. A furlong originally referred to the length of a plowed furrow.)

Problem 9DQ The quantity ? = 3.14159 c 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.

A certain fuel-efficient hybrid car gets gasoline mileage of 55.0 mpg (miles per gallon). (a) If you are driving this car in Europe and want to compare its mileage with that of other European cars, express this mileage in km/L (L = liter). Use the conversion factors in Appendix E. (b) If this car’s gas tank holds 45 L, how many tanks of gas will you use to drive 1500 km?

Problem 10DQ What are the units of volume? Suppose another student tells you that a cylinder of radius ?r ?and height ?h? has volume given by ??r?3?h?. Explain why this cannot be right.

Problem 10E The following conversions occur frequently in physics and are very useful. (a) Use 1 mi = 5280 ft and 1 h = 3600 s to convert 60 mph to units of ft/s. (b) The acceleration of a freely falling object is 32 ft/s2. Use 1 ft = 30.48 cm to express this acceleration in units of m/s2. (c) The density of water is 1.0 g/cm3. Convert this density to units of kg/m3.

Problem 11DQ Three archers each fire four arrows at a target. Joe’s four arrows hit at points 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.

Problem 11E Neptunium. In the fall of 2002, scientists at Los Alamos National Laboratory determined that the critical mass of neptunium-237 is about 60 kg. The critical mass of a fissionable material is the minimum amount that must be brought together to start a nuclear chain reaction. Neptunium-237 has a density of 19.5 g/cm3. What would be the radius of a sphere of this material that has a critical mass?

Problem 12DQ 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.

Problem 12E BIO (a) The recommended daily allowance (RDA) of the trace metal magnesium is 410 mg/day for males. Express this quantity in µg/day. (b) For adults, the RDA of the amino acid lysine is 12 mg per kg of body weight. How many grams per day should a 75-kg adult receive? (c) A typical multivitamin tablet can contain 2.0 mg of vitamin B2 (riboflavin), and the RDA is 0.0030 g/day. How many such tablets should a person take each day to get the proper amount of this vitamin, if he gets none from other sources? (d) The RDA for the trace element selenium is 0.000070 g/day. Express this dose in mg/day.

Problem 13DQ 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.

Problem 13E Figure 1.7 shows the result of unacceptable error in the stopping position of a train. (a) If a train travels 890 km from Berlin to Paris and then overshoots the end of the track by 10 m, what is the percent error in the total distance covered? (b) Is it correct to write the total distance covered by the train as 890,010 m? Explain.

Problem 14DQ One sometimes speaks of the “direction of time,” evolving from past to future. Does this mean that time is a vector quantity? Explain your reasoning.

Problem 14E With a wooden ruler, you measure the length of a rectangular piece of sheet metal to be 12 mm. With micrometer calipers, you measure the width of the rectangle to be 5.98 mm. Use the correct number of significant figures: What is (a) the area of the rectangle; (b) the ratio of the rectangle’s width to its length; (c) the perimeter of the rectangle; (d) the difference between the length and the width; and (e) the ratio of the length to the width?

Problem 15DQ 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?

Problem 15E A useful and easy-to-remember approximate value for the number of seconds in a year is ? X 107. Determine the percent error in this approximate value. (There are 365.24 days in one year.)

Problem 16DQ 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.

Problem 16E How many gallons of gasoline are used in the United States in one day? Assume that there are two cars for every three people, that each car is driven an average of 10,000 miles per year, and that the average car gets 20 miles per gallon.

Problem 17DQ (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)?

Problem 17E BIO A rather ordinary middle-aged man is in the hospital for a routine checkup. The nurse writes “200” on the patient’s medical chart but forgets to include the units. Which of these quantities could the 200 plausibly represent? The patient’s (a) mass in kilograms; (b) height in meters; (c) height in centimeters; (d) height in millimeters; (e) age in months.

Problem 18DQ If what must be true about the directions and magnitudes of and if C = A + B ? What must be true about the directions and magnitudes of and if C = 0?

Problem 18E How many kernels of corn does it take to fill a 2-L soft drink bottle?

Problem 19DQ If and are nonzero vectors, is it possible for both to be zero? Explain.

Problem 19E How many words are there in this book?

Problem 20DQ What does the scalar product of a vector with it-self, give? What about the vector product of a vector with itself?

Problem 20E BIO Four astronauts are in a spherical space station. (a) If, as is typical, each of them breathes about 500 cm3 of air with each breath, approximately what volume of air (in cubic meters) do these astronauts breathe in a year? (b) What would the diameter (in meters) of the space station have to be to contain all this air?

Problem 21DQ Let represent any nonzero vector. Why is a unit vector, and what is its direction? If ? is the angle thamakes with the + x -axis, explain why is called the direction cosine for that axis.

Problem 21E BIO How many times does a typical person blink her eyes in a lifetime?

Problem 22DQ Which of the following are legitimate mathematical operations: In each case, give the reason for your answer.

Problem 22E BIO How many times does a human heart beat during a person’s lifetime? How many gallons of blood does it pump? (Estimate that the heart pumps 50 cm3 of blood with each beat.)

Problem 23DQ Consider the vector products and 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 such that these two vector products are equal? If so, give an example.

Problem 23E In Wagner’s opera ?Das Rheingold?, the goddess Freia is ransomed for a pile of gold just tall enough and wide enough to hide her from sight. Estimate the monetary value of this pile. The density of gold is 19.3 g/cm3, and take its value to be about $10 per gram.

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

Problem 24E You are using water to dilute small amounts of chemicals in the laboratory, drop by drop. How many drops of water are in a 1.0-L bottle? ( Hint: Start by estimating the diameter of a drop of water.)

Problem 25DQ (a) If does it necessarily follow that A = 0 or B = 0? Explain. (b) If does it necessarily follow that A = 0 or B = 0 ? Explain.

Problem 25E How many pizzas are consumed each academic year by students at your school?

Problem 26DQ If for a vector in the ?xy? -plane, does it follow that Ax = -Ay? What can you say about Ax and Ay?

Problem 26E Hearing rattles from a snake, you make two rapid displacements of magnitude 1.8 m and 2.4 m. In sketches (roughly to scale), show how your two displacements might add up to give a resultant of magnitude (a) 4.2 in; (b) 0.6 m; (c) 3.0 m.

Problem 27E A postal employee drives a delivery truck along the route shown in Fig. E1.25. Determine the magnitude and direction of the resultant displacement by drawing a scale diagram. (See also Exercise 1.32 for a different approach.)

Problem 28E For the vectors and in Fig. E1.24, use a scale drawing to find the magnitude and direction of (a) the vector sum and (b) the vector difference Use your answers to find the magnitude and direction of (c) and (d) (See also Exercise 1.31 for a different approach.)

Problem 29E A spelunker is surveying a cave. She follows a passage 180 m straight west, then 210 m in a direction 45o east of south, and then 280 m at 30o east of north. After a fourth displacement, she finds herself back where she started. Use a scale drawing to determine the magnitude and direction of the fourth displacement. (See also Problem 1.61 for a different approach.)

Problem 30E Let ? be the angle that the vector makes with the + x -axis, measured counterclockwise from that axis. Find angle ? for a vector that has these components: (a) Ax = 2.00 m, Ay = - 1.00 m; (b) Ax = 2.00 m, Ay = 1.00 m; (c) Ax = - 2.00 m, Ay = 1.00 m; (d) Ax = - 2.00 m, Ay = - 1.00 m.

Problem 31E Compute the x - and y -components of the vectors and in Fig. E1.24.

Problem 32E Vector is in the direction 34.0o clockwise from the –y-axis. The x-component of is Ax = - 16.0 m. (a) What is the y -component of ? (b) What is the magnitude of ?

Problem 33E Vector has ?y?-component ?Ay? = +13.0 m. makes an angle of 32.0° counterclockwise from the +?y?-axis. (a) What is the ?x?-component o? (b) What is the magnitude of ?

Problem 34E A postal employee drives a delivery truck over the route shown in Fig. E1.25. Use the method of components to determine the magnitude and direction of her resultant displacement. In a vector-addition diagram (roughly to scale), show that the resultant displacement found from your diagram is in qualitative agreement with the result you obtained by using the method of components.

Problem 35E For the vectors in Fig. E1.24, use the method of components to find the magnitude and direction of (a) the vector sum (b) the vector sum (c) the vector difference (d) the vector difference

Problem 36E Find the magnitude and direction of the vector represented by the following pairs of components: (a) Ax = - 8.60 cm, Ay = 5.20 cm; (b) Ax = - 9.70 m, Ay = - 2.45 m; (c) Ax = 7.75 km, Ay = - 2.70 km.

Problem 37E A disoriented physics professor drives 3.25 km north, then 2.20 km west, and then 1.50 km south. Find the magnitude and direction of the resultant displacement, using the method of components. In a vector-addition diagram (roughly to scale), show that the resultant displacement found from your diagram is in qualitative agreement with the result you obtained by using the method of components.

Vector \(\overrightarrow{\boldsymbol{A}}\) is 2.80 cm long and is \(60.0^{\circ}\) above the \(x\)-axis in the first quadrant. Vector \(\overrightarrow{\boldsymbol{B}}\) is 1.90 cm long and is \(60.0^{\circ}\) below the \(x\)-axis in the fourth quadrant (Fig. E1.39). Use components to find the magnitude and direction of (a) \(\overrightarrow{\boldsymbol{A}}+\overrightarrow{\boldsymbol{B}}\) ; (b) \(\overrightarrow{\boldsymbol{A}}-\overrightarrow{\boldsymbol{B}}\) ; (c) \(\overrightarrow{\boldsymbol{B}}-\overrightarrow{\boldsymbol{A}}\) . In each case, sketch the vector addition or subtraction and show that your numerical answers are in qualitative agreement with your sketch. Figure E1.39

Problem 40E In each case, find the x - and y -components of vector (a)

Problem 42E Given two vectors (a) find the magnitude of each vector; (b) use unit vectors to write an expression for the vector difference and (c) find the magnitude and direction of the vector difference (d) In a vector diagram show and show that your diagram agrees qualitatively with your answer to part (c).

Problem 43E (a) Write each vector in Fig. E1.39 in terms of the unit vectors (b) Use unit vectors to express vector where (c) Find the magnitude and direction of

Problem 44E (a) Is the vector a unit vector? Justify your answer. (b) Can a unit vector have any components with magnitude greater than unity? Can it have any negative components? In each case justify your answer. (c) If where a? ? is a constant, determine the value of a that makes a unit vector.

Problem 41E Write each vector in Fig. E1.24 in terms of the unit vectors

Problem 45E For the vectors in Fig. E1.24, find the scalar products

Problem 46E (a) Find the scalar product of the vectors given in Exercise 1.38. (b) Find the angle between these two vectors.

Problem 47E Find the angle between each of these pairs of vectors:

Problem 48E Find the vector product (expressed in unit vectors) of the two vectors given in Exercise 1.38. What is the magnitude of the vector product?

Problem 49E For the vectors in Fig. E1.28, (a) Find the magnitude and direction of the vector product (b) find the magnitude and direction of

Problem 50E For the two vectors in Fig. E1.35, find the magnitude and direction of (a) the vector product (b) the vector product

Problem 51E For the two vectors . In Fig E1.43, (a) find the scalar product (b) find the magnitude and direction of the vector product

Problem 52E The vector A is 3.50 cm long and is directed into this pages. Vector Points from the lower right corner of this page to the upper left corner of this page. Define an appropriate right-handed coordinate system, and find the three components of the vector product , measured in cm3. In a diagram. show your coordinate system and the vectors .

Problem 53E Given two vectors and (a) find the magnitude of each vector; (b) use unit vectors to write an expression for the vector difference and (c) find the magnitude of the vector difference Is this the same as the magnitude of Explain.

Problem 54P An acre has a length of one furlong and a width one-tenth of its length. (a) How many acres are in a square mile? (b) How many square feet are in an acre? See Appendix E. (c) An acre-foot is the volume of water that would cover 1 acre of flat land to a depth of 1 foot. How many gallons are in 1 acre-foot?

Problem 55P 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 g/cm3), what is its radius expressed (a) in kilometers and (b) as a multiple of earth’s radius? Consult Appendix F for astronomical data.

Problem 56P 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 X 109 years? (d) By how many seconds would a hydrogen maser clock be off after a time interval equal to the age of the earth?

Problem 57P BIO Breathing Oxygen. The density of air under standard laboratory conditions is 1.29 kg/m3, and about 20% of that air consists of oxygen. Typically, people breathe about of air per breath. (a) How many grams of oxygen does a person breathe in a day? (b) If this air is stored uncompressed in a cubical tank, how long is each side of the tank?

Problem 58P A rectangular piece of aluminum is cm long and wide. (a) Find the area of the rectangle and the uncertainty in the area. (b) Verify that the fractional un-certainty in the area is equal to the sum of the fractional uncertainties in the length and in the width. (This is a general result.)

Problem 59P As you eat your way through a bag of chocolate chip cookies, you observe that each cookie is a circular disk with a diameter of 8.50 ± 0.02 cm and a thickness of 0.050 ± 0.005 cm. (a) Find the average volume of a cookie and the uncertainty in the volume. (b) Find the ratio of the diameter to the thickness and the uncertainty in this ratio.

BIO Biological tissues are typically made up of 98% water. Given that the density of water is \(1.0 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\), 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.

Problem 62P How many dollar bills would you have to stack to reach the moon? Would that be cheaper than building and launching a spacecraft? (?Hint?: Start by folding a dollar bill to see how many thicknesses make 1.0 mm.)

Problem 61P 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.)

Problem 63P How much would it cost to paper the entire United States (including Alaska and Hawaii) with dollar bills? What would he the cost to each person in the United States?

Stars in the Universe. Astronomers frequently say that there are more stars in the universe than there are grains of sand on all the beaches on the earth. (a) Given that a typical grain of sand is about 0.2 mm in diameter, estimate the number of grains of sand on all the earth’s beaches, and hence the approximate number of stars in the universe. It would be helpful to consult an atlas and do some measuring. (b) Given that a typical galaxy contains about 100 billion stars and there are more than 100 billion galaxies in the known universe, estimate the number of stars in the universe and compare this number with your result from part (a).

Problem 65P Two workers pull horizontally on a heavy box, but one pulls twice as hard as the other. The larger pull is directed at 25.0° west of north, and the resultant of these two pulls is 460.0 N directly northward. Use vector components to find the magnitude of each of these pulls and the direction of the smaller pull.

Problem 66P Three horizontal ropes pull on a large stone stuck in the ground, producing the vector forces shown in Fig. P1.60. Find the magnitude and direction of a fourth force on the stone that will make the vector sum of the four forces zero.

Problem 67P You are to program a robotic arm on an assembly line to move in the ?xy?-plane. Its first displacement is its second displacement is of magnitude 6.40 cm and direction 63.0° measured in the sense from the +?x- ? axis toward the ??? axis. The resultant of the two displacements should also have a magnitude of 6.40 cm, but a direction 22.0° measured in the sense from the +?x?-axis toward the +?y?-axis. (a) Draw the vector-addition diagram for these vectors, roughly to scale. (b)Find the components of (c) Find the magnitude and direction of

Problem 68P Emergency Landing. A plane leaves the airport in Galisteo and files 170 km at 68° east of north and then changes direction to fly 230 km at 48° 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?

Problem 71P A rocket fires two engines simultaneously. One produces a thrust of 480 N directly forward, while the other gives a 513-N thrust at 32.4° above the forward direction. Find the magnitude and direction (relative to the forward direction) of the resultant force that these engines exert on the rocket.

Problem 70P (a) Find the magnitude and direction of the vector that is the sum of the three vectors , , and in Fig. In a diagram, show how is formed from these three vectors. (b) Find the magnitude and direction of the vector In a diagram, show how is formed from these three vectors. Figure:

Problem 69P As noted in Exercise 1.26, a spelunker is surveying a cave. She follows a passage 180 m straight west, then 210 m in a direction 45o east of south, and then 280 m at 30o 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.

Problem 72P 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. P1.64). 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.

Problem 73P Dislocated Shoulder. A patient with a dislocated shoulder is put into a traction apparatus as shown in Fig. The pulls and have equal magnitudes and must combine to produce an outward traction force of 5.60 N on the patient’s arm. How large should these pulls be? Figure:

Problem 74P 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. P1.66). The directions are shown relative to north: 0o is north, 90o is east, 180o is south, and 270o 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.

Problem 77P A graphic artist is creating a new logo for her company’s website. In the graphics program she is using, each pixel in an image file has coordinates (?x?, y?), where the origin (0, 0) is at the upper left corner of the image, the +?x?-axis points to the right, and the +?y?-axis points down. Distances are measured in pixels. (a) The artist draws a line from the pixel location (10, 20) to the location (210, 200). She wishes to draw a second line that starts at (10, 20), is 250 pixels long, and is at an angle of 30° measured clockwise from the first line. At which pixel location should this second line end? Give your answer to the nearest pixel. (b) The artist now draws an arrow that connects the lower right end of the first line to the lower right end of the second line. Find the length and direction of this arrow. Draw a diagram showing all three lines.

Problem 76P Getting Back. An explorer in the dense jungles of equatorial Africa leaves his hut. He takes 40 steps northeast, then 80 steps 60° north of west, then 50 steps due south. Assume his steps all have equal length. (a) Sketch, roughly to scale, the three vectors and their resultant. (b) Save the explorer from becoming hopelessly lost in the jungle by giving him the displacement, calculated using the method of components, that will return him to his hut.

Problem 75P Equilibrium. We say an object is in ?equilibrium? if all the forces on it balance (add up to zero). Figure shows a beam weighing 124 N that is supported in equilibrium by a 100.0-N pull and a force F at the floor. The third force on the beam is the 124-N weight that acts vertically downward. (a) Use vector components to find the magnitude and direction of . (b) Check the reasonableness of your answer in part (a) by doing a graphical solution approximately to scale. Figure:

Problem 78P A ship leaves the island of Guam and sails 285 km at 40.0° north of west. In which direction must it now head and how far must it sail so that its resultant displacement will be 115 km directly east of Guam?

Problem 79P BIO Bones and Muscles. A physical therapy patient has a forearm that weighs 20.5 N and lifts a 112.0-N weight. These two forces are directed vertically downward. The only other significant forces on this forearm come from the biceps muscle (which acts perpendicular to the forearm) and the force at the elbow. If the biceps produces a pull of 232 N when the forearm is raised 43.0o above the horizontal, find the magnitude and direction of the force that the elbow exerts on the forearm. (The sum of the elbow force and the biceps force must balance the weight of the arm and the weight it is carrying, so their vector sum must be 132.5 N, upward.)

Problem 80P You decide to go to your favorite neighborhood restaurant. You leave your apartment, take the elevator 10 flights down (each flight is 3.0 m), and then walk 15 m south to the apartment exit. You then proceed 0.200 km east, turn north, and walk 0.100 km to the entrance of the restaurant. (a) Determine the dis-placement from your apartment to the restaurant. Use unit vector notation for your answer, clearly indicating your choice of coordinates. (b) How far did you travel along the path you took from your apartment to the restaurant, and what is the magnitude of the displacement you calculated in part (a)?

Problem 81P 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° west of north, and finally walk 1.00 km at 40.0° 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, check it with a graphical solution drawn roughly to scale.

Problem 82P A fence post is 52.0 m from where you are standing, in a direction 37.0° north of east. A second fence post is due south from you. What is the distance of the second post from you, if the distance between the two posts is 80.0 m?

Problem 83P A dog in an open field runs 12.0 m east and then 28.0 m in a direction 50.0o 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?

Problem 84P 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 60.0o west of north. Jane walks 16.0 m in a direction 30.0o 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?

Problem 85P John, Paul, and George are standing in a strawberry field. Paul is 14.0 m due west of John. George is 36.0 m from Paul, in a direction 37.0° south of east from Paul’s location. How far is George from John? What is the direction of George’s location from that of John?

Problem 86P 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 23.0o south of east. Karl’s tent is 32.0 m from yours, in the direction 37.0o north of east. What is the distance between Karl’s tent and Joe’s tent?

Problem 87P Vectors have scalar product - 6.00, and their vector product has magnitude + 9.00. What is the angle between these two vectors?

Problem 90P When two vectors and are drawn from a common point, the angle between them is ???. (a) Using vector techniques, show that the magnitude of their vector sum is given by (b) If and have the same magnitude, for which value of ?? will their vector sum have the same magnitude as or ?

Problem 89P Vector has magnitude 12.0 m and vector has magnitude 16.0 m. The scalar product is 90.0 m2. What is the magnitude of the vector product between these two vectors?

Problem 88P Bond Angle in Methane. In the methane molecule, CH4, 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 an adjacent C - H bond is in the direction. Calculate the angle between these two bonds.

Problem 91P A cube is placed so that one corner is at the origin and three edges are along the ?x -, y -, and ?z -axes of a coordinate system (Fig. P1.80). Use vectors to compute (a) the angle be-tween the edge along the ?z -axis (line ?ab?) and the diagonal from the origin to the opposite corner (line ?ad?), and (b) the angle between line a? c? (the diagonal of a face) and line ?ad?.

Problem 92P Vector has magnitude 6.00 m and vector has magnitude 3.00 m. the vector product between these two vectors has magnitude 12.0 m2. What are the two possible values for the scalar product of these two vectors? for each value of , draw a sketch that shows and and explain why the vector products in the two sketches are the same but the scalar products differ.

Problem 93P The scalar product of vectors is + 48.0 m2. Vector has magnitude 9.00 m and direction 28.0o west of south. If vector has direction 39.0o south of east, what is the magnitude of .

Problem 94P Obtain a unit vector perpendicular to the two vectors given in Exercise 1.41.

Problem 96P Two vectors have magnitudes A = 3.00 and B = 3.00. Their vector product is What is the angle between

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

Later in our study of physics we will encounter quantities represented by \((\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}) \cdot \overrightarrow{\boldsymbol{C}}\). (a) Prove that for any three vectors \(\overrightarrow{\boldsymbol{A}}, \overrightarrow{\boldsymbol{B}}\), and \(\overrightarrow{\boldsymbol{C}}, \overrightarrow{\boldsymbol{A}} \cdot(\overrightarrow{\boldsymbol{B}} \times \overrightarrow{\boldsymbol{C}})=(\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}) \cdot \overrightarrow{\boldsymbol{C}}\). (b) Calculate \((\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}) \cdot \overrightarrow{\boldsymbol{C}}\) for the three vectors \(\overrightarrow{\boldsymbol{A}}\) with magnitude A = 5.00 and angle \(\theta_A=26.0^{\circ}\) measured in the sense from the +x-axis toward the +y-axis, \(\overrightarrow{\boldsymbol{B}}\) with B = 4.00 and \(\theta_B=63.0^{\circ}\), and \(\overrightarrow{\boldsymbol{C}}\) with magnitude 6.00 and in the +z-direction. Vectors \(\vec{A}\) and \(\vec{B}\) are in the xy-plane.

Problem 98CP The length of a rectangle is given as ?L ± ?l and its width as ?W ± ?w?. (a) Show that the uncertainty in its area ?A is ?a = ?Lw + ?lW?. Assume that the uncertainties ?l and ?w are small, so that the product ?lw is very small and you can ignore it. (b) Show that the fractional uncertainty in the area is equal to the sum of the fractional uncertainty in length and the fractional uncertainty in width. (c) A rectangular solid has dimensions ?L ± ?l?, ?W ± ?w?, and ?H ± ?h?. Find the fractional uncertainty in the volume, and show that it equals the sum of the fractional uncertainties in the length, width, and height.

Completed Pass. At Enormous State University (ESU), the football team records its plays using vector displacements, 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, \(\hat{\boldsymbol{l}}\) is to the right, and \(\hat{\boldsymbol{J}}\) is downfield. Subsequent displacements of the receiver are \(+9.0 \hat{\imath}\) (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 \(-7.0 \hat{\boldsymbol{j}}\). How far and in which direction must the quarterback throw the ball? (Like the coach, you will be well advised to diagram the situation before solving it numerically.)

Problem 100CP 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 X 108 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.)

Problem 101CP 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 (1y), the distance that light travels in one year. One light-year equals 9.461 X 1015m. (a) Alkaid and Merak are 25.6o 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?