Vector A S is 2.80 cm long and is 60.0 above the x-axis in

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 mg>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 p = 3.14159c 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 pr3 h. Explain why this cannot be right.

Three archers each fire four arrows at a target. Joes 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 Moes arrows hit within 1 cm of a point 20 cm from the center, and Flos 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 1nd + ne + k n2 a unit vector? Is the vector 13.0nd 2.0ne2 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 C S = A S + B S , what must be true about the directions and magnitudes of A S and B S if C = A + B? What must be true about the directions and magnitudes of A S and B S if C = 0?

If A S and B S are nonzero vectors, is it possible for both A S ~ B S and A S : B S to be zero? Explain

What does A S ~ A S , the scalar product of a vector with itself, give? What about A S : A S , the vector product of a vector with itself?

Let A S represent any nonzero vector. Why is A S >A a unit vector, and what is its direction? If u is the angle that A S makes with the +x-axis, explain why 1A S >A2 ~ nd 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) A S ~ 1B S C S 2; (b) 1A S B S 2 : C S ; (c) A S ~ 1B S : C S 2; (d) A S : 1B S : C S 2; (e) A S : 1B S ~ C S 2? In each case, give the reason for your answer.

Consider the vector products A S : 1B S : C S 2 and 1A S : B S 2 : C S . 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 A S , B S , and C S such that these two vector products are equal? If so, give an example.

Show that, no matter what A S and B S are, A S ~ 1A S : B S 2 = 0. (Hint: Do not look for an elaborate mathematical proof. Consider the definition of the direction of the cross product.)

(a) If A S ~ B S = 0, does it necessarily follow that A = 0 or B = 0? Explain. (b) If A S : B S = 0, does it necessarily follow that A = 0 or B = 0? Explain.

If A S = 0 for a vector in the xy-plane, does it follow that Ax = -Ay? What can you say about Ax and Ay?

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.

. 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

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

The density of gold is 19.3 g>cm3 . What is this value in kilograms per cubic meter?

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.

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?

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

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 1 8 mile, and a fortnight is 14 days. A furlong originally referred to the length of a plowed furrow.)

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?

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>s 2 . Use 1 ft = 30.48 cm to express this acceleration in units of m>s 2 . (c) The density of water is 1.0 g>cm3 . Convert this density to units of kg>m3

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?

(a) The recommended daily allowance (RDA) of the trace metal magnesium is 410 mg>day for males. Express this quantity in mg>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

Bacteria. Bacteria vary in size, but a diameter of 2.0 mm is not unusual. What are the volume (in cubic centimeters) and surface area (in square millimeters) of a spherical bacterium of that size? (Consult Appendix B for relevant formulas.)

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 rectangles 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?

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

Express each approximation of p to six significant figures: (a) 22>7 and (b) 355>113. (c) Are these approximations accurate to that precision?

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

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.

How many times does a typical person blink her eyes in a lifetime?

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?

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 \(19.3 \ \mathrm{g/cm}^3\), and take its value to be about $10 per gram.

How many times does a human heart beat during a persons lifetime? How many gallons of blood does it pump? (Estimate that the heart pumps 50 cm3 of blood with each beat.)

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.)

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

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.)

A spelunker is surveying a cave. She follows a passage 180 m straight west, then 210 m in a direction 45 east of south, and then 280 m at 30 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.)

. Compute the x- and y-components of the vectors A S , B S , C S , and D S in Fig. E1.24.

Let u be the angle that the vector A S makes with the +x-axis, measured counterclockwise from that axis. Find angle u 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.

Vector A S has y-component Ay = +9.60 m. A S makes an angle of 32.0 counterclockwise from the +y-axis. (a) What is the x-component of A S ? (b) What is the magnitude of A S ?

Vector \(\vec A\) is in the direction \(34.0^\circ\) clockwise from the -y-axis. The x-component of \(\vec A\) is \(A_x = -16.0 \ \mathrm m\). (a) What is the y-component of \(\vec A\) ? (b) What is the magnitude of \(\vec A\) ?

For the vectors A S and B S in Fig. E1.24, use the method of components to find the magnitude and direction of (a) the vector sum A S + B S ; (b) the vector sum B S + A S ; (c) the vector difference A S B S ; (d) the vector difference B S A S .

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

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

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

Vector A S is 2.80 cm long and is 60.0 above the x-axis in the first quadrant. Vector B S is 1.90 cm long and is 60.0 below the x-axis in the fourth quadrant (Fig. E1.35). Use components to find the magnitude and direction of (a) A S + B S ; (b) A S B S ; (c) B S A S . In each case, sketch the vector addition or subtraction and show that your numerical answers are in qualitative agreement with your sketch.

In each case, find the x- and y-components of vector A S : (a) A S = 5.0nd 6.3ne; (b) A S = 11.2ne 9.91nd; (c) A S = -15.0nd + 22.4ne ; (d) A S = 5.0B S , where B S = 4nd 6ne.

. Write each vector in Fig. E1.24 in terms of the unit vectors nd and ne.

Given two vectors AS= 4.00nd + 7.00ne and BS= 5.00nd 2.00ne, (a) find the magnitude of each vector; (b) use unit vectors to write an expression for the vector difference A S B S ; and (c) find the magnitude and direction of the vector difference A S B S . (d) In a vector diagram show A S , B S , and A S B S , and show that your diagram agrees qualitatively with your answer to part (c).

(a) Write each vector in Fig. E1.39 in terms of the unit vectors nd and ne. (b) Use unit vectors to express vector C S , where C S = 3.00 A S 4.00 B S . (c) Find the magnitude and direction of C S .

You are given two vectors A S = -3.00nd + 6.00ne and B S = 7.00nd + 2.00ne. Let counterclockwise angles be positive. (a) What angle does A S make with the +x@axis? (b) What angle does B S make with the +x@axis? (c) Vector C S is the sum of A S and B S , so C S = A S + B S . What angle does C S make with the +x@axis?

You are given two vectors A S = -3.00nd + 6.00ne and B S = 7.00nd + 2.00ne. Let counterclockwise angles be positive. (a) What angle does A S make with the +x@axis? (b) What angle does B S make with the +x@axis? (c) Vector C S is the sum of A S and B S , so C S = A S + B S . What angle does C S make with the +x@axis?

(a) Find the scalar product of the vectors A S and B S given in Exercise 1.38. (b) Find the angle between these two vectors.

For the vectors A S , B S , and C S in Fig. E1.24, find the scalar products (a) A S ~ B S ; (b) B S ~ C S ; (c) A S ~ C S .

Find the vector product A S : B S (expressed in unit vectors) of the two vectors given in Exercise 1.38. What is the magnitude of the vector product?

Find the angle between each of these pairs of vectors: (a) A S = -2.00nd + 6.00ne and B S = 2.00nd 3.00ne (b) A S = 3.00nd + 5.00ne and B S = 10.00nd + 6.00ne (c) A S = -4.00nd + 2.00ne and B S = 7.00nd + 14.00ne

For the two vectors in Fig. E1.35, find the magnitude and direction of (a) the vector product A S : B S ; (b) the vector product B S : A S .

For the two vectors A S and D S in Fig. E1.24, find the magnitude and direction of (a) the vector product A S : D S ; (b) the vector product D S : A S .

For the two vectors A S and B S in Fig. E1.39, find (a) the scalar product A S ~ B S ; (b) the magnitude and direction of the vector product A S : B S .

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 g>cm3 , 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 g>cm3 ?

An acre has a length of one furlong 1 1 8 mi2 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?

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 earths 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 11.76 g>cm3 2, what is its radius expressed (a) in kilometers and (b) as a multiple of earths radius? Consult Appendix F for astronomical data.

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 * 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?

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 1 2 L 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?

A rectangular piece of aluminum is 7.60 { 0.01 cm long and 1.90 { 0.01 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.)

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.

Biological tissues are typically made up of 98% water. Given that the density of water is 1.0 * 103 kg>m3 , estimate the mass of (a) the heart of an adult human; (b) a cell with a diameter of 0.5 mm; (c) a honeybee

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.)

Two ropes in a vertical plane exert equal-magnitude forces on a hanging weight but pull with an angle of 72.0 between them. What pull does each rope exert if their resultant pull is 372 N directly upward?

Two workers pull horizontally on a heavy box, but one pulls twice as hard as the other. The larger pull is directed at 21.0 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.

Three horizontal ropes pull on a large stone stuck in the ground, producing the vector forces A S , B S , and C S 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

Three horizontal ropes pull on a large stone stuck in the ground, producing the vector forces A S , B S , and C S 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

Emergency Landing. A plane leaves the airport in Galisteo and flies 170 km at 68.0 east of north; then it changes direction to fly 230 km at 36.0 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?

Dislocated Shoulder. A patient with a dislocated shoulder is put into a traction apparatus as shown in Fig. P1.63. The pulls A S and B S have equal magnitudes and must combine to produce an outward traction force of 12.8 N on the patients arm. How large should these pulls be?

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.

You leave the airport in College Station and fly 23.0 km in a direction 34.0o 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?

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: 0 is north, 90 is east, 180 is south, and 270 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

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 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?

Getting Back. An explorer in Antarctica leaves his shelter during a whiteout. He takes 40 steps northeast, next 80 steps at 60 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

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 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?

A ship leaves the island of Guam and sails 285 km at 62.0 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?

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 43.0 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.)

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 displacement 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)?

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 32.0 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, 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 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 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 dont pitch your tents close together. Joes tent is 21.0 m from yours, in the direction 23.0 south of east. Karls tent is 32.0 m from yours, in the direction 37.0 north of east. What is the distance between Karls tent and Joes tent?

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 CH bonds is in the direction of nd + ne + k n, an adjacent CH bond is in the nd ne k n direction. Calculate the angle between these two bonds.

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

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 between 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 ac (the diagonal of a face) and line ad

Vector A S has magnitude 12.0 m, and vector B S has magnitude 16.0 m. The scalar product A S ~ B S is 112.0 m2 . What is the magnitude of the vector product between these two vectors?

Obtain a unit vector perpendicular to the two vectors given in Exercise 1.41

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 \(\vec B\)?

Two vectors A S and B S have magnitudes A = 3.00 and B = 3.00. Their vector product is A S : B S = -5.00k n + 2.00nd. What is the angle between A S and B S ?

You are given vectors A S = 5.0nd 6.5ne and B S = 3.5nd 7.0ne. A third vector, C S , lies in the xy-plane. Vector C S is perpendicular to vector A S , and the scalar product of C S with B S is 15.0. From this information, find the components of vector C S .

Later in our study of physics we will encounter quantities represented by 1A S : B S 2 ~ C S . (a) Prove that for any three vectors A S , B S , and C S , A S ~ 1B S : C S 2 = 1A S : B S 2 ~ C S . (b) Calculate 1A S : B S 2 ~ C S for vector A S with magnitude A = 5.00 and angle uA = 26.0 (measured from the +x-axis toward the +y-axis), vector B S with B = 4.00 and uB = 63.0, and vector C S with magnitude 6.00 and in the +z-direction. Vectors A S and B S are in the xy-plane

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.

You are a mechanical engineer working for a manufacturing company. Two forces, F S 1 and F S 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 F S 1 and F S 2 from 0o to 90o 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 u between the directions of the two forces as follows: U Resultant force (N) 0.0 8.00 45.0 7.43 60.0 7.00 90.0 5.83 (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. What is the magnitude of the resultant force in this case?

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: x y z Earth 0.3182 AU 0.9329 AU 0.0000 AU Mars 1.3087 AU -0.4423 AU -0.0414 AU With these coordinates, the sun is at the origin and the earths 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 * 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.)

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 \(-7.0 \hat{0}\). 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 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 * 1015 m. (a) Alkaid and Merak are 25.6o apart in the earths 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 mm3 ; (b) 2 m3 ; (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, (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.