Solved: Given two vectors (a) find the magnitude of each

The following conversions occur frequently in physics andare very useful. (a) Use and to con-vert 60 mph to units of (b) The acceleration of a freely fallingobject is Use to express this accelerationin units of (c) The density of water is Convertthis density to units of

Neptunium. In the fall of 2002, a group of scientists atLos 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 chain reaction. This element has a density of \(19.5\mathrm{\ g}/\mathrm{cm}^3\). What would be the radius of a sphere of this material that has a critical mass?

BIO (a) The recommended daily allowance (RDA) of thetrace metal magnesium is for males. Express thisquantity in (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 (riboflavin), and the RDA is How many such tablets should a person take each day to get the proper amount of this vitamin, assuming that he gets none from any other sources? (d) The RDA for the trace element selenium is Express this dose in

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 0.000070 g>day. mg>day. 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 towrite the total distance covered by the train as 890,010 m? Explain.

With a wooden ruler you measure the length of a rectangu- lar piece of sheet metal to be 12 mm. You use micrometer calipers to measure the width of the rectangle and obtain the value 5.98 mm. Give your answers to the following questions to the correct number of significant figures. (a) What is the area of the rectangle? (b) What is the ratio of the rectangles width to its length? (c) What is the perimeter of the rectangle? (d) What is the difference between the length and width? (e) What is 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 Determine the percent error in this approximate value. (There are 365.24 days in one year.)

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 mi per year, and that the average car gets 20 miles per gallon.

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

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

How many words are there in this book?

Four astronauts are in a spherical space station. (a) If, as is typical, each of them breathes about \(500\mathrm{\ cm}^3\) 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?

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

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

. In Wagners 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 and its value is about $10 per gram (although this varies).

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.)

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

Hearing rattles from a snake, you make two rapid dis- placements 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 m; (b) 0.6 m; (c) 3.0 m.

A postal employee drives a delivery truck along the route shown in Fig. E1.27. Determine the magnitude and direction of the resultant displacement by drawing a scale diagram. (See also Exer- cise 1.34 for a different approach to this same problem.)

For the vectors and in Fig. E1.28, 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 Exer- cise 1.35 for a different ap- proach to this problem.)

A spelunker is survey- ing a cave. She follows a pas- sage 180 m straight west, then 210 m in a direction east of south, and then 280 m at east of north. After a fourth unmea- sured 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 1.69 for a different approach to this problem.)

Let the angle be the angle that the vector makes with the measured counterclockwise from that axis. Find the angle for a vector that has the following components: (a) (b) (c) (d)

Compute the x- and y-components of the vectors \(\vec{A}, \vec{B}, \vec{C}\), and \(\vec{D}\) in Fig. E1.28.

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 \(\vec{A}\)?

Vector has y-component . makes an angle of counterclockwise from the y-axis. (a) What is the x-component of ? (b) What is the magnitude of ?

A postal employee drives a delivery truck over the route shown in Fig. E1.27. 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 using the method of components.

For the vectors and in Fig. E1.28, 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

Find the magnitude and direction of the vector represented by the following pairs of components: (a) Ax = -8.60 cm, B S A S A . S B S ; B S A S A ; S B S ; B S A S A S A S 32.0 + A S A Ay = +13.0 m S A S A S y-component A Ax = -16.0 m S -y-axis A 34.0 S D S C S B , S A , S , A Ay = -1.00 m. A x = -2.00 m, A y = 1.00 m; x = -2.00 m, Ay = 5.20 cm; Ax = -9.70 m, Ay = -2.45 m; Ax = 7.75 km, Ay = -2.70 km.

A disoriented physics professor drives 3.25 km north, then 2.90 km west, and then 1.50 km south. Find the magnitude and direc- tion of the resultant displacement, using the method of components. In a vector-addition diagram (roughly to scale), show that the result- ant displacement found from your diagram is in qualitative agree- ment with the result you obtained using the method of components.

Two ropes in a vertical plane exert equal-magnitude forces on a hanging weight but pull with an angle of 86.0 between them. What pull does each one exert if their resultant pull is 372 N directly upward?

Vector \(\vec{A}\) is 2.80 cm long and is above the x-axis in the first quadrant. Vector \(\vec{B}\) 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) \(\vec{A}+\vec{B}\); (b) \(\vec{A}-\vec{B}\); (c) \(\vec{B}-\vec{A}\). 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 \(\vec{A}\): (a) \(\vec{A}=5.0 \hat{\imath}-6.3 \hat{\jmath}\); (b) \(\vec{A}=11.2 \hat{\jmath}-9.91 \hat{\imath}\); (c) \(\vec{A}=-15.0 \hat{\imath}+22.4 \hat{\boldsymbol{J}}\); (d) \(\overrightarrow{\boldsymbol{A}}=5.0 \hat{\boldsymbol{B}} \text {, where } \overrightarrow{\boldsymbol{B}}=4 \hat{\imath}-6 \hat{\boldsymbol{\jmath}}\)

Write each vector in Fig. E1.28 in terms of the unit vectors \(\hat{i}\) and \(\hat{j}\).

Given two vectors and (a) find the magnitude of each vector; (b) write an expres- sion for the vector difference using unit vectors; (c) find the magnitude and direction of the vector difference (d) In a vector diagram show and and also show that your diagram agrees qualitatively with your answer in part (c).

(a) Write each vector in Fig. E1.43 in terms of the unit vectors and (b) Use unit vectors to express the vector where (c) Find the magnitude and direc- tion of

(a) Is the vector a unit vector? Jus- tify 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.

For the vectors \(\vec{A},\ \vec{B}\) and \(\vec{C}\) in Fig. E1.28, find the scalar products (a) \(\vec{A} \cdot \vec{B}\); (b) \(\vec{B} \cdot \vec{C}\); (c) \(\vec{A} \cdot \vec{C}\).

(a) Find the scalar product of the two vectors and given in Exercise 1.42. (b) Find the angle between these two vectors.

Find the angle between each of the following pairs of vectors: B S A S A (a) (b) (c)

Find the vector product \(\vec{A} \times \vec{B}\) (expressed in unit vectors) of the two vectors given in Exercise 1.42. What is the magnitude of the vector product?

For the vectors and in Fig. E1.28, (a) find the magni- tude and direction of the vector product (b) find the mag- nitude and direction of

For the two vectors in Fig. E1.39, (a) find the magnitude and direction of the vector product \(\vec{A} \times \vec{B}\); (b) find the magnitude and direction of \(\vec{B} \times \vec{A}\).

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

The vector \(\vec{A}\) is 3.50 cm long and is directed into this page. Vector \(\vec{B}\) 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 \(\vec{A} \times \vec{B}\) measured in \(\mathrm{cm}^{2}\). In a diagram, show your coordinate system and the vectors \(\overrightarrow{\boldsymbol{A}}, \overrightarrow{\boldsymbol{B}} \text {, and } \overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}\).

Given two vectors and do the following. (a) Find the mag- nitude of each vector. (b) Write an expression for the vector differ- ence using unit vectors. (c) Find the magnitude of the vector difference Is this the same as the magnitude of Explain.

An acre, a unit of land measurement still in wide use, 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?

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 what is its radius expressed (a) in kilometers and (b) as a multiple of earths radius? Consult Appendix F for astronomi- cal data.

The Hydrogen Maser. You can use the radio waves generated by a hydrogen maser as a standard of frequency. The fre- quency 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 hydro- gen 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 (d) By how many seconds would a hydrogen maser clock be off after a time interval equal to the age of the earth?

BIO Breathing Oxygen. The density of air under stan- dard laboratory conditions is 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 1 2 L 1.29 kg>m3 , 4.6 * 109 years? 11.76 g>cm3 2, 1 1 8 mi2 B S A S ? A S B S . A S B S B S 3.00n 1.00 n 3.00k N, A S -2.00n 3.00n 4.00k N A S : B S B . S A , S , cm2 A . S : B S , B S A S A S : B S . A S # B S ; B S A S B S : A S . A S : B S ; D S : A S . A S : D S ; D S A S A S : B S A S -4.00n 2.00n and B S 7.00n 14.00n A S 3.00n 5.00n and B S 10.00n 6.00n A S -2.00n 6.00n and B S 2.00n 3.00n 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 \(7.60 \pm 0.01\) cm long and \(1.90 \pm 0.01\) 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; see Challenge Problem 1.98.)

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 \(8.50 \pm 0.02\) cm and a thickness of \(0.050 \pm 0.005\) 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 \(1.0\times10^3\mathrm{\ kg}/\mathrm{m}^3\), 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 honey bee

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 for differ- ent elements, measured in atomic mass units; you can find the value of an atomic mass unit, or 1 u, in Appendix E.)

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.)

How much would it cost to paper the entire United States (including Alaska and Hawaii) with dollar bills? What would be 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 earths 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).

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.

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

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 meas- ured in the sense from the toward the The result- ant of the two displacements should also have a magnitude of 6.40 cm, but a direction measured in the sense 22.0 from the toward the (a) Draw the vector-addition diagram for these vectors, roughly to scale. (b) Find the components of (c) Find the magnitude and direction of

Emergency Landing. A plane leaves the airport in Galisteo and flies 170 km at \(68^{\circ}\) of north and then changes direction to fly 230 km at \(48^{\circ}\) 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?

As noted in Exercise 1.29, a spelunker is surveying a cave. She follows a passage 180 m straight west, then 210 m in a direction of south, and then 280 m at of north. After a fourth unmeasured 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 agree- ment with your numerical solution.

(a) Find the magnitude and direction of the vector that is the sum of the three vectors and in Fig. E1.28. In a dia- gram, show how is formed from these three vectors. (b) Find the magnitude and direction of the vector In a dia- gram, show how is formed from these three vectors.

A rocket fires two engines simultaneously. One produces a thrust of 480 N directly forward, while the other gives a 513-N thrust at 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.

A sailor in a small sailboat encounters shifting winds. She sails 2.00 km east, then 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. P1.72). 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.

Dislocated Shoulder. A patient with a dislocated shoulder is put into a traction apparatus as shown in Fig. P1.73. The pulls \(\overrightarrow{\boldsymbol{A}} \text { and } \overrightarrow{\boldsymbol{B}}\) 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?

On a training flight, a student pilot flies from Lincoln, Nebraska, to Clarinda, Iowa, then to St. Joseph, Missouri, and then to Manhattan, Kansas (Fig. P1.74). 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.

Equilibrium. We say an object is in equilibrium if all the forces on it balance (add up to zero). Figure P1.75 shows a beam weighing 124 N that is supported in equilibrium by a 100.0-N pull and a force 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.

Getting Back. An explorer in the dense jungles of equatorial Africa leaves his hut. He takes 40 steps northeast, then 80 steps \(60^{\circ}\) 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.

A graphic artist is creating a new logo for her companys website. In the graphics program she is using, each pixel in an image file has coordinates where the origin is at the upper left corner of the image, the points to the right, and the points down. Distances are measured in pixels. (a) The artist draws a line from the pixel location to the location She wishes to draw a second line that starts at is 250 pixels long, and is at an angle of 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.

A ship leaves the island of Guam and sails 285 km at \(40.0^{\circ}\) 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?

BIO Bones and Muscles. A patient in therapy has a forearm that weighs 20.5 N and that lifts a 112.0-N weight. These two forces have direction vertically downward. The only other significant forces on his forearm come from the biceps muscle (which acts perpendicularly to the forearm) and the force at the elbow. If the biceps produces a pull of 232 N when the forearm is raised 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 are hungry and decide to go to your favorite neigh- borhood fast-food restaurant. You leave your apartment and take the elevator 10 flights down (each flight is 3.0 m) and then go 15 m south to the apartment exit. You then proceed 0.2 km east, turn north, and go 0.1 km to the entrance of the restaurant. (a) Deter- mine the displacement from your apartment to the restaurant. Use unit vector notation for your answer, being sure to make clear 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 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.

A fence post is 52.0 m from where you are standing, in a direction north of east. A second fence post is due south from you. What is the distance of the second post from you, if the dis- tance between the two posts is 80.0 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 west of north. Jane walks 16.0 m in a direction 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?

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 south of east from Pauls location. How far is George from John? What is the direction of Georges location from that of John?

You are camping with two friends, 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 south of east. Karls tent is 32.0 m from yours, in the direction north of east. What is the distance between Karls tent and Joes tent?

Vectors and have scalar product and their vec- tor product has magnitude . What is the angle between these two vectors?

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 where one of the C-H bonds is in the direction of \(\hat{\imath}+\hat{\jmath}+\hat{k}\), an adjacent C-H bond is in the \(\hat{\imath}-\hat{\jmath}-\hat{k}\) direction. Calculate the angle between these two bonds.

Vector has magnitude 12.0 m and vector has magni- tude 16.0 m. The scalar product is . What is the mag- nitude of the vector product between these two vectors?

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 2A2 + B2 + 2AB cos f (b) If and have the same magnitude, for which value of will their vector sum have the same magnitude as or

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 sys- tem (Fig. P1.91). 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 \(\overrightarrow{A}\) has magnitude 6.00 m and vector \(\overrightarrow{B}\) has magnitude 3.00 m. The vector product between these two vectors has magnitude . What are the two possible values for the scalar product of these two vectors? For each value of \(\overrightarrow{A}\cdot\overrightarrow{B}\), draw a sketch that shows \(\overrightarrow{A}\) and \(\overrightarrow{B}\) and explain why the vector products in the two sketches are the same but the scalar products differ.

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 1.53.

You are given vectors \(\vec{A}=5.0 \hat{\imath}-6.5 \hat{\jmath}\) and \(\overrightarrow{\boldsymbol{B}}=-3.5 \hat{\imath}+7.0 \hat{\boldsymbol{\jmath}}\). 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 and have magnitudes and Their vector product is What is the angle between and

Later in our study of physics we will encounter quantities represented by (a) Prove that for any three vectors and (b) Calculate for the three vectors with magnitude and angle measured in the sense from the toward the with and and with magni- tude 6.00 and in the Vectors and are in the xy-plane.

The length of a rectangle is given as and its width as (a) Show that the uncertainty in its area A is 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 and 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 displace- ments, with the origin taken to be the position of the ball before the play starts. In a certain pass play, the receiver starts at +1.0n 5.0n, where the units are yards, is to the right, and n H h. L l, W w, a = Lw + lW. W w. L l B S A S +z-direction. C S B B = 4.00 uB = 63.0, S +y-axis, uA = 26.0 +x-axis A A = 5.00 S 1A S : B S 2 # C S A S # 1B S : C S 2 = 1A S : B S 2 # C S C . S B , S A , S , 1A S : B S 2 # C S . B S A ? S A S : B S -5.00k B = 3.00. N 2.00N. B A = 3.00 S A S C S . B S C S A S , C S C S B S -3.5n 7.0n. A S 5.0n 6.5n B S B 39.0 S A 28.0 S +48.0 m2 B S A S B S A S A S # B S 12.0 m2 B S A S B S A ? S B f S is downfield. Subsequent displacements of the receiver are (in motion before the snap), (breaks downfield), (zigs), and (zags). Meanwhile, the quarterback has dropped straight back to a position 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.)

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 touched down on the Martian surface, the positions of the earth and Mars were given by these coordinates: -7.0n. -6.0n 4.0n +12.0n 18.0n +9.0n +11.0n n xyz Earth 0.3182 AU 0.9329 AU 0.0000 AU Mars 1.3087 AU 0.4423AU 0.0414 AU Figure P1.91 x y z b c d a In these coordinates, the sun is at the origin and the plane of the earths orbit is the xy-plane. The earth passes through the 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 km, the average dis- tance from the earth to the sun. (a) In a diagram, show the posi- tions of the sun, the earth, and Mars on December 3, 1999. (b) Find the following distances in AU on December 3, 1999: (i) from the sun to the earth; (ii) from the sun to Mars; (iii) from 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 location at midnight on December 3, 1999. (When it is midnight at your loca- tion, the sun is on the opposite side of the earth from you.)

Navigating in the Big Dipper. All 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 P1.101 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 . (a) Alkaid and Merak are 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? 25.6o 9.461 * 1015 m 1.496 * 108 +x-axis Figure P1.101 Mizar 73 ly Megrez 81 ly Dubhe 105 ly Merak 77 ly Phad 80 ly Alioth 64 ly Alkaid 138 ly

The vector called the position vec- tor, points from the origin to an arbitrary point in space with coordinates Use what you know about vectors to prove the following: All points that satisfy the equation 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. An Bn Ck

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 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 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 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.)

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

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 \(cm^3\) 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?

Problem 7DQ What physical phenomena (other than a pendulum or cesium clock) could you use to define a time standard?

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

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

While driving in an exotic foreign land you see a speed limit sign on a highway that reads 180,000 furlongs per fortnight. How many miles per hour is this? (One furlong is \(1 / 8 \text { mile }\) and a fortnight is 14 days. A furlong originally referred to the length of a plowed furrow.) Equation Transcription: Text Transcription: 1/8 mile

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 \(55.0 \mathrm{mpg}\) (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 \(\mathrm{km} / \mathrm{L}\) ( \(\mathrm{L}=\) 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 \(1500 \mathrm{~km}\)?

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 ?r3h. Explain why this cannot be right.

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 \(32 \ \mathrm{ft} / \mathrm{s}^{2}\) .Use 1 ft = 30.48 cm to express this acceleration in units of \(\mathrm{m} / \mathrm{s}^{2}\). (c) The density of water is \(1.0 \mathrm{\ g} / \mathrm{cm}^{3}\). Convert this density to units of \(\mathrm{kg} / \mathrm{m}^{3}\).

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.

Neptunium. In the fall of 2002, a group of 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 chain reaction. This element has a density of \(19.5 \mathrm{\ g} / \mathrm{cm}^{3}\). 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.

With a wooden ruler you measure the length of a rectangular piece of sheet metal to be 12 mm. You use micrometer calipers to measure the width of the rectangle and obtain the value 5.98 mm. Give your answers to the following questions to the correct number of significant figures. (a) What is the area of the rectangle? (b) What is the ratio of the rectangle’s width to its length? (c) What is the perimeter of the rectangle? (d) What is the difference between the length and width? (e) What is the ratio of the length to the width?

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

A useful and easy-to-remember approximate value for the number of seconds in a year is \(\pi \times 10^{7}\). 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.

If \(\vec{C}\) is the vector sum of \(\vec{A}\) and \(\vec{B}\),\(\vec{C}=\vec{A}+\vec{B}\), what must be true about directions and magnitudes of \(\vec{A}\) and \(\vec{B}\) if \(C=A+B\)? What must be the true about the directions and magnitudes of \(\vec{A}\) and \(\vec{B}\) if \(c=0\)? Equation Transcription: Text Transcription: C A B C A + B A B C=A+B A B C=0

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

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

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

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. Equation Transcription: Text Transcription: Vec A dot Vec A Vec A times Vec A

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?

Let \(\vec{A}\) represent any nonzero vector. Why is \(\vec{A} / \mathrm{A}\) a unit vector, and what is its direction? If ? is the angle that \(\vec{A}\) makes with the \(+x \text {-axis }\), explain why ((\vec{A} / \mathrm{A})\)?î is called the direction cosine for that axis. Equation Transcription: Text Transcription: A A/A A +x-axis (A/A)

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

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. Equation Transcription: ); ); ); ); )? Text Transcription: vec{A}dot vec{B}-vec{C} vec{A}-vec{B}times vec{C} vec{A} cdot vec{B} times vec{C} vec{A} times vec{B} times vec{C} vec{A} times vec{B} cdot vec{C} ?

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.)

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

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 \(\vec{A} \text { and } \vec{B}\) 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 cross product.) Equation Transcription: and () Text Transcription: vec{A} text { and } vec{B} vec{A} cdot vec{A} times vec{B})=0

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.)

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

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

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

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.

A postal employee drives a delivery truck along the route shown in Fig. E1.27. Determine the magnitude and direction of the resultant displacement by drawing a scale diagram. (See also Exercise 1.34 for a different approach to this same problem.)

For the vectors \(\vec{A} \text { and } \vec{B}\) in Fig. E1.28, use a scale drawing to find the magnitude and direction of (a) the vector sum \(\vec{A}+\vec{B}\) and (b) the vector difference \(\vec{A}-\vec{B}\). Use you answers to find the magnitude and direction of ©\( -\vec{A}-\vec{B}\) and (d) \(\vec{B}-\vec{A}\). (See also Exercise 1.35 for a different approach to this problem.) Equation Transcription: and Text Transcription: vec{A} \text { and } \vec{B vec{A}+\vec{B} vec{A}-\vec{B -\vec{A}-\vec{B} vec{B}-\vec{A}

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 unmeasured 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 1.69 for a different approach to this problem.) Equation Transcription: Text Transcription: 45° degree 30° degree

Let the angle \(\boldsymbol{P}\) be the angle that the vector \(\vec{A}\) makes with the \(+x \text {-axis, }\)measured counterclockwise from that axis. Find the angle \(\boldsymbol{P}\) for a vector that has the following components: (a)\(\mathrm{A}_{\mathrm{x}}=2.00\mathrm{~m},\mathrm{~A}_{\mathrm{v}}=-1.00 \mathrm{~m}\) (b)\(\mathrm{A}_{\mathrm{x}}=2.00\mathrm{~m}, \mathrm{~A}_{\mathrm{y}}=1.00 \mathrm{~m}\) (c) \(A_{x}=-2.00 \mathrm{~m}, A_{y}=1.00 \mathrm{~m}\) (d) \(A_{x}=-2.00 \mathrm{~m}, A_{y}=-1.00 \mathrm{~m}\) Equation Transcription: Text Transcription: P A +x-axis P Ax=2.00 m, Ay = -1.00m; Ax=2.00 m, Ay = -1.00m; Ax=2.00 m, Ay = -1.00m; Ax=2.00 m, Ay = -1.00m;

Compute the \(x \text { - and } y \text { - }\) components of the vectors \(\vec{A}, \vec{B}, \vec{C}, \text { and } \vec{D}\)in Fig. E1.28. Equation Transcription: Text Transcription: x-and y- A,B,C and D

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

Vector \(\vec{A}\) has y-component \(A_{y}=+13.0 \mathrm{~m}\). \(\vec{A}\) makes an angle of \(32.0^{\circ}\) counterclockwise from the y-axis. (a) What is the x-component of \(\vec{A}\)? (b) What is the magnitude of \(\vec{A}\)?

A postal employee drives a delivery truck over the route shown in Fig. E1.27. 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.

For the vectors \(\vec{A} \text { and } \vec{B}\)in Fig. E1.28, use the method of components to find the magnitude and direction of (a) the vector \(sum\vec{A}+\vec{B}\); (b) the vectors sum \(\vec{B}+\vec{A}\); © the vector difference \(\vec{A}-\vec{B}\); (d) the vector difference \(\vec{B}-\vec{A}\). Equation Transcription: and Text Transcription: vec{A} \text { and } \vec{B} sum\vec{A}+\vec{B} vec{B}+\vec{A} vec{A}-\vec{B} vec{B}-\vec{A}

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.

Problem 38E Two ropes in a vertical plane exert equal-magnitude forces on a hanging weight but pull with an angle of 86.0° between them. What pull does each one exert if their resultant pull is 372 N directly upward?

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

In each case, find the ????- and ????-components of vector \(\vec{A}\) (a) \(\vec{A}=5.0 \hat{\imath}-6.3 \hat{j}\) (b) \(\vec{A}=11.2 \hat{\jmath}-9.91 \hat{i} ;\) (c) \(\vec{A}=-15.0 \hat{\imath}+22.4 \hat{j} ;\) (d) \(\vec{A}=5.0 \hat{B}, \text { where } \vec{B}=4 \hat{\imath}-6 \hat{\jmath}\) Equation Transcription: = 5.0 - 6.3; = 11.2- 9.91; = 15.0 + 22.4; = 5.0, where = 4- 6. Text Transcription: vec{A} vec{A}=5.0 \hat{\i}-6.3 \hat{j} vec{A}=11.2 \hat{\j}-9.91 \hat{i} vec{A}=-15.0 \hat{\i}+22.4 \hat{j} \vec{A}=5.0 \hat{B}, \text { where } \vec{B}=4 \hat{\i}-6 \hat{\j}

Write each vector in Fig. E1.28 in terms of the unit vectors \(\hat{i}\) and \(\hat{j}\) Equation Transcription: Text Transcription: \hat i \hat j

Given two vectors \(\vec{A}=4.00 \hat{i}+7.00 \hat{\jmath}\) and \(\vec{B}=5.00 \hat{i}-2.00 \hat{\jmath}\) (a) find the magnitude of each vector; (b) write an expression for the vector difference \(\vec{A}-\vec{B}\) using unit vectors; (c) find the magnitude and direction of the vector difference \(\vec{A}-\vec{B}\) (d) In a vector diagram show \(\vec{A}, \vec{B}\) and \(\vec{A}-\vec{B}\) also show that your diagram agrees qualitatively with your answer in part (c). Equation Transcription: = 4.00 + 7.00 = 5.00-2.00, - - , - Text Transcription: vec{A}=4.00 \hat{i}+7.00 \hat{\j} vec{B}=5.00 \hat{i}-2.00 \hat{\j} vec{A}-\vec{B}vec{A}-\vec{B} vec{A}, \vec{B} vec{A}-\vec{B}

1. Write each vector in Fig. E1.43 in terms of the unit vectors \(\hat{\imath} \text { and } \hat{\jmath}\). 2. (b) Use unit vectors to express the vector \(\(\vec{C}\) where \vec{C}=3.00 \vec{A}-4.00 \vec{B}\) 3. (c) Find the magnitude and direction of \(\vec{C}\) Equation Transcription: and = 3.00- 4.00 Text Transcription: hat{\imath} \text { and } \hat{\jmath} vec{C} vec{C}=3.00 \vec{A}-4.00 \vec{B} vec{C}

(a)Is the vector \((\hat{\imath}+\hat{\jmath}+\hat{k})\) a unit vector? Justify your answer. (b) Can a unit vector have any components with magnitude greater than unity? Can it have negative components? In each case, justify your answer. (c) If \(\vec{A}=a(3.0 \hat{i}+4.0 \hat{\jmath})\), where ???? is a constant, determine the value of ???? that makes \(\vec{A}\) a unit vector. Equation Transcription: ( ++) = (3.0+4.0) Text Transcription: Hat{\i}+\hat{\j}+\hat{k} vec{A}=a(3.0 \hat{i}+4.0 \hat{\j} vec{A}

For the vectors \(\vec{A}, \vec{B}, \text { and } \vec{C}\) in Fig. E1.28, find the scalar products (a) \(\vec{A} \cdot \vec{B}\); (b) \(\vec{B} \cdot \vec{C}\); (c) \(\vec{A} \cdot \vec{C}\). Equation Transcription: , ,and ; ; . Text Transcription: vec{A}, \vec{B}, \text { and } \vec{C} vec{A} \cdot \vec{B} vec{B} \cdot \vec{C} vec{A} \cdot \vec{C}

(a) Find the scalar product of the two vectors \(\overrightarrow{\boldsymbol{A}} \text { and } \overrightarrow{\boldsymbol{B}}\) given in Exercise 1.42. (b) Find the angle between these two vectors.

Find the angle between each of the following pairs of vectors: 1. \(\vec{A}=-2.00 \hat{\hat{i}}+6.00 \hat{\jmath} \text { and } \vec{B}=2.00 \hat{\mathbf{i}}-3.00 \hat{\jmath}\) 2. \(\vec{A}=3.00 \hat{\mathbf{i}}+5.00 \hat{\mathrm{j}} \text { and } \vec{B}=10.00 \hat{\mathbf{i}}+6.00 \hat{\mathrm{j}}\) 3. \(\vec{A}=-4.00 \hat{\imath}+2.00 \hat{\jmath} \text { and } \vec{B}=7.00 \hat{\hat{i}}+14.00 \hat{j}\) Equation Transcription: = -2.00+ 6.00 and = 2.00 - 3.00 = 3.00+ 5.00 and = 10.00+ 6.00 = -4.00+ 2.00 and = 7.00+ 14.00 Text Transcription: vec{A}=-2.00 \hat{\hat{i}}+6.00 \hat{\j} \text { and } \vec{B}=2.00 \hat{\bf{i}}-3.00 \hat{\j} vec{A}=3.00 \hat{\bf{i}}+5.00 \hat{\{j}} \text { and } \vec{B}=10.00 \hat{\bf{i}}+6.00 \hat{\{j} vec{A}=-4.00 \hat{\i}+2.00 \hat{\j} \text { and } \vec{B}=7.00 \hat{\hat{i}}+14.00 \hat{j}

Find the vector product \(\vec{A} \times \vec{B}\) (expressed in unit vectors) of the two vectors given in Exercise 1.42. What is the magnitude of the vector product? Equation Transcription: Text Transcription: vec{A} \times \vec{B}

For the vectors \(\vec{A}\) and \(\vec{D}\) in Fig. E1.28, (a) find the magnitude and direction of the vector product \(\vec{A} \times \vec{D}\); (b) find the magnitude and direction of \(\vec{D} \times \vec{A}\).

For the two vectors in Fig. E1.39, (a) find the magnitude and direction of the vector product \(\vec{A} \times \vec{B}\); (b) find the magnitude and direction of \(\vec{B} \times \vec{A}\).

For the two vectors \(\vec{A} \text { and } \vec{B}\) in Fig. E1.43, (a) find the scalar product \(\vec{A} \cdot \vec{B}\); (b) find the magnitude and direction of the vector product \(\vec{A} \times \vec{B}\). Equation Transcription: and ; . Text Transcription: vec{A} \text { and } \vec{B} vec{A} \cdot \vec{B} vec{A} \times \vec{B}

The vector \(\vec{A}\) is 3.50 cm long and is directed into this page. Vector \(\vec{B}\) 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 \(\vec{A} \times \vec{B}\) measured in \(\mathrm{cm}^{2}\). In a diagram, show your coordinate system and the vectors \(\vec{A}, \vec{B}, \text { and } \vec{A} \times \vec{B}\) Equation Transcription: and Text Transcription: vec{A} vec{B} vec{A} \times \vec{B} {cm}^{2} vec{A}, \vec{B}, \text { and } \vec{A} \times \vec{B}

Given two vectors \(\vec{A}=-2.00 \hat{\mathbf{i}}+3.00 \hat{\mathrm{j}}+4.00 \hat{\mathbf{k}} \text { and } \vec{B}=3.00 \hat{\mathbf{i}}+1.00 \hat{\mathrm{j}}-3.00 \hat{\mathbf{k}}\), do the following. (a) Find the magnitude of each vector. (b) Write an expression for the vector difference \(\vec{A}-\vec{B}\) using unit vectors. (c) Find the magnitude of the vector difference \(\vec{A}-\vec{B}\) Is this the same as the magnitude of \(\vec{B}-\vec{A}\)? Explain. Equation Transcription: = -2.00+4.00 and = 3.00+ 1.00- 3.00 Text Transcription: vec{A}=-2.00 \hat{\bf{i}}+3.00 \hat{\{j}}+4.00 \hat{\bf{k}} \text { and } \vec{B}=3.00 \hat{\bf{i}}+1.00 \hat{\{j}}-3.00 \hat{\bf{k} vec{A}-\vec{B} vec{A}-\vec{B} vec{B}-\vec{A}

An acre, a unit of land measurement still in wide use, has a length of one furlong \(\left(\frac{1}{8} \mathrm{mi}\right)\) 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? Equation Transcription: Text Transcription: (left(\frac{1}{8} \{mi}\right)

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?

BIO Breathing Oxygen. The density of air under standard laboratory conditions is \(1.29 \mathrm{~kg} / \mathrm{m}^{3}\) and about 20% of that air consists of oxygen. Typically, people breathe about \(\frac{1}{2} L\) ofair 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? Equation Transcription: 1.29kg/m3 L Text Transcription: 1.29{~kg}{m}^{3 frac{1}{2} L

A rectangular piece of aluminum is \(7.60 \pm 0.01 \mathrm{~cm}\) cm long and \(1.90 \pm 0.01\) 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; see Challenge Problem 1.98.) Equation Transcription: Text Transcription: 7.60 \pm 0.01 \{~cm} 1.90 \pm 0.01 \{~cm}

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.

Problem 60P BIO Biological tissues are typically made up of 98% water. Given that the density of water is 1.0 X 103 kg/m3, estimate the mass of (a) the heart of an adult human; (b) a cell with a diameter of 0.5 µm; (c) a honeybee.

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 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 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.

Three horizontal ropes pull on a large stone stuck in the ground, producing the vector forces \(\overrightarrow{\boldsymbol{A}}, \overrightarrow{\boldsymbol{B}} \text {, and } \overrightarrow{\boldsymbol{C}}\) shown in Fig. P1.66. Find the magnitude and direction of a fourth force on the stone that will make the vector sum of the four forces zero.

You are to program a robotic arm on an assembly line to move in the \(x y \text {-plane }\). Its first displacement is \(\vec{A}\) its second displacement is \(\vec{B}\) of magnitude 6.40 cm and direction \(6.30^{\circ}\) measured in the sense from the \(+x \text {-axis }\) toward the \(-y \text {-axis }\).The resultant \(\vec{C}=\vec{A}+\vec{B}\) of the two displacements should also have a magnitude of 6.40 cm, but a direction \(22.0^{\circ}\) measured in the sense from the \(+x \text {-axis }\) toward the \(+y \text {-axis. }\). (a) Draw the vector-addition diagram for these vectors, roughly to scale. (b) Find the components of \(\vec{A}\) (c) Find the magnitude and direction of \(\vec{A}\) Equation Transcription: 6.30° = + 22.0° Text Transcription: Xy-plane Vec A Vec B 6.30^circ +x-axis -y-axis vec{C}=\vec{A}+\vec{B} 22.0^{\circ} +x-axis -y-axis Vec A Vec A

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?

As noted in Exercise 1.29, a spelunker is surveying a cave. She follows a passage 180 m straight west, then 210 m in a direction \(45^{\circ}\) of south, and then 280 m at \(30^{\circ}\) of north. After a fourth unmeasured 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.

(a) Find the magnitude and direction of the vector \(\vec{R}\) that is the sum of the three vectors \(\vec{A}, \vec{B}, \text { and } \vec{C}\) in Fig. E1.28. In a diagram, show how \(\vec{R}\) is formed from these three vectors. (b) Find the magnitude and direction of the vector \(\vec{S}=\vec{C}-\vec{A}-\vec{B}\). In diagram, show how \(\vec{S}\) is formed from these three vectors. Equation Transcription: and = - - Text Transcription: vec{R} vec{A}, \vec{B},{ and } \vec{C} vec{R} vec{S}=\vec{C}-\vec{A}-\vec{B} vec{S}

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.

A sailor in a small sailboat encounters shifting winds. She sails \(2.00 \mathrm{~km}\) east, then \(3.50 \mathrm{~km}\) southeast, and then an additional distance in an unknown direction. Her final position is \(5.80 \mathrm{~km}\) directly east of the starting point (Fig. P1.72). 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. Equation Transcription: Text Transcription: 2.00 km 3.50 km 5.80 km

BIO Dislocated Shoulder. A patient with a dislocated shoulder is put into a traction apparatus as shown in Fig. P1.73. The pulls \(\vec{A} \text { and } \vec{B}\) 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? Equation Transcription: Text Transcription: vec{A} \text { and } \vec{B}

On a training flight, a student pilot flies from Lincoln, Nebraska, to Clarinda, Iowa, then to St. Joseph, Missouri, and then to Manhattan, Kansas (Fig. P1.74). 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. Equation Transcription: 0° 90° 180° 270° Text Transcription: (0^{\circ} 90^{\circ} 180^{\circ} 270^{\circ}

Equilibrium. We say an object is in equilibrium if all the forces on it balance (add up to zero). Figure P1.75 shows a beam weighing 124 N that is supported in equilibrium by a100.0-N pull and a force \(\vec{F}\) 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 \(\vec{F}\) (b) Check the reasonableness of your answer in part (a) by doing a graphical solution approximately to scale. Equation Transcription: Text Transcription: Vec f Vec f

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.

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 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 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 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?

Vectors \(\vec{A} \text { and } \vec{B}\) have scalar product6.00 and their vector product has magnitude +9.00. What is the angle between these two vectors? Equation Transcription: -6.00 +9.00 Text Transcription: Vec A and vec B -6.00 +9.00

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 where one of the \(\mathrm{C}-\mathrm{H}\) bonds is in the direction of \(\hat{\imath}+\hat{\jmath}+\hat{k}\), an adjacent \(\mathrm{C}-\mathrm{H}\) bond is in the \(\hat{\imath}-\hat{\jmath}-\hat{k}\) direction. Calculate the angle between these two bonds

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 \(90.0 \mathrm{~m}^{2}\). What is the magnitude of the vector product between these two vectors? Equation Transcription: Text Transcription: Vec A Vec B Vec A dot vec B 90.0 m2

When two vectors \(\vec{A}\) and \(\vec{B}\) are drawn from a common point, the angle between them is \(\phi\). (a) Using vector techniques, show that the magnitude of their vector sum is given by \(\sqrt{A^{2}+B^{2}+2 A B \cos \ \phi}\) (b) If \(\vec{A}\) and \(\vec{B}\) have the same magnitude, for which value of \(\phi\) will their vector sum have the same magnitude as \(\vec{A}\) or \(\vec{B}\)?

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. P1.91). 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 \(\vec{A}\) has magnitude 6.00 m and vector \(\vec{B}\) has magnitude 3.00 m. The vector product between these two vectors has magnitude \(12.0 \mathrm{~m}^{2}\). What are the two possible values for the scalar product of these two vectors? For each value of \(\vec{A} \cdot \vec{B}\) draw a sketch that shows \(\vec{A} \text { and } \vec{B}\)and explain why the vector products in the two sketches are the same but the scalar products differ. Equation Transcription: Text Transcription: vec{A} vec{B} 12.0 \{~m}^{2} \vec{A} \cdot \vec{B} \vec{A} \text { and } \vec{B}

Obtain a unit vector perpendicular to the two vectors given in Exercise 1.53.

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

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

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 \(+1.0 \hat{\imath}-5.0 \hat{1}\) where the units are yards, \(\hat{\imath}\) is to the right, and \(\hat{\jmath}\) is downfield. Subsequent displacements of the receiver are \(+9.0 \hat{i}\)(in motion before the snap), \(+11.0 \hat{j}\) (breaks downfield), \(-6.0 \hat{i}+4.0 \hat{j}\) (zigs), and \(+12.0 \hat{i}+18.0 \hat{\jmath}\)(zags). Meanwhile, the quarterback has dropped straight back to a position -7.0?. 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 it numerically.) Equation Transcription: +1.0- 5.0 +9.0 +11.0 -6.0 + 4.0 + 12.0 + 18.0 Text Transcription: +1.0 \hat{\i}-5.0 \hat{1} \hat{\i} \hat{\j} +9.0 hat î +11.0 hat ? -6.0î + 4.0 hat ? + 12.0î + 18.0 hat ?

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 touched down on the Martian surface, the positions of the earth and Mars were given by these coordinates: ???? ???? ???? _______________________________________ Earth 0.3182 AU 0.9329 AU 0.0000 AU Mars 1.3087 AU -0.4423 AU -0.0414 AU In these coordinates, the sun is at the origin and the plane of the earth’s orbit is the -plane. The earth passes through the +-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) In a diagram, show the positions of the sun, the earth, and Mars on December 3, 1999. (b) Find the following distances in AU on December 3, 1999: (i) from the sun to the earth; (ii) from the sun to Mars; (iii) from 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 location at midnight on December 3, 1999. (When it is midnight at your location,the sun is on the opposite side of the earth from you.) Equation Transcription: Text Transcription: 1.496 x 10^8km

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?

The vector \(\vec{r}=x \hat{\boldsymbol{i}}+y \hat{\boldsymbol{j}}+z \boldsymbol{k}\), called the position vector, points from the origin ( 0, 0, 0 ) to an arbitrary point in space with coordinates ( ????, ????, ???? ). Use what you know about vectors to prove the following: All points ( ????, ????, ???? ) that satisfy the equation A???? + B???? + C???? = 0, A, B, and C are constants, lie in a plane that passes through the origin and that is perpendicular to the vector \(4 \hat{\imath}+B \hat{\jmath}+C k\). Sketch this vector and the plane. Equation Transcription: = ++ and + B+ Text Transcription: \vec{r}=x \hat{\symbol{i}}+y \hat{\symbol{j}}+z \symbol{k} (x, y, z) (x, y, z) Ax+ By+Cz=0,A,B,and C 4 \hat{\i}+B \hat{\j}+C k

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.)

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 \(\vec{B}\)? Equation Transcription: Text Transcription: Vector A Vector B +48.0 m^2 Vector A 28.0deg Vector B 39.0deg Vector B

You are given vectors \(\vec{A}=5.0 \hat{i}-6.5 \hat{j}\) and \(\vec{B}=-3.5 \hat{i}+7.0 \hat{j}\). A third vector \(\vec{C}\) lies in the -plane. Vector 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 \(\vec{C}\). Equation Transcription: Text Transcription: Vector A=5.0i hat-6.5j hat Vector B=-3.5i hat+7.0j hat Vector C Vector C Vector A Vector C Vector B Vector C