P U Z Z L E R When this honebee gets back to its hive, it will tell the other bees how to return to the food it has found. moving in a special, ver precisel defined pattern, the bee conves to other workers the information the need to find a flower bed. ees communicate b speaking in vectors. What does the bee have to tell the other bees in order to specif where the flower bed is located relative to the hive? (E. Webber/Visuals Unlimited) c h a p t e r Vectors Chapter utline 3.1 Coordinate Sstems 3.2 Vector and Scalar Quantities 3.3 Some Properties of Vectors 3.4 Components of a Vector and Unit Vectors 58
3.1 Coordinate Sstems 59 We often need to work with phsical quantities that have both numerical and directional properties. s noted in Section 2.1, quantities of this nature are represented b vectors. This chapter is primaril concerned with vector algebra and with some general properties of vector quantities. We discuss the addition and subtraction of vector quantities, together with some common applications to phsical situations. Vector quantities are used throughout this tet, and it is therefore imperative that ou master both their graphical and their algebraic properties. (, ) 2.2 3.1 CRDINTE SYSTEMS Man aspects of phsics deal in some form or other with locations in space. In Chapter 2, for eample, we saw that the mathematical description of an object s motion requires a method for describing the object s position at various times. This description is accomplished with the use of coordinates, and in Chapter 2 we used the cartesian coordinate sstem, in which horizontal and vertical aes intersect at a point taken to be the origin (Fig. 3.1). Cartesian coordinates are also called rectangular coordinates. Sometimes it is more convenient to represent a point in a plane b its plane polar coordinates (r, ), as shown in Figure 3.2a. In this polar coordinate sstem, r is the distance from the origin to the point having cartesian coordinates (, ), and is the angle between r and a fied ais. This fied ais is usuall the positive ais, and is usuall measured counterclockwise from it. From the right triangle in Figure 3.2b, we find that sin /r and that cos /r. ( review of trigonometric functions is given in ppendi.4.) Therefore, starting with the plane polar coordinates of an point, we can obtain the cartesian coordinates, using the equations Furthermore, the definitions of trigonometr tell us that (3.1) (3.2) (3.3) (3.4) These four epressions relating the coordinates (, ) to the coordinates (r, ) appl onl when is defined, as shown in Figure 3.2a in other words, when positive is an angle measured counterclockwise from the positive ais. (Some scientific calculators perform conversions between cartesian and polar coordinates based on these standard conventions.) If the reference ais for the polar angle is chosen to be one other than the positive ais or if the sense of increasing is chosen differentl, then the epressions relating the two sets of coordinates will change. Quick Quiz 3.1 r cos r sin tan r 2 2 Would the honebee at the beginning of the chapter use cartesian or polar coordinates when specifing the location of the flower? Wh? What is the honebee using as an origin of coordinates? Q ( 3, 4) P (5, 3) Figure 3.1 Designation of points in a cartesian coordinate sstem. Ever point is labeled with coordinates (, ). sin θ = r cos θ = r tan θ = θ Figure 3.2 r (a) θ (b) (, ) (a) The plane polar coordinates of a point are represented b the distance r and the angle, where is measured counterclockwise from the positive ais. (b) The right triangle used to relate (, ) to (r, ). You ma want to read Talking pes and Dancing ees (1997) b ets Wckoff. r
60 CHPTER 3 Vectors EXMPLE 3.1 Polar Coordinates The cartesian coordinates of a point in the plane are (, ) ( 3.50, 2.50) m, as shown in Figure 3.3. Find the polar coordinates of this point. (m) Solution r 2 2 ( 3.50 m) 2 ( 2.50 m) 2 tan 2.50 m 3.50 m 0.714 4.30 m θ r 3.50, 2.50 (m) 216 Note that ou must use the signs of and to find that the point lies in the third quadrant of the coordinate sstem. That is, 216 and not 35.5. Figure 3.3 are given. Finding polar coordinates when cartesian coordinates 2.3 3.2 VECTR ND SCLR QUNTITIES s noted in Chapter 2, some phsical quantities are scalar quantities whereas others are vector quantities. When ou want to know the temperature outside so that ou will know how to dress, the onl information ou need is a number and the unit degrees C or degrees F. Temperature is therefore an eample of a scalar quantit, which is defined as a quantit that is completel specified b a number and appropriate units. That is, scalar quantit is specified b a single value with an appropriate unit and has no direction. ther eamples of scalar quantities are volume, mass, and time intervals. The rules of ordinar arithmetic are used to manipulate scalar quantities. If ou are getting read to pilot a small plane and need to know the wind velocit, ou must know both the speed of the wind and its direction. ecause direction is part of the information it gives, velocit is a vector quantit, which is defined as a phsical quantit that is completel specified b a number and appropriate units plus a direction. That is, vector quantit has both magnitude and direction. Figure 3.4 s a particle moves from to along an arbitrar path represented b the broken line, its displacement is a vector quantit shown b the arrow drawn from to. nother eample of a vector quantit is displacement, as ou know from Chapter 2. Suppose a particle moves from some point to some point along a straight path, as shown in Figure 3.4. We represent this displacement b drawing an arrow from to, with the tip of the arrow pointing awa from the starting point. The direction of the arrowhead represents the direction of the displacement, and the length of the arrow represents the magnitude of the displacement. If the particle travels along some other path from to, such as the broken line in Figure 3.4, its displacement is still the arrow drawn from to.
3.3 Some Properties of Vectors 61 (a) (b) (c) (a) The number of apples in the basket is one eample of a scalar quantit. Can ou think of other eamples? (Superstock) (b) Jennifer pointing to the right. vector quantit is one that must be specified b both magnitude and direction. (Photo b Ra Serwa) (c) n anemometer is a device meteorologists use in weather forecasting. The cups spin around and reveal the magnitude of the wind velocit. The pointer indicates the direction. (Courtes of Peet ros.compan, 1308 Doris venue, cean, NJ 07712) 2.4 In this tet, we use a boldface letter, such as, to represent a vector quantit. nother common method for vector notation that ou should be aware of is the use of an arrow over a letter, such as :. The magnitude of the vector is written either or. The magnitude of a vector has phsical units, such as meters for displacement or meters per second for velocit. 3.3 SME PRPERTIES F VECTRS Equalit of Two Vectors For man purposes, two vectors and ma be defined to be equal if the have the same magnitude and point in the same direction. That is, onl if and if and point in the same direction along parallel lines. For eample, all the vectors in Figure 3.5 are equal even though the have different starting points. This propert allows us to move a vector to a position parallel to itself in a diagram without affecting the vector. dding Vectors The rules for adding vectors are convenientl described b geometric methods. To add vector to vector, first draw vector, with its magnitude represented b a convenient scale, on graph paper and then draw vector to the same scale with its tail starting from the tip of, as shown in Figure 3.6. The resultant vector R is the vector drawn from the tail of to the tip of. This procedure is known as the triangle method of addition. For eample, if ou walked 3.0 m toward the east and then 4.0 m toward the north, as shown in Figure 3.7, ou would find ourself 5.0 m from where ou Figure 3.5 These four vectors are equal because the have equal lengths and point in the same direction. Figure 3.6 R = + When vector is added to vector, the resultant R is the vector that runs from the tail of to the tip of.
62 CHPTER 3 Vectors R = (3.0 m) 2 + (4.0 m) 2 = 5.0 m 4.0 ( ) θ = tan 1 = 53 3.0 Figure 3.7 3.0 m 4.0 m Vector addition. Walking first 3.0 m due east and then 4.0 m due north leaves ou R 5.0 m from our starting point. R = + + C + D Figure 3.8 Geometric construction for summing four vectors. The resultant vector R is b definition the one that completes the polgon. D C Commutative law started, measured at an angle of 53 north of east. Your total displacement is the vector sum of the individual displacements. geometric construction can also be used to add more than two vectors. This is shown in Figure 3.8 for the case of four vectors. The resultant vector R C D is the vector that completes the polgon. In other words, R is the vector drawn from the tail of the first vector to the tip of the last vector. n alternative graphical procedure for adding two vectors, known as the parallelogram rule of addition, is shown in Figure 3.9a. In this construction, the tails of the two vectors and are joined together and the resultant vector R is the diagonal of a parallelogram formed with and as two of its four sides. When two vectors are added, the sum is independent of the order of the addition. (This fact ma seem trivial, but as ou will see in Chapter 11, the order is important when vectors are multiplied). This can be seen from the geometric construction in Figure 3.9b and is known as the commutative law of addition: (3.5) When three or more vectors are added, their sum is independent of the wa in which the individual vectors are grouped together. geometric proof of this rule Commutative Law Figure 3.9 (a) In this construction, the resultant R is the diagonal of a parallelogram having sides and. (b) This construction shows that in other words, that vector addition is commutative. R = + (a) R = + (b)
3.3 Some Properties of Vectors 63 + ( + C) + C C ssociative Law ( + ) + C + C Figure 3.10 Geometric constructions for verifing the associative law of addition. for three vectors is given in Figure 3.10. This is called the associative law of addition: ( C) ( ) C (3.6) In summar, a vector quantit has both magnitude and direction and also obes the laws of vector addition as described in Figures 3.6 to 3.10. When two or more vectors are added together, all of them must have the same units. It would be meaningless to add a velocit vector (for eample, 60 km/h to the east) to a displacement vector (for eample, 200 km to the north) because the represent different phsical quantities. The same rule also applies to scalars. For eample, it would be meaningless to add time intervals to temperatures. ssociative law Negative of a Vector The negative of the vector is defined as the vector that when added to gives zero for the vector sum. That is, ( ) 0. The vectors and have the same magnitude but point in opposite directions. Subtracting Vectors The operation of vector subtraction makes use of the definition of the negative of a vector. We define the operation as vector added to vector : ( ) (3.7) The geometric construction for subtracting two vectors in this wa is illustrated in Figure 3.11a. nother wa of looking at vector subtraction is to note that the difference between two vectors and is what ou have to add to the second vector to obtain the first. In this case, the vector points from the tip of the second vector to the tip of the first, as Figure 3.11b shows. C = (a) Vector Subtraction (b) C = Figure 3.11 (a) This construction shows how to subtract vector from vector. The vector is equal in magnitude to vector and points in the opposite direction. To subtract from, appl the rule of vector addition to the combination of and : Draw along some convenient ais, place the tail of at the tip of, and C is the difference. (b) second wa of looking at vector subtraction. The difference vector C is the vector that we must add to to obtain.
64 CHPTER 3 Vectors EXMPLE 3.2 Vacation Trip car travels 20.0 km due north and then 35.0 km in a direction 60.0 west of north, as shown in Figure 3.12. Find the magnitude and direction of the car s resultant displacement. Solution In this eample, we show two was to find the resultant of two vectors. We can solve the problem geometricall, using graph paper and a protractor, as shown in Figure 3.12. (In fact, even when ou know ou are going to be carr- N ing out a calculation, ou should sketch the vectors to check our results.) The displacement R is the resultant when the two individual displacements and are added. To solve the problem algebraicall, we note that the magnitude of R can be obtained from the law of cosines as applied to the triangle (see ppendi.4). With 180 60 120 and R 2 2 2 2 cos, we find that R 2 2 2 cos (20.0 km) 2 (35.0 km) 2 2(20.0 km)(35.0 km)cos 120 W E 48.2 km Figure 3.12 S (km) 40 60.0 θ 20 R β (km) 20 0 Graphical method for finding the resultant displacement vector R. The direction of R measured from the northerl direction can be obtained from the law of sines (ppendi.4): sin sin R sin sin R 35.0 km 48.2 km 38.9 sin 120 0.629 The resultant displacement of the car is 48.2 km in a direction 38.9 west of north. This result matches what we found graphicall. Multipling a Vector b a Scalar If vector is multiplied b a positive scalar quantit m, then the product m is a vector that has the same direction as and magnitude m. If vector is multiplied b a negative scalar quantit m, then the product m is directed opposite. For eample, the vector 5 is five times as long as and points in the 1 same direction as ; the vector 3 is one-third the length of and points in the direction opposite. θ Figure 3.13 n vector ling in the plane can be represented b a vector ling along the ais and b a vector ling along the ais, where. 2.5 Quick Quiz 3.2 If vector is added to vector, under what condition does the resultant vector have magnitude? Under what conditions is the resultant vector equal to zero? 3.4 CMPNENTS F VECTR ND UNIT VECTRS The geometric method of adding vectors is not recommended whenever great accurac is required or in three-dimensional problems. In this section, we describe a method of adding vectors that makes use of the projections of vectors along coordinate aes. These projections are called the components of the vector. n vector can be completel described b its components. Consider a vector ling in the plane and making an arbitrar angle with the positive ais, as shown in Figure 3.13. This vector can be epressed as the
3.4 Components of a Vector and Unit Vectors 65 sum of two other vectors and. From Figure 3.13, we see that the three vectors form a right triangle and that. (If ou cannot see wh this equalit holds, go back to Figure 3.9 and review the parallelogram rule.) We shall often refer to the components of a vector, written and (without the boldface notation). The component represents the projection of along the ais, and the component represents the projection of along the ais. These components can be positive or negative. The component is positive if points in the positive direction and is negative if points in the negative direction. The same is true for the component. From Figure 3.13 and the definition of sine and cosine, we see that cos / and that sin /. Hence, the components of are cos (3.8) sin (3.9) These components form two sides of a right triangle with a hpotenuse of length. Thus, it follows that the magnitude and direction of are related to its components through the epressions 2 2 (3.10) (3.11) Note that the signs of the components and depend on the angle. For eample, if 120, then is negative and is positive. If 225, then both and are negative. Figure 3.14 summarizes the signs of the components when lies in the various quadrants. When solving problems, ou can specif a vector either with its components and or with its magnitude and direction and. tan 1 Components of the vector Magnitude of Direction of negative positive positive positive negative positive negative negative Quick Quiz 3.3 Can the component of a vector ever be greater than the magnitude of the vector? Figure 3.14 The signs of the components of a vector depend on the quadrant in which the vector is located. Suppose ou are working a phsics problem that requires resolving a vector into its components. In man applications it is convenient to epress the components in a coordinate sstem having aes that are not horizontal and vertical but are still perpendicular to each other. If ou choose reference aes or an angle other than the aes and angle shown in Figure 3.13, the components must be modified accordingl. Suppose a vector makes an angle with the ais defined in Figure 3.15. The components of along the and aes are cos and sin, as specified b Equations 3.8 and 3.9. The magnitude and direction of are obtained from epressions equivalent to Equations 3.10 and 3.11. Thus, we can epress the components of a vector in an coordinate sstem that is convenient for a particular situation. Unit Vectors Vector quantities often are epressed in terms of unit vectors. unit vector is a dimensionless vector having a magnitude of eactl 1. Unit vectors are used to specif a given direction and have no other phsical significance. The are used solel as a convenience in describing a direction in space. We shall use the smbols Figure 3.15 θ The component vectors of in a coordinate sstem that is tilted.
66 CHPTER 3 Vectors Position vector j i k i, j, and k to represent unit vectors pointing in the positive,, and z directions, respectivel. The unit vectors i, j, and k form a set of mutuall perpendicular vectors in a right-handed coordinate sstem, as shown in Figure 3.16a. The magnitude of each unit vector equals 1; that is, i j k 1. Consider a vector ling in the plane, as shown in Figure 3.16b. The product of the component and the unit vector i is the vector i, which lies on the ais and has magnitude. (The vector i is an alternative representation of vector.) Likewise, j is a vector of magnitude ling on the ais. (gain, vector j is an alternative representation of vector.) Thus, the unit vector notation for the vector is i j (3.12) For eample, consider a point ling in the plane and having cartesian coordinates (, ), as in Figure 3.17. The point can be specified b the position vector r, which in unit vector form is given b r i j (3.13) This notation tells us that the components of r are the lengths and. Now let us see how to use components to add vectors when the geometric method is not sufficientl accurate. Suppose we wish to add vector to vector, where vector has components and. ll we do is add the and components separatel. The resultant vector R is therefore R ( i j) ( i j) or R ( )i ( )j (3.14) ecause R R i R j, we see that the components of the resultant vector are R R (3.15) (a) z j Figure 3.16 i (b) (a) The unit vectors i, j, and k are directed along the,, and z aes, respectivel. (b) Vector i j ling in the plane has components and. Figure 3.17 r (,) The point whose cartesian coordinates are (, ) can be represented b the position vector r i j. R Figure 3.18 This geometric construction for the sum of two vectors shows the relationship between the components of the resultant R and the components of the individual vectors. R R
3.4 Components of a Vector and Unit Vectors 67 We obtain the magnitude of R and the angle it makes with the ais from its components, using the relationships R R 2 R 2 ( ) 2 ( ) 2 (3.16) tan R R (3.17) We can check this addition b components with a geometric construction, as shown in Figure 3.18. Remember that ou must note the signs of the components when using either the algebraic or the geometric method. t times, we need to consider situations involving motion in three component directions. The etension of our methods to three-dimensional vectors is straightforward. If and both have,, and z components, we epress them in the form i j z k (3.18) i j z k (3.19) The sum of and is R ( )i ( )j ( z z )k (3.20) Note that Equation 3.20 differs from Equation 3.14: in Equation 3.20, the resultant vector also has a z component R z z z. QuickLab Write an epression for the vector describing the displacement of a fl that moves from one corner of the floor of the room that ou are in to the opposite corner of the room, near the ceiling. Quick Quiz 3.4 If one component of a vector is not zero, can the magnitude of the vector be zero? Eplain. Quick Quiz 3.5 If 0, what can ou sa about the components of the two vectors? Problem-Solving Hints dding Vectors When ou need to add two or more vectors, use this step-b-step procedure: Select a coordinate sstem that is convenient. (Tr to reduce the number of components ou need to find b choosing aes that line up with as man vectors as possible.) Draw a labeled sketch of the vectors described in the problem. Find the and components of all vectors and the resultant components (the algebraic sum of the components) in the and directions. If necessar, use the Pthagorean theorem to find the magnitude of the resultant vector and select a suitable trigonometric function to find the angle that the resultant vector makes with the ais.
68 CHPTER 3 Vectors EXMPLE 3.3 The Sum of Two Vectors Find the sum of two vectors and ling in the plane and given b Solution Comparing this epression for with the general epression i j, we see that 2.0 m and that 2.0 m. Likewise, 2.0 m and 4.0 m. We obtain the resultant vector R, using Equation 3.14: or (2.0i 2.0j) m and (2.0i 4.0j) m R (2.0 2.0)i m (2.0 4.0)j m (4.0i 2.0j) m R 4.0 m R 2.0 m The magnitude of R is given b Equation 3.16: R R 2 R 2 (4.0 m) 2 ( 2.0 m) 2 20 m 4.5 m We can find the direction of R from Equation 3.17: tan R R 2.0 m 4.0 m 0.50 Your calculator likel gives the answer 27 for tan 1 ( 0.50). This answer is correct if we interpret it to mean 27 clockwise from the ais. ur standard form has been to quote the angles measured counterclockwise from the ais, and that angle for this vector is 333. EXMPLE 3.4 The Resultant Displacement particle undergoes three consecutive displacements: d 1 (15i 30j 12k) cm, d 2 (23i 14j 5.0k) cm, and d 3 ( 13i 15j) cm. Find the components of the resultant displacement and its magnitude. Solution Rather than looking at a sketch on flat paper, visualize the problem as follows: Start with our fingertip at the front left corner of our horizontal desktop. Move our fingertip 15 cm to the right, then 30 cm toward the far side of the desk, then 12 cm verticall upward, then 23 cm to the right, then 14 cm horizontall toward the front edge of the desk, then 5.0 cm verticall toward the desk, then 13 cm to the left, and (finall!) 15 cm toward the back of the desk. The mathematical calculation keeps track of this motion along the three perpendicular aes: R d 1 d 2 d 3 (15 23 13)i cm (30 14 15)j cm (12 5.0 0)k cm (25i 31j 7.0k) cm The resultant displacement has components R 31 cm, and R z 7.0 cm. Its magnitude is R R 2 R 2 R 2 z (25 cm) 2 (31 cm) 2 (7.0 cm) 2 R 25 cm, 40 cm EXMPLE 3.5 Taking a Hike hiker begins a trip b first walking 25.0 km southeast from her car. She stops and sets up her tent for the night. n the second da, she walks 40.0 km in a direction 60.0 north of east, at which point she discovers a forest ranger s tower. (a) Determine the components of the hiker s displacement for each da. Solution If we denote the displacement vectors on the first and second das b and, respectivel, and use the car as the origin of coordinates, we obtain the vectors shown in Figure 3.19. Displacement has a magnitude of 25.0 km and is directed 45.0 below the positive ais. From Equations 3.8 and 3.9, its components are cos( 45.0 ) (25.0 km)(0.707) 17.7 km (km) 20 10 W R N S Car 0 45.0 20 30 40 50 10 60.0 20 Tent E Tower (km) sin( 45.0 ) (25.0 km)(0.707) 17.7 km Figure 3.19 R. The total displacement of the hiker is the vector
3.4 Components of a Vector and Unit Vectors 69 The negative value of indicates that the hiker walks in the negative direction on the first da. The signs of and also are evident from Figure 3.19. The second displacement has a magnitude of 40.0 km and is 60.0 north of east. Its components are cos 60.0 (40.0 km)(0.500) sin 60.0 (40.0 km)(0.866) 20.0 km 34.6 km (b) Determine the components of the hiker s resultant displacement R for the trip. Find an epression for R in terms of unit vectors. Solution The resultant displacement for the trip R has components given b Equation 3.15: R 17.7 km 20.0 km 37.7 km R 17.7 km 34.6 km 16.9 km In unit vector form, we can write the total displacement as R (37.7i 16.9j) km Eercise Determine the magnitude and direction of the total displacement. nswer 41.3 km, 24.1 north of east from the car. EXMPLE 3.6 Let s Fl wa! commuter airplane takes the route shown in Figure 3.20. First, it flies from the origin of the coordinate sstem shown to cit, located 175 km in a direction 30.0 north of east. Net, it flies 153 km 20.0 west of north to cit. Finall, it flies 195 km due west to cit C. Find the location of cit C relative to the origin. Solution It is convenient to choose the coordinate sstem shown in Figure 3.20, where the ais points to the east and the ais points to the north. Let us denote the three consecutive displacements b the vectors a, b, and c. Displacement a has a magnitude of 175 km and the components a a cos(30.0 ) (175 km)(0.866) 152 km a a sin(30.0 ) (175 km)(0.500) 87.5 km Displacement b, whose magnitude is 153 km, has the components b b cos(110 ) (153 km)( 0.342) 52.3 km b b sin(110 ) (153 km)(0.940) 144 km Finall, displacement c, whose magnitude is 195 km, has the components c c cos(180 ) (195 km)( 1) 195 km c c sin(180 ) 0 Therefore, the components of the position vector R from the starting point to cit C are R a b c 152 km 52.3 km 195 km C R (km) 250 c 200 150 100 50 b a W 20.0 110 N E S 30.0 (km) 50 100 150 200 R a b c 87.5 km 144 km 0 95.3 km 232 km In unit vector notation, R ( 95.3i 232j) km. That is, the airplane can reach cit C from the starting point b first traveling 95.3 km due west and then b traveling 232 km due north. Eercise Find the magnitude and direction of R. Figure 3.20 The airplane starts at the origin, flies first to cit, then to cit, and finall to cit C. nswer 251 km, 22.3 west of north.
70 CHPTER 3 Vectors R = + R R Figure 3.21 (a) (b) (a) Vector addition b the triangle method. (b) Vector addition b the parallelogram rule. θ Figure 3.22 The addition of the two vectors and gives vector. Note that i and j, where and are the components of vector. SUMMRY Scalar quantities are those that have onl magnitude and no associated direction. Vector quantities have both magnitude and direction and obe the laws of vector addition. We can add two vectors and graphicall, using either the triangle method or the parallelogram rule. In the triangle method (Fig. 3.21a), the resultant vector R runs from the tail of to the tip of. In the parallelogram method (Fig. 3.21b), R is the diagonal of a parallelogram having and as two of its sides. You should be able to add or subtract vectors, using these graphical methods. The component of the vector is equal to the projection of along the ais of a coordinate sstem, as shown in Figure 3.22, where cos. The component of is the projection of along the ais, where sin. e sure ou can determine which trigonometric functions ou should use in all situations, especiall when is defined as something other than the counterclockwise angle from the positive ais. If a vector has an component and a component, the vector can be epressed in unit vector form as i j. In this notation, i is a unit vector pointing in the positive direction, and j is a unit vector pointing in the positive direction. ecause i and j are unit vectors, i j 1. We can find the resultant of two or more vectors b resolving all vectors into their and components, adding their resultant and components, and then using the Pthagorean theorem to find the magnitude of the resultant vector. We can find the angle that the resultant vector makes with respect to the ais b using a suitable trigonometric function. QUESTINS 1. Two vectors have unequal magnitudes. Can their sum be zero? Eplain. 2. Can the magnitude of a particle s displacement be greater than the distance traveled? Eplain. 3. The magnitudes of two vectors and are 5 units and 2 units. Find the largest and smallest values possible for the resultant vector R. 4. Vector lies in the plane. For what orientations of vector will both of its components be negative? For what orientations will its components have opposite signs? 5. If the component of vector along the direction of vector is zero, what can ou conclude about these two vectors? 6. Can the magnitude of a vector have a negative value? Eplain. 7. Which of the following are vectors and which are not: force, temperature, volume, ratings of a television show, height, velocit, age? 8. Under what circumstances would a nonzero vector ling in the plane ever have components that are equal in magnitude? 9. Is it possible to add a vector quantit to a scalar quantit? Eplain.
Problems 71 PRLEMS 1, 2, 3 = straightforward, intermediate, challenging = full solution available in the Student Solutions Manual and Stud Guide WE = solution posted at http://www.saunderscollege.com/phsics/ = Computer useful in solving problem = Interactive Phsics = paired numerical/smbolic problems Section 3.1 WE Coordinate Sstems 1. The polar coordinates of a point are r 5.50 m and 240. What are the cartesian coordinates of this point? 2. Two points in the plane have cartesian coordinates (2.00, 4.00) m and ( 3.00, 3.00) m. Determine (a) the distance between these points and (b) their polar coordinates. 3. If the cartesian coordinates of a point are given b (2, ) and its polar coordinates are (r, 30 ), determine and r. 4. Two points in a plane have polar coordinates (2.50 m, 30.0 ) and (3.80 m, 120.0 ). Determine (a) the cartesian coordinates of these points and (b) the distance between them. 5. fl lands on one wall of a room. The lower left-hand corner of the wall is selected as the origin of a twodimensional cartesian coordinate sstem. If the fl is located at the point having coordinates (2.00, 1.00) m, (a) how far is it from the corner of the room? (b) what is its location in polar coordinates? 6. If the polar coordinates of the point (, ) are (r, ), determine the polar coordinates for the points (a) (, ), (b) ( 2, 2), and (c) (3, 3). Section 3.2 Section 3.3 Vector and Scalar Quantities Some Properties of Vectors 7. n airplane flies 200 km due west from cit to cit and then 300 km in the direction 30.0 north of west from cit to cit C. (a) In straight-line distance, how far is cit C from cit? (b) Relative to cit, in what direction is cit C? 8. pedestrian moves 6.00 km east and then 13.0 km north. Using the graphical method, find the magnitude and direction of the resultant displacement vector. 9. surveor measures the distance across a straight river b the following method: Starting directl across from a tree on the opposite bank, she walks 100 m along the riverbank to establish a baseline. Then she sights across to the tree. The angle from her baseline to the tree is 35.0. How wide is the river? 10. plane flies from base camp to lake, a distance of 280 km at a direction 20.0 north of east. fter dropping off supplies, it flies to lake, which is 190 km and 30.0 west of north from lake. Graphicall determine the distance and direction from lake to the base camp. 11. Vector has a magnitude of 8.00 units and makes an angle of 45.0 with the positive ais. Vector also has a magnitude of 8.00 units and is directed along the neg- WE WE ative ais. Using graphical methods, find (a) the vector sum and (b) the vector difference. 12. force F 1 of magnitude 6.00 units acts at the origin in a direction 30.0 above the positive ais. second force F 2 of magnitude 5.00 units acts at the origin in the direction of the positive ais. Find graphicall the magnitude and direction of the resultant force F 1 + F 2. 13. person walks along a circular path of radius 5.00 m. If the person walks around one half of the circle, find (a) the magnitude of the displacement vector and (b) how far the person walked. (c) What is the magnitude of the displacement if the person walks all the wa around the circle? 14. dog searching for a bone walks 3.50 m south, then 8.20 m at an angle 30.0 north of east, and finall 15.0 m west. Using graphical techniques, find the dog s resultant displacement vector. 15. Each of the displacement vectors and shown in Figure P3.15 has a magnitude of 3.00 m. Find graphicall (a), (b), (c), (d) 2. Report all angles counterclockwise from the positive ais. 3.00 m 3.00 m 30.0 Figure P3.15 Problems 15 and 39. 16. rbitraril define the instantaneous vector height of a person as the displacement vector from the point halfwa between the feet to the top of the head. Make an order-of-magnitude estimate of the total vector height of all the people in a cit of population 100 000 (a) at 10 a.m. on a Tuesda and (b) at 5 a.m. on a Saturda. Eplain our reasoning. 17. roller coaster moves 200 ft horizontall and then rises 135 ft at an angle of 30.0 above the horizontal. It then travels 135 ft at an angle of 40.0 downward. What is its displacement from its starting point? Use graphical techniques. 18. The driver of a car drives 3.00 km north, 2.00 km northeast (45.0 east of north), 4.00 km west, and then
72 CHPTER 3 Vectors 3.00 km southeast (45.0 east of south). Where does he end up relative to his starting point? Work out our answer graphicall. Check b using components. (The car is not near the North Pole or the South Pole.) 19. Fo Mulder is trapped in a maze. To find his wa out, he walks 10.0 m, makes a 90.0 right turn, walks 5.00 m, makes another 90.0 right turn, and walks 7.00 m. What is his displacement from his initial position? Section 3.4 Components of a Vector and Unit Vectors 20. Find the horizontal and vertical components of the 100-m displacement of a superhero who flies from the top of a tall building following the path shown in Figure P3.20. 30.0 100 m Figure P3.20 21. person walks 25.0 north of east for 3.10 km. How far would she have to walk due north and due east to arrive at the same location? 22. While eploring a cave, a spelunker starts at the entrance and moves the following distances: She goes 75.0 m north, 250 m east, 125 m at an angle 30.0 north of east, and 150 m south. Find the resultant displacement from the cave entrance. 23. In the assembl operation illustrated in Figure P3.23, a robot first lifts an object upward along an arc that forms one quarter of a circle having a radius of 4.80 cm and Figure P3.23 WE ling in an east west vertical plane. The robot then moves the object upward along a second arc that forms one quarter of a circle having a radius of 3.70 cm and ling in a north south vertical plane. Find (a) the magnitude of the total displacement of the object and (b) the angle the total displacement makes with the vertical. 24. Vector has,, and z components of 4.00, 6.00, and 3.00 units, respectivel. Calculate the magnitude of and the angles that makes with the coordinate aes. 25. vector has an component of 25.0 units and a component of 40.0 units. Find the magnitude and direction of this vector. 26. map suggests that tlanta is 730 mi in a direction 5.00 north of east from Dallas. The same map shows that Chicago is 560 mi in a direction 21.0 west of north from tlanta. ssuming that the Earth is flat, use this information to find the displacement from Dallas to Chicago. 27. displacement vector ling in the plane has a magnitude of 50.0 m and is directed at an angle of 120 to the positive ais. Find the and components of this vector and epress the vector in unit vector notation. 28. If 2.00i 6.00j and 3.00i 2.00j, (a) sketch the vector sum C and the vector difference D. (b) Find solutions for C and D, first in terms of unit vectors and then in terms of polar coordinates, with angles measured with respect to the ais. 29. Find the magnitude and direction of the resultant of three displacements having and components (3.00, 2.00) m, ( 5.00, 3.00) m, and (6.00, 1.00) m. 30. Vector has and components of 8.70 cm and 15.0 cm, respectivel; vector has and components of 13.2 cm and 6.60 cm, respectivel. If 3C 0, what are the components of C? 31. Consider two vectors 3i 2j and i 4j. Calculate (a), (b), (c), (d), (e) the directions of and. 32. bo runs 3.00 blocks north, 4.00 blocks northeast, and 5.00 blocks west. Determine the length and direction of the displacement vector that goes from the starting point to his final position. 33. btain epressions in component form for the position vectors having polar coordinates (a) 12.8 m, 150 ; (b) 3.30 cm, 60.0 ; (c) 22.0 in., 215. 34. Consider the displacement vectors (3i 3j) m, (i 4j) m, and C ( 2i 5j) m. Use the component method to determine (a) the magnitude and direction of the vector D C and (b) the magnitude and direction of E C. 35. particle undergoes the following consecutive displacements: 3.50 m south, 8.20 m northeast, and 15.0 m west. What is the resultant displacement? 36. In a game of merican football, a quarterback takes the ball from the line of scrimmage, runs backward for 10.0 ards, and then sidewas parallel to the line of scrimmage for 15.0 ards. t this point, he throws a forward
Problems 73 pass 50.0 ards straight downfield perpendicular to the line of scrimmage. What is the magnitude of the football s resultant displacement? 37. The helicopter view in Figure P3.37 shows two people pulling on a stubborn mule. Find (a) the single force that is equivalent to the two forces shown and (b) the force that a third person would have to eert on the mule to make the resultant force equal to zero. The forces are measured in units of newtons. F 2 = 80.0 N 75.0 60.0 Figure P3.37 F 1 = 120 N 38. novice golfer on the green takes three strokes to sink the ball. The successive displacements are 4.00 m to the north, 2.00 m northeast, and 1.00 m 30.0 west of south. Starting at the same initial point, an epert golfer could make the hole in what single displacement? 39. Find the and components of the vectors and shown in Figure P3.15; then derive an epression for the resultant vector in unit vector notation. 40. You are standing on the ground at the origin of a coordinate sstem. n airplane flies over ou with constant velocit parallel to the ais and at a constant height of 7.60 10 3 m. t t 0, the airplane is directl above ou, so that the vector from ou to it is given b P 0 (7.60 10 3 m)j. t t 30.0 s, the position vector leading from ou to the airplane is P 30 (8.04 10 3 m)i (7.60 10 3 m)j. Determine the magnitude and orientation of the airplane s position vector at t 45.0 s. 41. particle undergoes two displacements. The first has a magnitude of 150 cm and makes an angle of 120 with the positive ais. The resultant displacement has a magnitude of 140 cm and is directed at an angle of 35.0 to the positive ais. Find the magnitude and direction of the second displacement. 42. Vectors and have equal magnitudes of 5.00. If the sum of and is the vector 6.00 j, determine the angle between and. 43. The vector has,, and z components of 8.00, 12.0, and 4.00 units, respectivel. (a) Write a vector epression for in unit vector notation. (b) btain a unit vector epression for a vector one-fourth the length of pointing in the same direction as. (c) btain a unit vector epression for a vector C three times the length of pointing in the direction opposite the direction of. 44. Instructions for finding a buried treasure include the following: Go 75.0 paces at 240, turn to 135 and walk 125 paces, then travel 100 paces at 160. The angles are measured counterclockwise from an ais pointing to the east, the direction. Determine the resultant displacement from the starting point. 45. Given the displacement vectors (3i 4j 4k) m and (2i 3j 7k) m, find the magnitudes of the vectors (a) C and (b) D 2, also epressing each in terms of its,, and z components. 46. radar station locates a sinking ship at range 17.3 km and bearing 136 clockwise from north. From the same station a rescue plane is at horizontal range 19.6 km, 153 clockwise from north, with elevation 2.20 km. (a) Write the vector displacement from plane to ship, letting i represent east, j north, and k up. (b) How far apart are the plane and ship? 47. s it passes over Grand ahama Island, the ee of a hurricane is moving in a direction 60.0 north of west with a speed of 41.0 km/h. Three hours later, the course of the hurricane suddenl shifts due north and its speed slows to 25.0 km/h. How far from Grand ahama is the ee 4.50 h after it passes over the island? 48. (a) Vector E has magnitude 17.0 cm and is directed 27.0 counterclockwise from the ais. Epress it in unit vector notation. (b) Vector F has magnitude 17.0 cm and is directed 27.0 counterclockwise from the ais. Epress it in unit vector notation. (c) Vector G has magnitude 17.0 cm and is directed 27.0 clockwise from the ais. Epress it in unit vector notation. 49. Vector has a negative component 3.00 units in length and a positive component 2.00 units in length. (a) Determine an epression for in unit vector notation. (b) Determine the magnitude and direction of. (c) What vector, when added to vector, gives a resultant vector with no component and a negative component 4.00 units in length? 50. n airplane starting from airport flies 300 km east, then 350 km at 30.0 west of north, and then 150 km north to arrive finall at airport. (a) The net da, another plane flies directl from airport to airport in a straight line. In what direction should the pilot travel in this direct flight? (b) How far will the pilot travel in this direct flight? ssume there is no wind during these flights.
74 CHPTER 3 Vectors WE 51. Three vectors are oriented as shown in Figure P3.51, where 20.0 units, 40.0 units, and C 30.0 units. Find (a) the and components of the resultant vector (epressed in unit vector notation) and (b) the magnitude and direction of the resultant vector. Start 100 m End 300 m 200 m 45.0 45.0 60 30 150 m C Figure P3.57 Figure P3.51 52. If (6.00i 8.00j) units, ( 8.00i 3.00j) units, and C (26.0i 19.0j) units, determine a and b such that a b C 0. DDITINL PRLEMS 53. Two vectors and have precisel equal magnitudes. For the magnitude of to be 100 times greater than the magnitude of, what must be the angle between them? 54. Two vectors and have precisel equal magnitudes. For the magnitude of to be greater than the magnitude of b the factor n, what must be the angle between them? 55. vector is given b R 2.00i 1.00j 3.00k. Find (a) the magnitudes of the,, and z components, (b) the magnitude of R, and (c) the angles between R and the,, and z aes. 56. Find the sum of these four vector forces: 12.0 N to the right at 35.0 above the horizontal, 31.0 N to the left at 55.0 above the horizontal, 8.40 N to the left at 35.0 below the horizontal, and 24.0 N to the right at 55.0 below the horizontal. (Hint: Make a drawing of this situation and select the best aes for and so that ou have the least number of components. Then add the vectors, using the component method.) 57. person going for a walk follows the path shown in Figure P3.57. The total trip consists of four straight-line paths. t the end of the walk, what is the person s resultant displacement measured from the starting point? 58. In general, the instantaneous position of an object is specified b its position vector P leading from a fied origin to the location of the object. Suppose that for a certain object the position vector is a function of time, given b P 4i 3j 2t j, where P is in meters and t is in seconds. Evaluate dp/dt. What does this derivative represent about the object? 59. jet airliner, moving initiall at 300 mi/h to the east, suddenl enters a region where the wind is blowing at 100 mi/h in a direction 30.0 north of east. What are the new speed and direction of the aircraft relative to the ground? 60. pirate has buried his treasure on an island with five trees located at the following points: (30.0 m, 20.0 m), (60.0 m, 80.0 m), C( 10.0 m, 10.0 m), D(40.0 m, 30.0 m), and E( 70.0 m, 60.0 m). ll points are measured relative to some origin, as in Figure P3.60. Instructions on the map tell ou to start at and move toward, but to cover onl one-half the distance between and. Then, move toward C, covering one-third the distance between our current location and C. Net, move toward D, covering one-fourth the distance between where ou are and D. Finall, move toward E, covering one-fifth the distance between ou and E, stop, and dig. (a) What are the coordinates of the point where the pirate s treasure is buried? (b) Re- E C Figure P3.60 D
nswers to Quick Quizzes 75 arrange the order of the trees, (for instance, (30.0 m, 20.0 m), (60.0 m, 80.0 m), E( 10.0 m, 10.0 m), C(40.0 m, 30.0 m), and D( 70.0 m, 60.0 m), and repeat the calculation to show that the answer does not depend on the order of the trees. 61. rectangular parallelepiped has dimensions a, b, and c, as in Figure P3.61. (a) btain a vector epression for the face diagonal vector R 1. What is the magnitude of this vector? (b) btain a vector epression for the bod diagonal vector R 2. Note that R 1, ck, and R 2 make a right triangle, and prove that the magnitude of R 2 is a 2 b 2 c 2. a z b 62. point ling in the plane and having coordinates (, ) can be described b the position vector given b r i j. (a) Show that the displacement vector for a particle moving from ( 1, 1 ) to ( 2, 2 ) is given b d ( 2 1 )i ( 2 1 )j. (b) Plot the position vectors r 1 and r 2 and the displacement vector d, and verif b the graphical method that d r 2 r 1. 63. point P is described b the coordinates (, ) with respect to the normal cartesian coordinate sstem shown in Figure P3.63. Show that (, ), the coordinates of this point in the rotated coordinate sstem, are related to (, ) and the rotation angle b the epressions cos sin sin cos P R 1 R 2 c α Figure P3.61 Figure P3.63 NSWERS T QUICK QUIZZES 3.1 The honebee needs to communicate to the other honebees how far it is to the flower and in what direction the must fl. This is eactl the kind of information that polar coordinates conve, as long as the origin of the coordinates is the beehive. 3.2 The resultant has magnitude when vector is oriented in the same direction as vector. The resultant vector is 0 when vector is oriented in the direction opposite vector and. 3.3 No. In two dimensions, a vector and its components form a right triangle. The vector is the hpotenuse and must be longer than either side. Problem 61 etends this concept to three dimensions. 3.4 No. The magnitude of a vector is equal to 2 2 2 z. Therefore, if an component is nonzero, cannot be zero. This generalization of the Pthagorean theorem is left for ou to prove in Problem 61. 3.5 The fact that 0 tells ou that. Therefore, the components of the two vectors must have opposite signs and equal magnitudes:,, and z z.