REVIEW & SUMMARY 55 We net evlute ech term with Eq. 3-24, finding the direction with the right-hnd rule. For the first term here, the ngle f etween the two vectors eing crossed is 0. For the other terms, f is 90.We find c 6(0) 9( ĵ) 8( kˆ ) 12î 12î 9 ĵ 8 kˆ. (Answer) This vector c is perpendiculr to oth nd, fct ou cn check showing tht c = 0 nd c = 0; tht is, there is no component of c long the direction of either or. In generl A cross product gives perpendiculr vector, two perpendiculr vectors hve ero dot product, nd two vectors long the sme is hve ero cross product. Additionl emples, video, nd prctice ville t WilePLUS Review & Summr Sclrs nd Vectors Sclrs, such s temperture, hve mgnitude onl. The re specified numer with unit (10 C) nd oe the rules of rithmetic nd ordinr lger. Vectors, such s displcement, hve oth mgnitude nd direction (5 m, north) nd oe the rules of vector lger. Adding Vectors Geometricll Two vectors nd m e dded geometricll drwing them to common scle nd plcing them hed to til. The vector connecting the til of the first to the hed of the second is the vector sum s. To sutrct from, reverse the direction of to get ; then dd to.vector ddition is commuttive nd oes the ssocitive lw ( ) c ( c ). Components of Vector The (sclr) components nd of n two-dimensionl vector long the coordinte es re found dropping perpendiculr lines from the ends of onto the coordinte es. The components re given cos u nd sin u, (3-5) where u is the ngle etween the positive direction of the is nd the direction of. The lgeric sign of component indictes its direction long the ssocited is. Given its components, we cn find the mgnitude nd orienttion (direction) of the vector using 2 2 2 nd tn (3-6) Unit-Vector Nottion Unit vectors î, ĵ, nd kˆ hve mgnitudes of unit nd re directed in the positive directions of the,, nd es, respectivel, in right-hnded coordinte sstem (s defined the vector products of the unit vectors). We cn write vector in terms of unit vectors s î ĵ kˆ, (3-7) in which î, ĵ, nd kˆ re the vector components of nd,, nd re its sclr components. (3-2) (3-3) To dd vectors in com- Adding Vectors in Component Form ponent form, we use the rules r r r. (3-10 to 3-12) Here nd re the vectors to e dded, nd r is the vector sum. Note tht we dd components is is.we cn then epress the sum in unit-vector nottion or mgnitude-ngle nottion. Product of Sclr nd Vector The product of sclr s nd vector v is new vector whose mgnitude is sv nd whose direction is the sme s tht of v if s is positive, nd opposite tht of v if s is negtive. (The negtive sign reverses the vector.) To divide v s, multipl v 1/s. The Sclr Product The sclr (or dot) product of two vectors nd is written nd is the sclr quntit given cos f, (3-20) in which f is the ngle etween the directions of nd. A sclr product is the product of the mgnitude of one vector nd the sclr component of the second vector long the direction of the first vector. Note tht, which mens tht the sclr product oes the commuttive lw. In unit-vector nottion, ( î ĵ kˆ ) ( î ĵ kˆ ), (3-22) which m e epnded ccording to the distriutive lw. The Vector Product The vector (or cross) product of two vectors nd is written nd is vector c whose mgnitude c is given c sin f, (3-24) in which f is the smller of the ngles etween the directions of nd. The direction of c is perpendiculr to the plne defined nd nd is given right-hnd rule, s shown in Fig. 3-19. Note tht ( ), which mens tht the vector product does not oe the commuttive lw. In unit-vector nottion, ( î ĵ kˆ ) ( î ĵ kˆ ), (3-26) which we m epnd with the distriutive lw.
56 CHAPTER 3 VECTORS Questions 1 Cn the sum of the mgnitudes of two vectors ever e equl to the mgnitude of the sum of the sme two vectors If no, wh not If es, when 2 The two vectors shown in Fig. 3-21 lie in n plne. Wht re the signs of the nd components, respectivel, of () d 1 d2, () d 1 d2, nd (c) d 2 d1 3 Being prt of the Gtors, the Universit of Florid golfing tem must pl on putting green with n lligtor pit. Figure 3-22 shows n overhed view of one putting chllenge of the tem; n coordinte sstem is superimposed. Tem memers must putt from the origin to the hole, which is t coordintes (8 m, 12 m), ut the cn putt the golf ll using onl one or more of the following displcements, one or more times (8 m)î (6 m)ĵ, d2 (6 m)ĵ, (8 m)î. d 1 The pit is t coordintes (8 m, 6 m). If tem memer putts the ll into or through the pit, the memer is utomticll trnsferred to Florid Stte Universit, the rch rivl. Wht sequence of displcements should tem memer use to void the pit nd the school trnsfer 4 Eqution 3-2 shows tht the ddition of two vectors nd is commuttive. Does tht men sutrction is commuttive, so tht 5 Which of the rrngements of es in Fig. 3-23 cn e leled right-hnded coordinte sstem As usul, ech is lel indictes the positive side of the is. ( ) (d ) ( ) (e ) Figure 3-23 Question 5. d 2 d 1 d 3 Figure 3-21 Question 2. Hole Gtor pit Figure 3-22 Question 3. (c ) ( f ) 6 Descrie two vectors nd such tht () c nd c; () ; (c) c nd 2 2 c 2. 7 If d ( c), does () ( d) c ( ), () ( ) d c, nd (c) c ( d) 8 If c, must equl c 9 If F q( v B ) nd v is perpendiculr to B, then wht is the direction of B in the three situtions shown in Fig. 3-24 when constnt q is () positive nd () negtive F F v v (1) (2) (3) Figure 3-24 Question 9. 10 Figure 3-25 shows vector nd four other vectors tht hve the sme D B mgnitude ut differ in orienttion. () Which of those other four vectors hve the sme dot product with A () A Which hve negtive dot product with A C 11 In gme held within threedimensionl E me, ou must move Figure 3-25 Question 10. our gme piece from strt, t coordintes (0, 0, 0), to finish, t coordintes ( 2 cm, 4 cm, 4 cm). The gme piece cn undergo onl the displcements (in centimeters) given elow. If, long the w, the gme piece lnds t coordintes ( 5 cm, 1 cm, 1 cm) or (5 cm, 2 cm, 1 cm), ou lose the gme. Which displcements nd in wht sequence will get our gme piece to finish p 7î 2ĵ 3kˆ r 2î 3ĵ 2kˆ q 2î ĵ 4kˆ s 3î 5ĵ 3 kˆ. 12 The nd components of four vectors,, c, nd d re given elow. For which vectors will our clcultor give ou the correct ngle u when ou use it to find u with Eq. 3-6 Answer first emining Fig. 3-12, nd then check our nswers with our clcultor. 3 3 c 3 c 3 3 3 d 3 d 3. 13 Which of the following re correct (meningful) vector epressions Wht is wrong with n incorrect epression () A ( B C ) (f) A ( B C) () A ( B C ) (g) 5 A (c) A ( B C ) (h) 5 ( B C) (d) A ( B C ) (i) 5 ( B C) (e) A ( B C ) (j) ( A B ) ( B C ) A F v
PROBLEMS 57 Prolems SSM Tutoring prolem ville (t instructor s discretion) in WilePLUS nd WeAssign Worked-out solution ville in Student Solutions Mnul WWW Worked-out solution is t Numer of dots indictes level of prolem difficult ILW Interctive solution is t Additionl informtion ville in The Fling Circus of Phsics nd t flingcircusofphsics.com http//www.wile.com/college/hllid Module 3-1 Vectors nd Their Components 1 SSM Wht re () the component nd () the component of vector in the plne if its direction is 250 counterclockwise from the positive direction of the is nd its mgnitude is 7.3 m r 2 A displcement vector r in the plne is 15 m long nd directed t ngle u 30 in Fig. 3-26. Determine () the component nd () the component of the vector. Figure 3-26 Prolem 2. 3 SSM The component of vector A is 25.0 m nd the component is 40.0 m. () Wht is the mgnitude of A () Wht is the ngle etween the direction of A nd the positive direction of 4 Epress the following ngles in rdins () 20.0, () 50.0, (c) 100. Convert the following ngles to degrees (d) 0.330 rd, (e) 2.10 rd, (f) 7.70 rd. 5 A ship sets out to sil to point 120 km due north. An unepected storm lows the ship to point 100 km due est of its strting point. () How fr nd () in wht direction must it now sil to rech its originl destintion 6 In Fig. 3-27, hev piece of mchiner is rised sliding it distnce d 12.5 m long plnk oriented t ngle u 20.0 to the horiontl. How fr is it moved () verticll nd () horiontll 7 Consider two displcements, one of mgnitude 3 m nd nother Figure 3-27 Prolem 6. of mgnitude 4 m. Show how the displcement vectors m e comined to get resultnt displcement of mgnitude () 7 m, () 1 m, nd (c) 5 m. Module 3-2 Unit Vectors, Adding Vectors Components 8 A person wlks in the following pttern 3.1 km north, then 2.4 km west, nd finll 5.2 km south. () Sketch the vector digrm tht represents this motion. () How fr nd (c) in wht direction would ird fl in stright line from the sme strting point to the sme finl point 9 Two vectors re given (4.0 m)î (3.0 m)ĵ (1.0 m)kˆ nd ( 1.0 m)î (1.0 m)ĵ (4.0 m)kˆ. In unit-vector nottion, find (), (), nd (c) third vector c such tht c 0. 10 Find the (), (), nd (c) components of the sum r of the displcements c nd d whose components in meters re c 7.4, c 3.8, c 6.1; d 4.4, d 2.0, d 3.3. 11 SSM () In unit-vector nottion, wht is the sum if (4.0 m) î (3.0 m) ĵ nd ( 13.0 m) î (7.0 m) ĵ Wht re the () mgnitude nd (c) direction of d 12 A cr is driven est for distnce of 50 km, then north for 30 km, nd then in direction 30 est of north for 25 km. Sketch the vector digrm nd determine () the mgnitude nd () the ngle of the cr s totl displcement from its strting point. 13 A person desires to rech point tht is 3.40 km from her present loction nd in direction tht is 35.0 north of est. However, she must trvel long streets tht re oriented either north south or est west. Wht is the minimum distnce she could trvel to rech her destintion 14 You re to mke four stright-line moves over flt desert floor, strting t the origin of n coordinte sstem nd ending t the coordintes ( 140 m, 30 m). The component nd component of our moves re the following, respectivel, in meters (20 nd 60), then ( nd 70), then ( 20 nd c ), then ( 60 nd 70). Wht re () component nd () component c Wht re (c) the mgnitude nd (d) the ngle (reltive to the positive direction of the is) of the overll displcement 15 SSM ILW WWW The two vec- tors nd in Fig. 3-28 hve equl mgnitudes of 10.0 m nd the ngles re 1 30 nd 2 105. Find the () nd () components of their 2 vector sum r, (c) the mgnitude of r, nd (d) the ngle r mkes with the positive direction of the is. 16 For the displcement vectors 1 (3.0 m)î (4.0 m)ĵ nd O (5.0 m)î ( 2.0 m)ĵ, give in Figure 3-28 Prolem 15. () unit-vector nottion, nd s () mgnitude nd (c) n ngle (reltive to ). Now give î in (d) unit-vector nottion, nd s (e) mgnitude nd (f) n ngle. 17 ILW Three vectors,, nd c ech hve mgnitude of 50 m nd lie in n plne. Their directions reltive to the positive direction of the is re 30, 195, nd 315, respectivel.wht re () the mgnitude nd () the ngle of the vector c, nd (c) the mgnitude nd (d) the ngle of c Wht re the (e) mgnitude nd (f) ngle of fourth vector d such tht ( ) (c d ) 0 18 In the sum A B C, vector A hs mgnitude of 12.0 m nd is ngled 40.0 counterclockwise from the direction, nd vector C hs mgnitude of 15.0 m nd is ngled 20.0 counterclockwise from the direction. Wht re () the mgnitude nd () the ngle (reltive to ) of B 19 In gme of lwn chess, where pieces re moved etween the centers of squres tht re ech 1.00 m on edge, knight is moved in the following w (1) two squres forwrd, one squre rightwrd; (2) two squres leftwrd, one squre forwrd; (3) two squres forwrd, one squre leftwrd. Wht re () the mgnitude nd () the ngle (reltive to forwrd ) of the knight s overll displcement for the series of three moves
58 CHAPTER 3 VECTORS 20 An eplorer is cught in whiteout (in which the snowfll is so thick tht the ground cnnot e distinguished from the sk) while returning to se cmp. He ws supposed to trvel due north for 5.6 km, ut when the snow clers, he discovers tht he ctull trveled 7.8 km t 50 north of due est. () How fr nd () in wht direction must he now trvel to rech se cmp 21 An nt, cred the Sun on hot Tes fternoon, drts over n plne scrtched in the dirt. The nd components of four consecutive drts re the following, ll in centimeters (30.0, 40.0), (, 70.0), ( 20.0, c ), ( 80.0, 70.0). The overll displcement of the four drts hs the components ( 140, 20.0). Wht re () nd () c Wht re the (c) mgnitude nd (d) ngle (reltive to the positive direction of the is) of the overll displcement 22 () Wht is the sum of the following four vectors in unitvector nottion For tht sum, wht re () the mgnitude, (c) the ngle in degrees, nd (d) the ngle in rdins E 6.00 m t 0.900 rd G 4.00 m t 1.20 rd 23 If is dded to C 3.0î 4.0ĵ, the result is vector in the positive direction of the is, with mgnitude equl to tht of C. Wht is the mgnitude of B 24 Vector A, which is directed long n is, is to e dded to vector B, which hs mgnitude of 7.0 m.the sum is third vector tht is directed long the is, with mgnitude tht is 3.0 times tht of A.Wht is tht mgnitude of A 25 Osis B is 25 km due est of osis A. Strting from osis A, cmel wlks 24 km in direction 15 south of est nd then wlks 8.0 km due north. How fr is the cmel then from osis B 26 Wht is the sum of the following four vectors in () unitvector nottion, nd s () mgnitude nd (c) n ngle B F 5.00 m t 75.0 H 6.00 m t 210 A (2.00 m)î (3.00 m)ĵ B 4.00 m, t 65.0 C ( 4.00 m)î ( 6.00 m)ĵ D 5.00 m, t 235 27 If d1 d 2 5d 3, d 1 d 2 3d 3, nd d3 2î 4ĵ, then wht re, in unit-vector nottion, () d1 nd () d2 28 Two eetles run cross flt snd, strting t the sme point. Beetle 1 runs 0.50 m due est, then 0.80 m t 30 north of due est. Beetle 2 lso mkes two runs; the first is 1.6 m t 40 est of due north. Wht must e () the mgnitude nd () the direction of its second run if it is to end up t the new loction of eetle 1 29 Tpicl ckrd nts often crete network of chemicl trils for guidnce. Etending outwrd from the nest, tril rnches (ifurctes) repetedl, with 60 etween the rnches. If roming nt chnces upon tril, it cn tell the w to the nest t n rnch point If it is moving w from the nest, it hs two choices of pth requiring smll turn in its trvel direction, either 30 leftwrd or 30 rightwrd. If it is moving towrd the nest, it hs onl one such choice. Figure 3-29 shows tpicl nt tril, with lettered stright sections of 2.0 cm length nd smmetric ifurction of 60. Pth v is prllel to the is. Wht re the () mgnitude nd () ngle (reltive to the positive direction of the superimposed is) of n nt s displcement from the nest (find it in the figure) if the nt enters the tril t point A Wht re the (c) mgnitude nd (d) ngle if it enters t point B 30 e f Here re two vectors d (4.0 m)î (3.0 m)ĵ nd (6.0 m)î (8.0 m)ĵ. Wht re () the mgnitude nd () the ngle (reltive to î) of Wht re (c) the mgnitude nd (d) the ngle of Wht re (e) the mgnitude nd (f) the ngle of ; (g) the mgnitude nd (h) the ngle of ; nd (i) the mgnitude nd (j) the ngle of (k) Wht is the ngle etween the directions of nd 31 In Fig. 3-30, vector with mgnitude of 17.0 m is directed t ngle 56.0 counterclockwise from the is. Wht re the components () nd () of the vector A second coordinte sstem is inclined ngle 18.0 with respect to the first. Wht re the components (c) nd (d) in this primed coordinte sstem ' ' ' c g A h i 32 In Fig. 3-31, cue of edge length sits with one corner t the origin of n coordinte sstem. A od digonl is line tht etends from one corner to nother through the center. In unit-vector nottion, wht is the od digonl tht etends from the corner t () coordintes (0, Figure 3-31 Prolem 32. 0, 0), () coordintes (, 0, 0), (c) coordintes (0,, 0), nd (d) coordintes (,, 0) (e) Determine the v w m j k u Figure 3-29 Prolem 29. O ' Figure 3-30 Prolem 31. l p s t ' r n ' o q B
PROBLEMS 59 ngles tht the od digonls mke with the djcent edges. (f) Determine the length of the od digonls in terms of. Module 3-3 Multipling Vectors 33 For the vectors in Fig. 3-32, with 4, 3, nd c 5, wht re () the mgnitude nd () the direction of, (c) the mgnitude nd (d) the direction of c, nd (e) the mgnitude nd (f) the direction of c (The is c is not shown.) 34 Two vectors re presented s 3.0î 5.0ĵ nd 2.0î 4.0ĵ. Find (), (), (c) ( ), nd Figure 3-32 (d) the component of long the direction of. (Hint For (d), consider Eq. 3-20 Prolems 33 nd 54. nd Fig. 3-18.) 35 Two vectors, r nd s, lie in the plne. Their mgnitudes re 4.50 nd 7.30 units, respectivel, nd their directions re 320 nd 85.0, respectivel, s mesured counterclockwise from the positive is.wht re the vlues of () r s nd () r s 36 If d nd, then wht is (d 1 1 3î 2ĵ 4kˆ d d 2) (d 1 4d 2 5î 2ĵ kˆ 2) 37 Three vectors re given 3.0î 3.0ĵ 2.0kˆ, 1.0î 4.0ĵ 2.0kˆ, nd c 2.0î 2.0ĵ 1.0kˆ. Find () ( c ), () ( c ), nd (c) ( c ). 38 For the following three vectors, wht is 3C (2A B ) 39 Vector hs mgnitude of 6.00 units, vector hs mgnitude of 7.00 units, nd A B hs vlue of 14.0. Wht is the ngle etween the directions of A nd B 40 Displcement d1 is in the plne 63.0 from the positive direction of the is, hs positive component, nd hs mgnitude of 4.50 m. Displcement d2 is in the plne 30.0 from the positive direction of the is, hs positive component, nd hs mgnitude 1.40 m. Wht re () d, () d1 1 d 2 d 2, nd (c) the ngle etween d nd d 41 SSM ILW WWW Use the definition of sclr product, cos, nd the fct tht to cl- culte the ngle etween the two vectors given 3.0î 3.0ĵ 3.0kˆ nd 2.0î 1.0ĵ 3.0kˆ. 42 In meeting of mimes, mime 1 goes through displcement d1 (4.0 m)î (5.0 m)ĵ nd mime 2 goes through displcement d. Wht re () d, () d, (c) (d 1 d 2) 1 1 2 ( 3.0 m)î (4.0 m)ĵ d 2 d 2 d 2, nd (d) the component of d1 long the direction of c d2 (Hint For (d), see Eq. 3-20 nd Fig. 3-18.) 43 SSM ILW The three vectors in Fig. 3-33 hve mgnitudes 3.00 m, 4.00 m, nd c 10.0 m nd ngle 30.0. Wht re () the component nd () the component of ; (c) the component nd (d) the com- Figure 3-33 Prolem 43. A 2.00î 3.00ĵ 4.00kˆ B 3.00î 4.00ĵ 2.00kˆ C 7.00î 8.00ĵ A 1 2 B ponent of ; nd (e) the component nd (f) the component of c If c p q, wht re the vlues of (g) p nd (h) q 44 In the product F qv B, tke q 2, v 2.0î 4.0ĵ 6.0kˆ nd F 4.0î 20ĵ 12kˆ. Wht then is B in unit-vector nottion if B B Additionl Prolems 45 Vectors A nd B lie in n plne. A hs mgnitude 8.00 nd ngle 130 ; B hs components B 7.72 nd B 9.20. () Wht is 5A B Wht is 4A 3B in () unit-vector nottion nd (c) mgnitude-ngle nottion with sphericl coordintes (see Fig. 3-34) (d) Wht is the ngle etween the directions of A nd 4A 3B (Hint Think it efore ou resort to clcultion.) Wht is A 3.00kˆ in (e) unit-vector nottion nd (f) mgnitudengle nottion with sphericl coordintes φ Figure 3-34 Prolem 45. 46 Vector hs mgnitude of 5.0 m nd is directed est. Vector hs mgnitude of 4.0 m nd is directed 35 west of due north. Wht re () the mgnitude nd () the direction of Wht re (c) the mgnitude nd (d) the direction of (e) Drw vector digrm for ech comintion. 47 Vectors A nd B lie in n plne. A hs mgnitude 8.00 nd ngle 130 ; B hs components B 7.72 nd B 9.20. Wht re the ngles etween the negtive direction of the is nd () the direction of A, () the direction of the product A B, nd (c) the direction of A (B 3.00kˆ ) 48 Two vectors nd hve the components, in meters, 3.2, 1.6, 0.50, 4.5. () Find the ngle etween the directions of nd.there re two vectors in the plne tht re perpendiculr to nd hve mgnitude of 5.0 m. One, vector c, hs positive component nd the other, vector d, negtive component. Wht re () the component nd (c) the component of vector c, nd (d) the component nd (e) the component of vector d 49 SSM A silot sets out from the U.S. side of Lke Erie for point on the Cndin side, 90.0 km due north. The silor, however, ends up 50.0 km due est of the strting point. () How fr nd () in wht direction must the silor now sil to rech the originl destintion 50 Vector d1 is in the negtive direction of is, nd vector d2 is in the positive direction of n is. Wht re the directions of () d2/4 nd () d1/( 4) Wht re the mgnitudes of products (c) d nd (d) d1 (d 1 d 2 2/4) Wht is the direction of the vector resulting from (e) d nd (f) d2 1 d 2 d 1 Wht is the mgnitude of the vector product in (g) prt (e) nd (h) prt (f) Wht re the (i) mgnitude nd (j) direction of d1 (d 2/4)
60 CHAPTER 3 VECTORS 51 Rock fults re ruptures long which opposite fces of rock hve slid pst ech other. In Fig. 3-35, points A nd B coincided efore the rock in the foreground slid down to the right. The net displcement 9 is long the plne of the fult. The horiontl component of AB 9AB is the strike-slip AC. The component of AB tht is 9 directed down the plne of the fult is the dip-slip AD. () Wht is the mgnitude of the net displcement AB 9 if the strike-slip is 22.0 m nd the dip-slip is 17.0 m () If the plne of the fult is inclined t ngle 52.0 to the horiontl, wht is the verticl component of AB 9 Strike-slip Dip-slip A D φ C B Fult plne Figure 3-35 Prolem 51. 52 Here re three displcements, ech mesured in meters d1 4.0î 5.0ĵ 6.0kˆ, d2 1.0î 2.0ĵ 3.0kˆ, nd d. () Wht is r d 1 d 2 3 4.0î 3.0ĵ 2.0kˆ d 3 () Wht is the ngle etween r nd the positive is (c) Wht is the component of d1 long the direction of d2 (d) Wht is the component of d1 tht is perpendiculr to the direction of d2 nd in the plne of d1 nd d2 (Hint For (c), consider Eq. 3-20 nd Fig. 3-18; for (d), consider Eq. 3-24.) 53 SSM A vector of mgnitude 10 units nd nother vector of mgnitude 6.0 units differ in directions 60. Find () the sclr product of the two vectors nd () the mgnitude of the vector product. 54 For the vectors in Fig. 3-32, with 4, 3, nd c 5, clculte (), (), nd (c) c c. 55 A prticle undergoes three successive displcements in plne, s follows d1, 4.00 m southwest; then d2, 5.00 m est; nd finll d3, 6.00 m in direction 60.0 north of est. Choose coordinte sstem with the is pointing north nd the is pointing est.wht re () the component nd () the component of d1 Wht re (c) the component nd (d) the component of d2 Wht re (e) the component nd (f) the component of d3 Net, consider the net displcement of the prticle for the three successive displcements. Wht re (g) the component, (h) the component, (i) the mgnitude, nd ( j) the direction of the net displcement If the prticle is to return directl to the strting point, (k) how fr nd (l) in wht direction should it move 56 Find the sum of the following four vectors in () unit-vector nottion, nd s () mgnitude nd (c) n ngle reltive to. P 10.0 m, t 25.0 counterclockwise from Q 12.0 m, t 10.0 counterclockwise from R 8.00 m, t 20.0 clockwise from S 9.00 m, t 40.0 counterclockwise from 57 SSM If B is dded to A, the result is 6.0î 1.0 ĵ. If B is sutrcted from A, the result is 4.0î 7.0 ĵ.wht is the mgnitude of A 58 A vector d hs mgnitude of 2.5 m nd points north. Wht re () the mgnitude nd () the direction of Wht re (c) 4.0d the mgnitude nd (d) the direction of 3.0d 59 A hs the mgnitude 12.0 m nd is ngled 60.0 counterclockwise from the positive direction of the is of n coordinte sstem. Also, B (12.0 m)î (8.00 m)ĵ on tht sme coordinte sstem. We now rotte the sstem counterclockwise out the origin 20.0 to form n sstem. On this new sstem, wht re () A nd () B, oth in unit-vector nottion 60 If 2c, 4c, nd c 3î 4ĵ, then wht re () nd () 61 () In unit-vector nottion, wht is r c if 5.0î 4.0ĵ 6.0 kˆ, 2.0î 2.0ĵ 3.0 kˆ, nd c 4.0î 3.0ĵ 2.0 kˆ () Clculte the ngle etween r nd the positive is. (c) Wht is the component of long the direction of (d) Wht is the component of perpendiculr to the direction of ut in the plne of nd (Hint For (c), see Eq. 3-20 nd Fig. 3-18; for (d), see Eq. 3-24.) 62 A golfer tkes three putts to get the ll into the hole. The first putt displces the ll 3.66 m north, the second 1.83 m southest, nd the third 0.91 m southwest. Wht re () the mgnitude nd () the direction of the displcement needed to get the ll into the hole on the first putt 63 Here re three vectors in meters d1 3.0î 3.0ĵ 2.0kˆ d2 2.0î 4.0ĵ 2.0kˆ d3 2.0î 3.0ĵ 1.0kˆ. Wht results from () d () d nd (c) d1 (d 2 1 (d 2 1 (d 2 d 3), d 3), d 3) 64 SSM WWW A room hs dimensions 3.00 m (height) 3.70 m 4.30 m. A fl strting t one corner flies round, ending up t the digonll opposite corner. () Wht is the mgnitude of its displcement () Could the length of its pth e less thn this mgnitude (c) Greter (d) Equl (e) Choose suitle coordinte sstem nd epress the components of the displcement vector in tht sstem in unit-vector nottion. (f) If the fl wlks, wht is the length of the shortest pth (Hint This cn e nswered without clculus. The room is like o. Unfold its wlls to fltten them into plne.) 65 A protester crries his sign of protest, strting from the origin of n coordinte sstem, with the plne horiontl. He moves 40 m in the negtive direction of the is, then 20 m long perpendiculr pth to his left, nd then 25 m up wter tower. () In unit-vector nottion, wht is the displcement of the sign from strt to end () The sign then flls to the foot of the tower. Wht is the mgnitude of the displcement of the sign from strt to this new end 66 Consider in the positive direction of, in the positive direction of, nd sclr d. Wht is the direction of /d if d is () positive nd () negtive Wht is the mgnitude of (c) nd (d) /d Wht is the direction of the vector resulting from (e) nd (f) (g) Wht is the mgnitude of the vector product in (e) (h) Wht is the mgnitude of the vector product in (f) Wht re (i) the mgnitude nd (j) the direction of /d if d is positive