Chapter 1: Problems (1.87 - 1.93)

1.87 The two vectors $\vec{A}$ and $\vec{B}$ are drawn from a common point, $\vec{C} = \vec{A} + \vec{B}$. a) Show that if $C^2 = A^2 + B^2$, the angle between the vectors $\vec{A}$ and $\vec{B}$ is $90^\circ$. b) Show that if $C^2 < A^2 + B^2$, the angle between the vectors $\vec{A}$ and $\vec{B}$ is greater than $90^\circ$. c) Show that if $C^2 > A^2 + B^2$, the angle between the vectors $\vec{A}$ and $\vec{B}$ is between $0^\circ$ and $90^\circ$.

1.88 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

$\sqrt{A^2 + B^2 +2AB\cos{\phi}}$

b) if $\vec{A}$ and $\vec{B}$ have the same magnitude, for which value of $\phi$ will their vector sum have the same magnitude of $\vec{A}$ or $\vec{B}$? c) Derive a result analogous to that in part (a) for the magnitude of the vector difference $\vec{A} - \vec{B}$. d) If $\vec{A}$ and $\vec{B}$ have the same magnitude, for what value of $\phi$ will $\vec{A} - \vec{B}$ have this same magnitude?

1.89 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. 1.35). Use vectors to compute a) the angle betwee the edge along the z-axis (line ab) and the diagonal from the origin to the opposite corner (line ad); b) the angle between line ac (the diagonal of a face) and line ad.

1.90 Obtain a unit vector perpendicular to the two vectors given in Problem 1.85.

1.91 You are given vectors $\vec{A} = 5.0\hat{i} - 6.5\hat{j}$ and $\vec{B} = -3.5\hat{i} + 7.0{j}$. A third vector $\vec{C}$ lies in the xy-plane. Vector $\vec{C}$. 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 $\vec{C}$.

1.92 Two vectors $\vec{A}$ and $\vec{B}$have magnitude A = 3.00 and B = 3.00. Their vector product is $\vec{A} \times \vec{B} = -5.00\hat{k} + 2.00\hat{i}$. What is the angle between $\vec{A}$ and $\vec{B}$?

1.93 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 $\vec{A}$, $\vec{B}$, 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-axis toward the +y-axis, $\vec{B}$ with B = 4.00 and $\theta_B = 63.0^\circ$, and $\vec{C}$ with magnitude 6.00 and in the +z-direction. Vectors $\vec{A}$ and $\vec{B}$ are in the xy-plane.

Chapter 1: Problems (1.82 - 1.86)

1.82 For the two vectors $\vec{A}$ and $\vec{B}$ in Fig. 1.30, a) find the scalar product $\vec{A} \cdot \vec{B}$; b) find the magnitude and direction of the vector product $\vec{A} \times \vec{B}$.

1.83 Figure 1.8c shows a parallelogram based on the two vectors $\vec{A}$ and $\vec{B}$. a) Show that the magnitude of the cross product of these two vectors is equal to the area of the parallelogram. (Hint: area = base $\times$ height.) b) What is the angle between the cross product and the plane of the parallelogram?

1.84 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 cm$^2$. In a diagram, show your coordinate system and the vectors $\vec{A}$, $\vec{B}$, and $\vec{A} \times \vec{B}$.

1.85 Given two vectors $\vec{A} = -2.00\hat{i} + 3.00\hat{j} + 4.00\hat{k}$ and $\vec{B} = 3.00\hat{i} + 1.00\hat{j} - 3.00\hat{k}$, do the following. a) Find the magnitude of each vector. b) Write an expression for the vector difference $\vec{A} - \vec{B}$. Is this the same as the magnitude of $\vec{B} - \vec{A}$? Explain.

1.86 Bond Angle in Methane. In the methane molecule, 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 $\hat{i} + \hat{j} + \hat{k}$, an adjacent C – H bond is in the direction of $\hat{i} - \hat{j} - \hat{k}$ direction. Calculate the angle between these two bonds.

Chapter 1: Problems (1.77 - 1.81)

1.77 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 $43^\circ$ 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)

1.78 You are hungry and decide to go to your favorite neighborhood 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) Determine the displacement from your apartment to the restaurant. Use unit-vector notation for your answer, being sure to make clear 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)?

1.79 You are canoeing on a lake. Starting at your camp on the shore, you travel 240 m in the direction $32^\circ$ south of east to reach a store to purchase supplies. You know the distance because you have located both your camp and the store on a map of the lake. On the return trip you travel distance B in the direction $48^\circ$ north of west, distance C in the direction $62^\circ$ south of west, and then you are back at your camp. You measure the directions of travel with your compass, but you don't know the distances. Since you are curious to know the total distance you rowed, use vector methods to calculate the distances B and C.

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

1.81 Vectors $\vec{A}$ and $\vec{B}$ are drawn from a common point. Vector $\vec{A}$ has magnitude A and angle $\theta_A$ measured in the sense from the +x-axis to the +y-axis. The corresponding quantities for vector $\vec{B}$ are B and $\theta_B$. Then $\vec{A} = A\cos\theta_A\hat{i} + A\sin\theta_A\hat{j}$, $\vec{B} = B\cos\theta_B\hat{i} + B\sin\theta_B\hat{j}$, and $\phi = |\theta_B - \theta_A|$ is the angle between $\vec{A}$ and $\vec{B}$. a) Derive Eq. (1.18) from Eq. (1.21). b) Derive Eq. (1.22) from Eq. (1.27).

Chapter 1: Problems (1.72 - 1.76)

1.72 On a training flight, a student pilot flies from Lincoln, Nebraska to Clarinda, Iowa, and then to St. Joseph, Missouri, and then to Manhattan, Kansas (Fig. 1.33). 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; b) the direction (relative to north) she must fly to get there. Illustrate your solution with a vector diagram.

1.73 A graphic artist is creating a new logo for her company's Web site. In the graphics program she is using, each pixel in an image file has coordinates (x, y), where the origin (0, 0) is at the upper-left corner of the image, the +x-axis points to the right, and the +y-axis points down. Distances are measured in pixels. a) The artist draws a line from the pixel location (10, 20) to the location (210, 200). She wishes to draw a second line that starts at (10, 20), is 250 pixels long, and is at an angle of $30^\circ$ 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.

1.74 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 him 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.

1.75 A ship leaves the island of Guam and sails 285 km at $40.0^\circ$ 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?

1.76 A boulder of weight w rests on a hillside that rises at a constant angle $\alpha$ above the horizontal, as shown in Fig. 1.34. Its weight is a force on the boulder that has direction vertically downward. a) In terms of $\alpha$ and w, what is the component of the weight of the boulder in the direction parallel to the surface of the hill? b) What is the component of the weight in the direction perpendicular to the surface of the hill? c) An air conditioner unit is fastened to a roof that slopes upward at an angle of $35.0^\circ$. In order that the unit not slide down the rood, the component of the unit's weight parallel to the roof cannot exceed 550 N. What is the maximum allowed weight of
the unit?

Chapter 1: Problems (1.67 - 1.71)

1.67 You are to program a robotic arm on an assembly line to move in the xy-plane. Its first displacement is $\vec{A}$; its second displacement is $\vec{B}$, of magnitude 6.40 cm and direction $63^\circ$ measured in the sense from the +x-axis toward the -y-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-axis toward the +y-axis. a) Draw the vector addition diagram for these vectors, roughly to scale. b) Find the components of $\vec{A}$. c) Find the magnitude of $\vec{A}$.

1.68 a) Find the magnitude and direction of the vector $\vec{R}$ that is the sum of the three vectors $\vec{A}, \vec{B},$ and $\vec{C}$ in Fig. 1.28. In a diagram, show how $\vec{R}$ is formed from the three vectors in Fig. 1.28. b) Find the magnitude and direction of the vector $\vec{S} = \vec{C} - \vec{A} - \vec{B}$. In a diagram, show how $\vec{S}$ is formed from the three vectors in Fig. 1.28.

1.69 As noted in Exercise 1.33, a spelunker is surveying a cace. She follows a passage 180 m straight west, then 210 m in a direction $45^\circ$ east of south, then 280 m at $36^\circ$ east 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.

1.70 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. 1.32). 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.

1.71 A cross-country skier skis 2.80 km in the direction $45.0^\circ$ west of south, then 7.40 km in the direction $22.0^\circ$ south of west. a) Show these displacements in a diagram. b) How far is the skier from the starting point?

Chapter 1: Problems (1.62 - 1.66)

1.62 Iron has a property such that a $1.00 m^3$ volume has a mass of $7.86 \times 10^3 kg$ (density equals $7.86 \times 10^3 kg/m^3$). You want to manufacture iron into cubes and spheres. Fina a) the length of the side of a cube of iron that has a mass of 200.0 g; b) the radius of a solid sphere of iron that has a mass of 200.0 g.

1.63 a) Estimate the number of dentists in your city. You will need to consider the number of people in your city, how often they need to go to the dentist, how oftern they actually go, how many hours a typical dental procedure (filling, root canal, and so on) takes, and how many hours a dentist works in a week. b) Using your local telephone directory, check to see whether your estimate was roughly correct.

1.64 Physicists, mathematicians, and others often deal with large numbers. The number $10^100$ has been given the whimsical name googol by mathematicians. Let us compare some large numbers in physics with the googol. (Note: This problem requires numerical values that you can find in the appendices of the book, with which you should become familiar.) a) Approximately how many atoms make up our planet? For simplicity, assume the average atomic mass of the atoms to be 14 g/mol. Avogadro's number gives the number of atoms in a mole. b) Approximately how many neutrons are in a neutron star? Neutron stars are composed almost entirely of neutrons and have approximately twice the mass of the sun. c) In the leading theory of the origin of the universe that we can now observe occupied, at a very early time, a sphere whose radius was approximately equal to the present distance of the earth to the sun. At that time the universe had a density (mass divided by volume) of $10^{15} g/cm^3$. Assuming that $\frac{1}{3}$ of the particles were protons, $\frac{1}{3}$ of the particles were neutrons, and the remaining $\frac{1}{3}$ were electrons, how many particles then made up the universe?

1.65 Three horizontal ropes pull on a large stone stuck in the grounding, producing the vector forces $\vec{A}$, $\vec{B}$, and $\vec{C}$ shown in Fig. 1.31. Find the magnitude and direction of a fourth force on the stone that will make the vector sum of the four forces zero.

1.66 Emergency Landing. A plane leaves the airport in Galisto and flies 170 km at $68^\circ$ east of north and then changes direction to fly 230 km at $48^\circ$ south of east, after which it makes and 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?

Chapter 1: Problems (1.57 - 1.61)

1.57 An acre, a unit of land measurement still in wide use, has a length of one furlong (1/8 mi) 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 one acre of flat land to a depth of one foot. How many gallons are in an acre-foot?

1.58 An estate on the California coast was offered for sale for $4,950,000. The total area of the estate was 102 acres (see problem 1.57). a) Considering the price of the estate to be proportional to its area, what was the cost of one square meter of the estate? b) What would be the price of a portion of the estate the size of a postage stamp (7/8 in. by 1.0 in.)?

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

1.60 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 different elements, measure in atomic mass units; you can find the value of an atomic mass unit, or 1 u, in Appendix F.)

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

Chapter 1: Exercises (1.51 - 1.56)

1.51 a) Find the scalar product of the two vectors $\vec{A}$ and $\vec{B}$ given in Exercise 1.47. b) Find the angle between these two vectors.

1.52 Find the angle between each of the following pairs of vectors:
a) $\vec{A} = -2.00\hat{i} + 6.00\hat{j}$ and $\vec{B} = 2.00\hat{i} - 3.00\hat{j}$
b) $\vec{A} = 3.00\hat{i} + 5.00\hat{j}$ and $\vec{B} = 10.00\hat{i} + 6.00\hat{j}$
c) $\vec{A} = -4.00\hat{i} + 2.00\hat{j}$ and $\vec{B} = 7.00\hat{i} + 14.00\hat{j}$

1.53 Assuming a right-handed coordinate system, find the direction of the +z-axis in a) Fig. 1.15a; b) Fig. 1.15b.

1.54 For the two vectors in Fig. 1.27, a) find the magnitude and direction of the vector product $\vec{A} \times \vec{B}$; b) find the magnitude and direction of $\vec{B} \times \vec{A}$.

1.55 Find the vector product $\vec{A} \times \vec{B}$ (expressed in unit vectors) of the two vectors given in Exercise 1.47. What is the magnitude of the vector product?

1.56 For the two vectors in Fig. 1.29, a) find the magnitude and direction of the vector product $\vec{A} \times \vec{B}$; b) find the magnitude and direction of $\vec{B} \times \vec{A}$.

Chapter 1: Exercises (1.46 - 1.50)

1.46 a) Write each vector in Fig. 1.30 in terms of the unit vectors $\hat{i}$ and $\hat{j}$. b) Use unit vectors to express the vector $\vec{C}$, where $\vec{C} = 3.00\vec{A} - 4.00\vec{B}$. c) Find the magnitude and direction of $\vec{C}$.

1.47 Given two vectors $\vec{A} = 4.00\hat{i} + 3.00\hat{j}$ and $\vec{B} = 5.00\hat{i} - 2.00\hat{j}$, 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 $\vec{A}-\vec{B}$ and also show that your diagram agrees qualitatively with your answer in part (c).

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

1.49 a) Use vector components to prove that two vectors commute for both addition and the scalar product. b) Use vector components to prove that two vectors anticommute for the vector product. That is, prove that $\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}$.

Section 1.10 Products of Vectors

1.50 For the vectors $\vec{A}, \vec{B},$ and $\vec{C}$ in Fig. 1.28, find the scalar products a) $\vec{A} \cdot \vec{B}$; b) $\vec{B} \cdot \vec{C}$; c) $\vec{A} \cdot \vec{C}$.

Chapter 1: Exercises (1.41 - 1.45)

1.41 A disoriented physics professor drives 325 km north, then 4.75 km west, and then 1.50 km south. Find the magnitude and direction of the resultant displacement, using the method of components.

1.42 Vector $\vec{A}$ has components $A_{x} = 1.30 cm, A_{y} = 2.25 cm$; vector $\vec{B}$ has components $B_{x} = 4.10 cm, B_{y} = -3.75 cm$. Find a) the components of the vector sum $\vec{A} + \vec{B}$; b) the magnitude and direction of $\vec{A} + \vec{B}$; c) the components of the vector difference $\vec{B} - \vec{A}$; b) the magnitude and direction of $\vec{B} - \vec{A}$;

1.43 Vector $\vec{A}$ is 2.80 cm long and is $60.0^{\circ}$ 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. 1.29). 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.

Section 1.9 Unit Vectors

1.44 Write each vector in Fig. 1.27 in terms of the unit vectors $\hat{i}$ and $\hat{j}$.

1.45 Write each vector in Fig. 1.28 in terms of the unit vectors $\hat{i}$ and $\hat{j}$.