Chapter 2. Vectors. 1 vv 2 c 2.

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1 Chpter 2 Vetors CHAPTER 2 VECTORS In the first hpter on Einstein s speil theor of reltivit, we sw how muh we ould lern from the simple onept of uniform motion. Everthing in the speil theor n e derived from (1) the ide tht ou nnot detet our own uniform motion, nd (2) the eistene of rel lok tht runs slow ftor 1 vv 2 2. We re now out to stud more omplited kinds of motion where either the speed, the diretion of motion, or oth, re hnging. Our work with non-uniform motion will e sed to lrge etent on onept disovered Glileo out 300 ers efore Einstein developed the speil theor of reltivit. It is interesting tht fter studing omple forms of motion for over 300 ers, we still hd so muh to lern out simple uniform motion. But the histor of siene is like tht. Mjor disoveries often our when we see the simple underling fetures fter long struggle with omple situtions. If our gol is to present sientifi ides in the orderl progression from the simple to the omple, we must epet tht the historil order of their disover will not neessril follow the sme route. Glileo ws studing the motion of projetiles, tring to predit where nnon lls would lnd. He devised set of eperiments involving nnon lls rolling long slightl inlined plnes. These eperiments effetivel slowed down the tion, llowing Glileo to see the w the speed of flling ojet hnged s the ojet fell. To eplin his results Glileo invented the onept of elertion nd pointed out tht the simple feture of projetile motion is tht projetiles move with onstnt or uniform elertion. We n think of this s one step up in ompleit from the uniform motion disussed in the previous hpter. To stud motion tod, we hve mn tools tht were not ville to Glileo. In the lortor we n slow down the tion, or stop it, using stroe photogrphs or television mers. To desrie nd nlze motion we hve numer of mthemtil tools, prtiulrl the onept of vetors nd the sujet of lulus. And to predit motion, to predit not onl where nnon lls lnd ut lso the trjetor of sperft on mission to photogrph the solr sstem, we now hve digitl omputers. As we enter the stud of more omple forms of motion, ou will notie shift in the w ides re presented. Throughout the tet, our gol is to onstrut modern view of nture strting s muh s possile from the si underling ides. In our stud of speil reltivit, the underling ide, the priniple of reltivit, is more urtel epressed in terms of our eperiene fling in jet thn it is n forml set of equtions. As result we were le to etrt the ontent of the theor in series of disussions tht drew upon our eperiene. In most other topis in phsis, ommon eperiene is either not ver helpful or downright misleding. If ou hve driven r, ou know where the elertor pedl is loted nd hve some ide out wht

2 2-2 Vetors elertion is. But unless ou hve lred lerned it in phsis ourse, our view of elertion will er little reltionship to the onept of elertion developed Glileo nd now used phsiists. It is perhps unfortunte tht we use the word elertion in phsis, for we often hve to spend more time dismntling the students previous notions of elertion thn we do uilding the onept s used in phsis. And sometimes we fil. The phsil ides tht we will stud re often simpl epressed in terms of mthemtil onepts like vetor, derivtive, or n integrl. This does not men tht we will drop phsil intuition nd rel on mthemtis. Insted we will use them oth to our est dvntge. In some emples, the phsil sitution is ovious, nd n e used to provide insight into the relted mthemtis. The est w, for emple, to otin solid grip on lulus is to see it pplied to phsis prolems. On the other hnd, the onept of vetor, whose mthemtil properties re esil developed, is n etremel powerful tool for eplining mn phenomen in phsis. VECTORS In this hpter we will stud the vetor s mthemtil ojet. The ide is to hve the onept of vetors in our g of mthemtil tools red for use in our stud of more omple motion, red to e pplied to the ides of veloit, elertion nd lter, fore nd momentum. In sense, we will develop new mth for vetors. We will egin with definition of displement vetors, nd will then eplin how two vetors re dded. From this, we will develop set of rules for the rithmeti of vetors. In some ws, the rules re the sme s those for numers, ut in other ws the re different. We will see tht most of the rules of rithmeti ppl to vetors nd tht lerning the vetor onvention is reltivel simple. Displement Vetors A displement vetor is mthemtil w of epressing the seprtion or displement etween two ojets. To see wht is involved in desriing the seprtion etween ojets, onsider mp suh s the one in Figure (1), whih shows the position of the two ities, New York nd Boston. If we re driving on wellmrked rods, it is suffiient, when plnning trip, to know tht these two ities re seprted distne of 190 miles. However, the pilot of smll plne fling from New York to Boston in fog must know in wht diretion to fl; he must lso know tht Boston is loted t n ngle of 54 degrees est of north from New York. Corning, NY Boston Pittsurgh New York Figure 1 Displement vetors. Boston nd Corning, N. Y., hve equl displements from New York nd Pittsurgh, respetivel. These displements re loted t different prts of the mp, ut the re the sme displement.

3 Arithmeti of Vetors 2-3 The sttement tht Boston is loted distne of 190 miles nd t n ngle of 54 degrees est of north from New York provides suffiient informtion to llow pilot leving New York to reh Boston in the thikest fog. The seprtion or displement etween the two ities is ompletel desried giving oth the distne nd the diretion. Looking gin t Figure (1), we see tht Corning, N.Y., is loted 190 miles, t n ngle of 54 degrees est of north, from Pittsurgh. The ver sme instrutions, trvel 190 miles t n ngle of 54 degrees, will tke pilot from either Pittsurgh to Corning or New York to Boston. If we s tht these instrutions define wht we men the word displement, then we see tht Corning hs the sme displement from Pittsurgh s Boston does from New York. (For our disussion we will ignore the effets of the urvture of the erth.) The displement itself is ompletel desried when we give oth the distne nd diretion, nd does not depend upon the point of origin. The displement we hve een disussing n e represented grphill n rrow pointing in the diretion of the displement (54 degrees est of north), nd whose length represents the distne (190 mi). An rrow tht represents displement is lled displement vetor, or simpl vetor. One thing ou should note is tht vetor tht defines distne nd diretion does not depend on its point of origin. In Figure (1) we hve drwn two rrows; ut the oth represent the sme displement, nd thus re the sme vetor. Arithmeti of Vetors Suppose tht pilot flies from New York to Boston nd then to Bufflo. To his originl displement from New York to Boston he dds displement from Boston to Bufflo. Wht is the sum of these two displements? After these displements he will e 300 miles from New York t n ngle 57 degrees west of north, s shown in Figure (2). This is the net displement from New York, whih is wht we men the sum of the first two displements. If the pilot flies to five different ities, he is dding together five displements, whih we n represent the vetors,,, d, nd e shown in Figure (3). (An rrow pled over smol is used to indite tht the smol represents vetor.) Sine the pilot s net displement from his point of origin, represented the old vetor, is simpl the sum of his previous five displements, we will s tht the old vetor is the sum of the other five vetors. We will write this sum s ( d + e ), ut rememer tht the ddition of vetors is defined grphill s illustrted in Figure (3). If the numers 405 nd 190 re dded, the nswer is 595. But, s seen in Figure (2), if ou dd the vetor representing the 405-mile displement from Boston to Bufflo to the vetor representing the l90-mile displement from New York to Boston, the result is vetor representing 300-mile displement. Clerl, there is differene etween dding numers nd vetors. The plus sign etween two numers hs different mening from tht of the plus sign etween two vetors. d Bufflo 300 mi 405 mi mi New York Boston e d e Figure 2 Addition of vetors. The vetor sum of the displement from New York to Boston plus the displement from Boston to Bufflo is the displement from New York to Bufflo. Figure 3 The sum of five displements,,, d, nd e equls the vetor +++d+e.

4 2-4 Vetors Although vetors differ from numers, some similrities etween the two n e noted, prtiulrl with regrd to the rules of rithmeti. First, we will review the rules of rithmeti for numers, nd then see whih of these rules lso ppl to vetors. Rules for Numer Arithmeti 1. Commuttive lw. In dding two numers, nd, the order of ddition mkes no differene. + = + 2. Assoitive lw. In dding three or more numers,,, nd, we hve ( + ) + = + ( + ) Tht is, if we first dd to, nd then dd, we get the sme result s if we hd dded to the sum ( + ). Rules for Vetor Arithmeti 1. The ommuttive lw implies tht + = + Figure (4 ) verifies this rule grphill. The reder should e le to see tht + nd + re the sme vetors. Figure 4 2. The ssoitive lw pplied to vetors would impl The negtive of numer is defined + ( ) = 0 where ( ) is the negtive of. 4. Sutrtion is defined s the ddition of the negtive numer. = + ( ) These rules re so ovious when pplied to numers tht it is hrd to relize tht the re rules. Let us ppl the foregoing rules to vetors, using the method of ddition of displements. (+)+ = +(+) From Figure (5 ) ou should onvine ourself tht this lw works. + ( ) + + ( ) + = = + ( ) ( + ) + ( ) + + ( ) + Figure 5

5 The negtive of vetor is defined + = 0 The onl w to get zero displement is to return to the point of origin. Thus, the negtive of vetor is vetor of the sme length ut pointing in the opposite diretion (Figure 6). Figure 6 4. The sutrtion of vetors is now es. If we wnt -, we just find +. Tht is = + To sutrt, we just dd the negtive vetor s shown in Figure (7). Figure 7 ( ) Multiplition of Vetor Numer Suppose we multipl vetor the numer 5. Wht do we men the result 5? Let us gin tr to follow the rules of rithmeti to nswer this question. In rithmeti we were tught tht 5=++++ Let us tr the sme rule for vetors. 5 = With this definition we see tht 5 is vetor in the sme diretion s ut five times s long (see Figure 8). Figure 8 5 We m lso multipl vetor negtive numer (see Figure 9); the minus sign just turns the vetor round. For emple, 3 =3 = + + ( 3) Figure 9 When we multipl vetor positive numer, we merel hnge the length of the vetor; multiplition negtive numer hnges the length nd reverses the diretion.

6 2-6 Vetors Mgnitude of Vetor Often we will wnt to disuss onl the length or mgnitude of vetor, regrdless of the diretion in whih it is pointing. For emple, if we represent the displement of Boston from New York the vetor s, then the mgnitude of s (the length of this displement) is 190 miles. We use vertil r on eh side of the vetor to represent the mgnitude ; thus, we write s = 190 mi (see Figure 10). s = 190 mi s Boston Emple 1 The vetor s strts from point nd we would like to redrw it strting from point, s shown in Figure (11). Solution: We wnt to drw line through tht is prllel to s. This n e done with strightedge nd tringle s shown in Figure (12). s Figure 10 New York Vetor Equtions Just s we n solve lgeri equtions involving numers, we n do the sme for vetors. Suppose, for emple, we would like to find the vetor in the vetor eqution Figure 12 s Ruler or stright edge = Solving this eqution the sme w we would n other, we get = 1/2 3/2 Grphill, we find (1/2),, nd ( 3/2); we then vetorill dd these quntities together to get the vetor. Ple the stright edge nd tringle so tht one side of the tringle lies long the stright edge nd the other long the vetor s. Then slide the tringle long the stright edge until the side of the tringle tht ws originll long s now psses through. Drw this line through. If nothing hs slipped, the line will e prllel to s s shown in Figure (13). Grphil Work In the erl setions of this tet, we shll do fir mount of grphil work with vetors. As we n see from the previous emples, the min prolem in grphil work is to move vetor urtel from one prt of the pge to nother. This is esil done with plsti tringle nd ruler s desried in the following emple. Figure 13 s

7 2-7 We now hve the diretion of s strting from. Thus, we hve onl to put in the length. This is most esil done mrking the length of s on the edge of piee of pper nd reproduing this length, strting from s shown in Figure (14). Eerise 2 Assoitive lw Use the ter out pge 2-20 for the vetors of Figure (16), find () + (in lk); () ( +)+ (in red); s () ( +) (in lk); (d) +(+) (in lue). s Figure 16 Figure 14 B eing reful, using shrp penil, nd prtiing, ou should hve no diffiult in performing urte nd rpid grphil work. The prtie n e gined doing Prolems 1 through 5. (Note tht it is essentil to distinguish vetor from numer. Therefore, when ou re solving prolems or working on lortor eperiment, it is reommended tht ou lws ple n rrow over the smol representing vetor.) Eerise 1 Commuttive lw The vetors,,nd of Figure 15 re shown enlrged on the ter out pge Using tht pge for our work, find () ++ (in lk); () ++ (in red); () ++ (in lue). (Lel ll our work.) Does the ommuttive lw work? Figure 15 Eerise 3 Sutrtion Use the ter out pge 2-21 for the three vetors,, nd shown in Figure (17), find the following vetors grphill, leling our results. () + () () (d) ( )+( ) (e) + Figure 17 Eerise 4 Equtions Suppose tht phsil lw is given the vetor eqution P i =P f Suppose tht P f is the sum of two vetors; tht is, P f = P f1 + P f2 Given the two vetors P i nd P f1 (Figure 18), find P f2. (These vetors re found on the ter out pge 2-22.) p i Figure 18 p f1

8 112 mi 2-8 Vetors Eerise 5 Assume tht the vetors P f, P f1, nd P f2 re relted the vetor lw: P f = P f1 + P f2 In ddition, the mgnitudes of the vetors re relted P f = Pf1 + Pf2 If ou re given P f nd onl the diretion of P f1 (Figure 19), find P f1 nd P f2 grphill. (These vetors re found on the ter out pge 2-22.) Figure 19 p f diretion of p f1 COMPONENTS Another w to work with vetors, one tht is espeill onvenient for solving numeril prolems, is through the use of oordinte sstem nd omponents. To illustrte this method, suppose we were giving instrutions to pilot on how to fl from New York to Boston. One w, whih we hve mentioned, would e to tell the pilot oth the diretion nd the distne she must fl, s fl t n ngle of 54 degrees est of north for distne of 190 miles. But we ould lso tell her fl 132 miles due est nd then fl 112 miles due north. This seond routing, whih desries the displement in terms of its esterl nd northerl omponents, s illustrted in Figure (20), is less diret, ut will lso led the pilot to Boston. We n use the sme lternte tehnique to desrie vetor drwn on piee of pper. In Figure (20), we drew two lines to indite esterl nd northerl diretions. We hve drwn the sme lines in Figure (21), ut now we will s tht these lines represent the nd diretions. The lines themselves re lled the nd es, respetivel, nd form wht is lled oordinte sstem. Just s the displement from New York to Boston hd oth n esterl nd northerl omponent, the vetor in Figure (21) hs oth n nd omponent. In ft, the vetor is just the sum of its omponent vetors nd : = + (1) north Boston 190 mi 54 New York 154 mi Figure 20 Two ws to reh Boston from New York. est θ Figure 21 Component vetors. The sum of the omponent vetors nd is equl to the vetor.

9 2-9 Trigonometr n e used to find the length or mgnitude of the omponent vetors; we get = os θ (2) = sinθ (3) Often we will represent the mgnitude of omponent vetor not using the rrow, s ws done in the foregoing equtions. (The equl sign with three rs,, simpl mens tht is defined to e the sme smol s.) It is ommon terminolog to ll the mgnitude of omponent vetor simpl the omponent; for emple, is lled the omponent of the vetor. Addition of Vetors Adding Components An importnt use of omponents is s mens for hndling vetors numerill rther thn grphill. We will show how this works using n emple of the ddition of vetors dding omponents. Consider the three vetors shown in Figure (22). Sine eh vetor is the vetor sum of its individul omponents vetors, we hve Eqution 4 gives us new w to dd vetors, s illustrted in Figure (23). Previousl we would hve dded the vetors,, nd diretl, s shown in Figure (24). The new rule shows how we n first dd the omponents ( + + ) s shown in Figure (23), then seprtel dd the omponents ( + + ) s shown in Figure (23), nd then dd these vetor sums vetorill, s shown in Figure (23), to get the vetor ( + + ). () () () Figure 23 ( + + ) ( + + ) ( + + ) ( + + ) ( + + ) = + = + = + B dding ll three vetors,, nd together, we get + + = ( + ) + ( + ) + ( + ) The right-hnd side of this eqution m e rerrnged to give ( + + ) + + = ( + + ) + ( + + ) (4) Figure 24 Figure 22

10 2-10 Vetors The dvntge of using omponents is tht we n numerill dd or sutrt the lengths of vetors tht point in the sme diretion. Thus, to dd 500 vetors, we would ompute the lengths of ll the omponents nd dd (or sutrt) these together. We would then dd the lengths of the omponents, nd finll, we would vetorill dd the resulting nd omponents. Sine the nd omponents re t right ngles, we m find the totl length nd finl diretion using the Pthgoren theorem nd trigonometr, s shown in Figure (25). θ Figure 25 It is not neessr to lws hoose the omponents horizontll nd the omponents vertill. We m hoose oordinte sstem (', ') tilted t n ngle, s shown in Figure (26). To use the lnguge of the mthemtiin, ' is the omponent of (or projetion of) in the diretion '. We see tht the vetor sum of ll the omponent vetors still dds up to the vetor itself. Figure 26 ' oordinte sstem (', ') ' ' ' 2 = tn θ = ' ' = ' + ' Eerise 6 Imgine ou re given the vetors,, nd nd the two sets of oordinte es ( 1, 1 ) nd ( 2, 2 ) shown in Figure (27). Using the vetors found on the ter out pge 2-23 Figure 27 ) Find (++) diret ddition of vetors. ) Choose 1 nd 1 s our oordinte es. Find (in red) the 1 nd 1 omponents of,,. Then (i) Find ( ) (ii) Find ( ) (iii) Find ( ) + ( ). How does this ompre with (++)? ) Repet prt () for the oordinte is ( 2, 2 ). Vetor Equtions in Component Form Often we will run into sitution where we hve vetor eqution of the form = + ut ou hve to solve the eqution using omponents. This is es to do, euse to go from vetor eqution to omponent equtions, just rewrite the eqution three (or two) times, one for eh omponent. The ove eqution eomes = + = + z = z + z

11 2-11 VECTOR MULTIPLICATION We hve seen how the rules work for vetor ddition, sutrtion, nd the multiplition of vetor numer. Does it mke n sense to multipl two vetors together? In onsidering the multiplition of the two vetors, the first question to nswer is: wht is the result? Wht kind of thing do we get if we multipl vetor pointing est vetor pointing north? Do we get vetor pointing in some third diretion? Do we get numer tht does not point? Or do we get some quntit more omple thn vetor? And perhps more importnt question wh would one wnt to multipl two vetors together? We will see in the stud of phsis tht there re vrious resons wh we will wnt to multipl vetors, nd we n get vrious nswers. One kind of multiplition produes numer; this is lled slr multiplition or the dot produt. We will see emples of slr multiplition shortl. A few hpters lter we will enounter the vetor ross produt where the result of the multiplition of two vetors is itself vetor, one tht points in diretion perpendiulr to the two vetors eing multiplied together. Finll there is form of multiplition tht leds to quntit more omple thn vetor, n ojet lled tensor or mtri. A tensor is n ojet tht mintins the diretionl nture of oth vetors involved in the produt. Tensors re useful in the forml mthemtil desription of the si lws of phsis, ut re not needed nd will not e used in this tet. The nmes slr, vetor, nd tensor desrie hierrh of mthemtil quntities. Slrs re numers like, 1, 3, nd -7, tht hve mgnitude ut do not point nwhere. Vetors hve oth mgnitude nd diretion. Tensors hve the si properties of oth vetors used to onstrut them. In ft there re higher rnk tensors tht hve the properties of 3, 4, or more vetors. People working with Einstein s generlized grvittionl theor hve to work ll the time with tensors. One of the remrkle disoveries of the twentieth entur is tht there is lose reltionship etween the mthemtil properties of slrs, vetors, nd tensors, nd the phsil properties of the vrious elementr prtiles. Lter on we will disuss prtiles suh s the π meson now used in ner reserh, the photon whih is the prtile of light ( em of light is em of photons), nd the grviton, the prtile hpothesized to e responsile for the grvittionl fore. It turns out tht the phsil properties of the π meson resemle the mthemtil properties of slr, the properties of the photon re desried vetor (we will see this lter in the tet), nd it requires tensor to desrie the grviton (tht is wh people working with grvittionl theories hve to work with tensors). One of the surprises of phsis nd mthemtis is tht there re prtiles like the eletron, proton nd neutron, the si onstituents of toms, tht re not desried slrs, vetors, or tensors. To desrie these prtiles, new kind of mthemtil ojet hd to e invented n ojet lled the spinor. The spinor desriing the eletron hs properties hlf w etween slr nd vetor. No one knew out the eistene of spinors until the disover ws fored the need to eplin the ehvior of eletrons. In this tet we will not go into the mthemtis of spinors, ut we will enounter some of the unusul properties tht spinors hve when we stud the ehvior of eletrons in toms. In ver rel sense the spinor nture of eletrons is responsile for the periodi tle of elements nd the entire field of hemistr. In this tet we n disuss gret mn phsil onepts using onl slrs or vetors, nd the two kinds of vetor produts tht give slr or vetor s result. We will first disuss the slr or dot produt whih is some ws is lred fmilir onept, nd then the vetor or ross produt whih pls signifint role lter in the tet.

12 2-12 Vetors The Slr or Dot Produt In slr produt, we strt with two vetors, multipl them together, nd get numer s result. Wht kind of mthemtil proess does tht involve? The Pthgoren theorem provides prt of the nswer. Suppose tht we hve vetor whose nd omponents re nd s shown in Figure (28). Then the mgnitude or length of the vetor is given the Pthgoren theorem s 2 = Figure 28 (4) In some sense 2 is the produt of the vetor with itself, nd the nswer is numer tht is equl to the squre of the length of the vetor. Now suppose tht we use different oordinte sstem ', ' shown in Figure (29) ut hve the sme vetor. In this new oordinte sstem the length of the vetor is given the formul To formlize this onept, we will define the slr produt of the vetor with itself s eing the squre of the length of. We will denote the slr produt using the dot smol to denote slr multiplition: Slr produt of with itself 2 (6) From Equtions (4) nd (6) we hve in the (, ) oordinte sstem = (7) In the ( ', ' ) oordinte sstem we get = ' 2 + ' 2 (8) The ft tht the length of the vetor is the sme in oth oordinte sstems mens tht this slr or dot produt of with itself hs the sme vlue even though the omponents or piees 2, or ', ' re different. In more forml lnguge, we n s tht the slr produt is unhnged, or invrint under hnges in the oordinte sstem. Bsill we n s tht there is phsil mening to the quntit (i.e. the length of the vetor) tht does not depend upon the oordinte sstem used to mesure the vetor. 2 = Figure 29 2 ' + ' ' ' 2 ' ' (5) Eerise 7 Find the dot produt for vetor with omponents,, z in three dimensionl spe. How does the Pthgoren theorem enter in this se? The omponents ' nd ' re different from nd, ut we know tht the length of hs not hnged, thus 2 must e the sme in Equtions (4) nd (5). We hve found quntit 2 whih hs the sme vlue in ll oordinte sstems even though the piees 2 nd 2 hnge from one oordinte sstem to nother. This is the ke propert of wht we will ll the slr produt.

13 2-13 The emple of lulting ove gives us lue to guessing more generl definition of dot or slr produts when we hve to del with the produt of two different vetors nd. As guess let us tr s definition + (9) or in three dimensions + + z z (10) This definition of dot produt does not represent the length of either or ut perhps hs the speil propert tht its vlue is independent of the hoie of oordinte sstem, just s hd the sme vlue in n oordinte sstem. To find out we need to lulte the quntit + + z z in nother oordinte sstem nd see if we get the sme nswer. We will do simple se to show tht this is true, nd leve the more generl se to the reder. Figure 30 Suppose we hve two vetors nd seprted n ngle q s shown in Figure (30). Let the lengths nd e denoted nd respetivel. Choosing oordinte sstem (, ) where the is lines up with, we hve =, =0 = os θ, = sin θ nd the dot produt, Eqution 9, gives θ = + = os θ +0 = = os θ = sin θ ' ' Figure 31 θ = sin θ = os θ =0 = Net hoose oordinte sstem, rotted from, n ngle 90 θ s shown in Figure (31). Here lies long the is nd the dot produt is given = + = os θ +0 Agin we get the result = os θ (11) Eqution (11) holds no mtter wht oordinte sstem we use, s ou n see working the following eerise. Eerise 8 Choose oordinte sstem, where the is is n ngle φ elow the horizontl s shown in Figure (32). First lulte the omponents,,, nd then show tht ou still get + = os θ Figure 32 '' φ '' To do this prolem, ou need the following reltionships. sin θ + φ = sin θ os φ + os θ sin φ os θ + φ = os θ os φ sin θ sin φ sin 2 φ + os 2 φ = 1 θ for n ngle φ ) (This prolem is muh messier thn the emple we did.)

14 2-14 Vetors Interprettion of the Dot Produt When nd re the sme vetor, then we hd = 2 whih is just the squre of the length of the vetor. If nd re different vetors ut prllel to eh other, then θ = 0, os θ = 1, nd we get = θ In other words the dot produt of prllel vetors is just the produt of the lengths of the vetors. Another etreme is when the vetors re perpendiulr to eh other. In this se θ =90, os θ =0 nd = 0. The dot θ = 90 produt of perpendiulr vetors is zero. In sense the dot produt of two vetors mesures the prllelism of the vetors. If the two vetors re prllel, the dot produt is equl to the full produt. If the re perpendiulr, we get nothing. If the re t some intermedite ngle, we get numer etween nd zero. Inresing θ more, we see tht if the vetors re seprted n ngle etween 90 nd 180 s in Figure (33), then the os θ nd the dot produt re negtive. A negtive dot produt indites n nti-prllelism. The etreme se is θ = 180 where =. Figure 33 Here osθ is negtive. Phsil Use of the Dot Produt We hve seen tht the dot produt is given the simple formul = os θ nd it hs the speil propert tht + + z z hs the sme vlue in n oordinte sstem even though the omponents, et., re different in different oordinte sstems. The ft tht is the sme numer in different oordinte sstems mens tht it is trul numer with no dependene on diretion. Tht is wht we men slr quntit. This is speil propert euse is mde up of the vetors nd tht do depend upon diretion nd whose vlues do hnge when we go to different oordinte sstems. In phsis there re quntities like displements, veloities v, fores F tht ll ehve like vetors. All point somewhere nd hve omponents tht depend upon our hoie of diretion. Yet we will del with other quntities like energ whih does not point nwhere. Energ hs mgnitude ut no diretion. Yet our formuls for energ involve the vetors, v, nd F. How n we onstrut numers or slrs from vetors? The nswer is - tke dot or slr produts of the vetors. This is the mthemtil reson wh most of our formuls for energ will involve dot produts.

15 2-15 Vetor Cross Produt The other kind of vetor produt we will use in this ourse is the vetor ross produt where we multipl two vetors nd together to get third vetor. The nottion is = (12) where the nme ross produt omes from the ross we ple etween the vetors we re multipling together. When ou first enounter the ross produt, it does not seem prtiulrl intuitive. But we use it so muh in lter hpters tht ou will get quite used to it. Perhps the est proedure is to skim over this mteril now, nd refer k to it lter when we strt using it in vrious phsis pplitions. To define the ross produt =, we hve to define not onl the mgnitude ut lso the diretion of the resulting vetor. Strting with two vetors nd pointing in different diretions s in Figure 34, wht unique diretion is there for to point? Should point hlf w etween nd, or should it e loser to euse is longer thn? No, there is nothing prtiulrl unique or ovious out n of the diretions in the plne defined nd. The onl trul unique diretion is perpendiulr to this plne. We will s tht points in this unique diretion s shown in Figure 35. The diretion perpendiulr to the plne of nd is not quite unique. The vetor ould point either up or down s indited the solid or dotted vetor in Figure 35. To selet etween these two hoies, we use wht is lled the right hnd rule whih n e stted s follows: Point the fingers of our right hnd in the diretion of the first of the two vetors in the ross produt (in this se the vetor ). Then url our fingers until the point in the diretion of the seond vetor (in this se ), s shown in Figure 36. If ou orient our right hnd so tht this urling is phsill possile, then our thum will point in the diretion of the ross produt vetor. Figure 34 = θ Figure 35 Eerise 9 Wht diretion would the vetor point if ou used our left hnd rther thn our right hnd in the ove rule? We sid tht the vetor ross produt ws not prtiulrl intuitive onept when ou first enounter it. In the ove eerise, ou see tht if ident ou use our left hnd rther thn our right hnd, = will point the other w. One n resonl wonder how ross produt ould pper in n lw of phsis, for wh would nture prefer right hnd rules over left hnded rules. It seems unelievle tht n si onept should involve nthing s ritrr s the right hnd rule. There re two nswers to this prolem. One is tht in most ses, nture hs no preferene for right hndedness over left hndedness. In these ses it turns out tht n lw of phsis tht involves right hnd rules turns out to involve n even numer of them so tht n phsil predition does not depend upon whether ou used right hnd rule or left hnd rule, s long s ou use the sme rule throughout. Sine there re more right hnded people thn left hnded people, the right hnd rule hs een hosen s the stndrd onvention. = = Figure 36 Right hnd rule for the vetor ross produt.

16 2-16 Vetors Eerise 10 There is left nd right hndedness in the diretion of the threds on srew or olt. In Figure (37) we show srew with right hnded thred. B this, we men tht if we turn the srew in the diretion tht we n url the fingers of our right hnd, the srew will move through wood in the diretion tht the thum of our right hnd points. Figure 37 Right hnded thred. In Figure (37), we hve left hnd thred. If we turn the srew in the diretion we n url the fingers of our left hnd, the srew will move through the diretion pointed our left thum. Until 1956 it ws elieved tht the si lws of phsis did not distinguish etween left nd right hndedness. The ft tht there re more right hnded thn left hnded people, or tht the DNA used living orgnisms hd right hnded spirl struture (like right hnded thred) ws simpl n historil ident. But then in 1956 it ws disovered tht the elementr prtile lled the neutrino ws fundmentll left hnded. Neutrinos spin like top. If neutrino is pssing ou nd ou point the thum of our left hnd in the diretion the neutrino is moving, the fingers of our left hnd url in the diretion tht the neutrino is spinning. Or we m s tht the neutrino turns in the diretion of left hnded thred, s shown in Figure 38. diretion of motion neutrino diretion of rottion Figure 38 The neutrino is inherentl left hnded ojet. When one psses ou, it spins in the diretion tht the threds on left hnded srew turn. left-hnded srew Figure 37 Left hnded thred For this eerise find some srews nd olts, nd determine whether the threds re right hnded or left hnded. Mnufturers use one kind of thred predomintel over the other. Whih is the predominnt thred? Cn ou lote emples of the other kind of thred? (The est ple to look for the other kind of thred is in the mehnism of some wter fuets. Cn ou find wter fuet where one side uses right hnd thred nd the other left hnd thred? If ou find one, determine whih is the right nd whih the left hnd thred.) Another prtile, lled the nti-neutrino, is right hnded. If ou point the thum of our right hnd in the diretion of motion of n nti-neutrino, the fingers of our right hnd n url in the diretion tht the ntineutrino rottes. T.D. Lee nd N.C. Yng reeived the 1957 Noel prize in phsis for their disover tht some si phenomen of phsis n e used to distinguish etween left nd right hndedness. The ide of right or left hndedness in the lws of phsis will pper in severl of our lter disussions of the si lws of phsis. The point for now is tht hving quntit like the vetor ross produt tht uses the right hnd onvention m e useful tool to distinguish etween left nd right hndedness.

17 2-17 Eerise 11 Go k to Figure 34 where we show the vetors nd, nd drw the vetor ' =. Use the right hnd rule s we stted it to determine the diretion of '. From our result, deide wht hppens when ou reverse the order in whih ou write the vetors in ross produt. Whih of the rithmeti rules does this violte? Mgnitude of the Cross Produt Now tht we hve the right hnd rule to determine the diretion of =, we now need to speif the mgnitude of. Figure 39 A lue s to onsistent definition of the mgnitude of is the ft tht when nd re prllel, the do not define plne. In this speil se there is n entire plne perpendiulr to oth nd, s shown in Figure 39. Thus there is n infinite numer of diretions tht ould point nd still e perpendiulr to oth nd. We n void this mthemtil miguit onl if hs zero mgnitude when nd re prllel. We do not re where points if it hs no length. Figure 40 θ plne of vetors perpendiulr to nd The simplest formul for the mgnitude =, tht is relted to the produt of nd, et hs zero length when nd re prllel is = = sinθ (13) where nd re the lengths of nd respetivel, nd θ is the ngle etween them. Eqution 13 is the definition we will use for the mgnitude of the vetor ross produt. In Eqution 13, we see tht not onl is the ross produt zero when the vetors re prllel, ut is mimum when the vetors re perpendiulr. In the sense tht the dot produt ws mesure of the prllelism of the vetors nd, the ross produt is mesure of their perpendiulrit. If nd re perpendiulr, then the length of is just the produt. As the vetors eome prllel the length of redues to zero. Component Formul for the Cross Produt Sometimes one needs the formul for the omponents of = epressed in terms of the omponents of nd. The result is mess, nd is rememered onl those who frequentl use ross produts. The nswer is = z - z = z - z z = - (14) These formuls re not so d if ou re doing omputer lultion nd ou re letting the omputer evlute the individul omponents. Eerise 12 Assume tht points in the diretion nd is in the plne s shown in Figure 41. B the right hnd rule, will point long the z is s shown. Use Eqution 14 to lulte the mgnitude of z nd ompre our result with Eqution 13. z θ Figure 41

18 2-18 Vetors RIGHT HANDED COORDINATE SYSTEM Notie in Figure 41, we hve drwn n (,, z) oordinte sstem where z rises up from the plne. We ould hve drwn z down nd still hve three perpendiulr diretions. Wh did we selet the upwrd diretion for z? The nswer is tht the oordinte sstem shown in Figure 41 is right hnd oordinte sstem, defined s follows. Point the fingers of our right hnd in the diretion of the first oordinte is (). Then url our fingers towrd the seond oordinte is (). If ou hve oriented our right hnd so tht ou n url our fingers this w, then our thum points in the diretion of the third oordinte is (z). The importne of using right hnded oordinte sstem is tht Eqution 14 for the ross produt epressed s omponents works onl for right hnded oordinte sstem. If ident ou used left hnded oordinte sstem, the signs in the eqution would e reversed. Eerise 13 Deide whih of the (,, z) oordinte sstems re right hnded nd whih re left hnded. z z z () () () z (d) (e) z Figure 42

19 Ter out pge 2-19 Figure 15 Vetors for Eerise 1, pge 7. Find () + + (in lk); () + + () + + (in red); (in lue).

20 2-20 Vetors Ter out pge Figure 16 Vetors for Eerise 2, pge 7. Find () + (in lk); () ( + ) + (in red); () ( + ) (d) + ( + ) (in lk); (in lue).

21 Ter out pge 2-21 Figure 17 Vetors for Eerise 3, pge 7. Find () + () - () - (d) ( - ) + ( - ) (e) + -

22 2-22 Vetors Ter out pge Figure 18 Vetors for Eerise 4, pge 7. P i = P f Suppose tht P f is the sum of two vetors; tht is, P f = P f1 + P f2 Given the two vetors P i nd P f1 (Figure 18), find P f2. p i p f1 Figure 19 Vetors for Eerise 5, pge 8. P f = P f1 + P f2 In ddition, the mgnitudes of the vetors re relted P f 2 = Pf1 2 + Pf2 2 If ou re given P f nd onl the diretion of P f1, find P f1 nd P f2 grphill. p f diretion of p f1

23 Ter out pge 2-23 Figure 27 Vetors for Eerise 6, pge 10. () Find + + diret ddition of vetors. () Choose 1 nd 1 s our oordinte es. Find (in red) the 1 nd 1 omponents of,,. Then 1 (i) Find (ii) Find (iii) Find ( ) + ( ). How does this ompre with ( + + )? 2 1 () Repet prt B for the oordinte is ( 2, 2 ). (ou n use the k side of this pge.) 2

24 2-24 Vetors Ter out pge Figure 27 Vetors for Eerise 6, pge 10, repeted. () Find + + diret ddition of vetors. () Choose 1 nd 1 s our oordinte es. Find (in red) the 1 nd 1 omponents of,,. Then 1 (i) Find (ii) Find (iii) Find ( ) + ( ). How does this ompre with ( + + )? 2 1 () Repet prt B for the oordinte is ( 2, 2 ). 2

25 2-25 Inde A Arithmeti of vetors. See lso Vetor Addition 2-3 Assoitive lw 2-4 Commuttive lw 2-4 Multiplition numer 2-5 Negtive of 2-5 Slr or dot produt 2-12 Sutrtion of 2-5 Vetor ross produt 2-15 Assoitive lw, Eerise 2-7 C Commuttive lw, eerise on 2-7 Components, vetor Formul for ross produt 2-17 Introdution to 2-8 Coordinte sstem, right hnded 2-18 Cross produt Component formul for 2-17 Disussion of 2-15 Mgnitude of 2-17 D Displement vetors Introdution to 2-2 Dot produt Definition of 2-12 Interprettion 2-14 E Equtions, vetor In omponent form 2-10 M Mgnitude of Vetor 2-6 Multiplition of vetors B numer 2-5 Slr or dot produt 2-12 Vetor ross produt 2-15 R Right hnded oordinte sstem 2-18 S Slr dot produt 2-12 Sutrtion of vetors 2-7 V Vetor Addition 2-3 Addition omponents 2-9 Components 2-8 Equtions Eerise on 2-7, 2-8 In omponent form 2-10 Mgnitude of 2-6 Multiplition 2-11 Multiplition, ross produt 2-15 Formul for 2-17 Mgnitude of 2-17 Multiplition, slr or dot produt 2-12 Interprettion of 2-14 Vetors 2-2 Arithmeti of 2-3 Assoitive lw 2-4 Commuttive lw 2-4 Displement 2-2 Multiplition numer 2-5 Negtive of 2-5 Sutrtion of 2-5 X X-Ch 2 Eerise Eerise Eerise Eerise Eerise Eerise Eerise Eerise Eerise Eerise Eerise Eerise Eerise