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Components nd Elements Bsic Mnipultions Vector unctions Dot nd Cross Product 6Chpter 6: Vectors 147
Chpter 6: Vectors 6.1 Introduction You cn describe quntities such s temperture, time nd length using single number. or emple, ou might find tht n thlete strted jump t time t 8. seconds nd tht he covered distnce l 7.9 metres. or quntities like velocit, ccelertion nd force things re different. Besides mgnitude, these quntities lso hve direction. Emples re wind speed of 10 m/s coming from the West nd n upwrd pulling force of 100 N. Note tht the direction indictors ("from the West" nd "upwrd") re essentil here. The nnouncement tht the wind speed is 10 m/s mens nothing if ou wnt to know if our biccle ride through the flt Dutch lndscpe will be plin siling or not. In order to describe quntit tht hs both mgnitude nd direction, ou cn mke use of vector. This mthemticl object nd the mnipultions tht m be pplied to it, will be discussed in the net sections. irst brief remrk on the different nottions for vectors. Generll, vectors re indicted b letter. Usull lowercse is used, but ou m lso come cross vectors indicted b cpitl or even Greek letter. To distinguish vectors from norml numbers ou cn plce little rrow over the letter, for emple:. However, in this book nother nottion will be used: letter printed in boldfce:. The dvntge of this nottion is tht it is consistent with the nottion tht previls within mtri clcultion (see Chpter 7). The mgnitude (the length) of vector is smbolied in this book b the corresponding normll printed letter:. In other mthemticl tetbooks ou m find for this the nottion. 149
Mths in Motion 6. A Grphicl Introduction A vector is defined epressl b mgnitude nd direction. So other things such s the position in spce, do not pl prt in this definition. You cn represent vector grphicll b mens of n rrow. The point of the rrow indictes the direction, wheres its length stnds for the mgnitude. In this grphicl representtion the dding up of two vectors nd b goes s follows: Plce the strting point of vector b t the end point of vector. You will rrive t the sum vector, or resultnt c + b b drwing n rrow from the strting point of to the end point of b (see the left figure). b c c b The tringle tht is formed b the vectors, b nd c could be etended to prllelogrm. The result is shown in the right figure. Here, the vectors nd b hve been drwn in such w tht the both strt in the sme point. or the resultnt c ou cn now esil verif tht: c + b b + Check tht if nd b hve the sme direction, then the resultnt c lso hs tht direction, nd tht if both vectors lso hve the sme length (so if b ) ou cn write the sum vector s: c + This nottion shows tht vector m be multiplied b number. If this number is positive, differentl sied vector will rise tht hs the sme direction s the originl one. If, on the other hnd, the number is negtive, the resulting vector will point in the opposite direction. 150
Chpter 6: Vectors In the figure below vector b hs been drwn (this is the vector b, multiplied b the number 1). The sum of nd b gives the resultnt d: d b b d + ( b) b This shows tht not onl the dding up of two vectors hs mening but so hs the subtrction of two vectors. 6.3 Components, Elements nd Unit Vectors In order to ctull clculte with vectors it is necessr tht ou cn epress their mgnitude nd direction in numericl vlues. or this ou need reference frme of some sort. In generl, ou will use right-hnded oriented rectngulr coordinte sstem. or the ske of simplicit, in the figure below twodimensionl sitution hs been drwn. However, ll results cn be esil epnded into three dimensions. Becuse position is not one of vector's chrcteristics, the position of vector hs been chosen in such w tht the beginning of the rrow coincides with the origin. is is Perpendiculr projection of the vector's rrowhed onto the is cretes the soclled component of. You cn construct the vector's nd component ( 151
Mths in Motion nd ) nlogousl. In this two-dimensionl cse, the component is so-clled ero vector 0 ( vector with ero mgnitude nd n undefined direction). Check tht the sum of the components produces the vector itself: + + In vector clculus jrgon one ss tht vector hs been decomposed into its components, nd. It will be cler tht the lengths of the components re normtive for both the vector's mgnitude nd direction. Determining these lengths goes s follows: Suppose vector i lies long the is. The vector strts in the origin nd hs its rrowhed pointing towrds the positive prt of the is. Suppose lso tht the length of this vector is defined s 1. Such vector is clled unit vector. is j i is You cn now interpret component s multiple of unit vector i: i Here, is the multipliction fctor. Suppose tht lso long the nd is unit vectors hve been defined (j nd k respectivel). Then, in the sme w s for the component, ou get: j nd k 15
Chpter 6: Vectors So now ou cn lso write tht vector is the sum of its components s follows: i + j + k You cll the multipliction fctors, nd the elements of the vector. It is common prctice to use the following nottion: After decomposing vector into its components ou obtin the mgnitude b using the three-dimensionl form of the Pthgoren Theorem: + + You cn specif the direction of with the three ngles α, β nd γ the vector mkes with the positive, nd is respectivel. The figure below once gin shows two-dimensionl sitution. You cn think of the positive is (not drwn here) s stnding perpendiculr to the pper (pointing out from the pper). Check tht the ngle γ in this cse is 90 o. is β j α i is 153
Mths in Motion If ou know the elements, nd of vector, nd ou cn thus clculte the mgnitude, the following pplies for the cosines of the ngles α, β nd γ : cos α cos β cosγ These re the so-clled direction cosines. In order to find the ngles α, β nd γ, ou hve to determine the rccosines of the frctions. The ngles re defined on the intervl from 0 o to 180 o (from 0 to π rdins), Plese note tht if for vector the ngles with respect to two es hve been given, the ngle with respect to the third is is fied. Mthemticll, this cn be eplined s follows: Rewrite the definitions of the direction cosines: cosα cos β cosγ With this ou cn write the squred mgnitude ( ) of the vector s: + + (cos α + cos β + cos γ ) This epression is correct onl if the following reltion between the direction ngles eists: cos α + cos β + cos γ 1 Emple 6.3.1 or force cting on bod, the following elements re given: 5 N, 10 N nd 10 N. You clculte the mgnitude of the force s follows: + + (5) + (10) + ( 10) 15 154
Chpter 6: Vectors The ngles the force vector mkes with the positive, nd is respectivel, follow from the direction cosines: cosα α rccos 5 rccos 15 o 70.5 cos β β rccos 10 rccos 15 o 48. cosγ γ rccos 10 rccos 131.8 15 o Eercises 6.3 1. Suppose the velocit vector v of point moving in the plne hs the elements v 5 m/s nd v 7 m/s. Wht is the mgnitude of the velocit nd wht ngles does the velocit vector mke with the positive nd is respectivel?. Suppose point moving in the plne. Wht re the elements v nd v of the velocit vector v, if it is given tht v 4 m/s, α 30 o nd β 60 o? 3. Wht re the elements of force vector, if it is given tht the mgnitude of the vector is 5 N nd tht the ngles tht the vector mkes with the positive, nd is re α 90 o, β 45 o nd γ 45 o respectivel? 6.4 Bsic Mnipultions Now tht the elements hve given the vector quntittive (numericl) chrcter, it is possible to further work out the clcultion rules. Equting Two vectors nd b re equl to ech other if the hve the sme mgnitude nd point in the sme direction. This is the cse onl if the elements of both vectors re the sme. So: b b nd b nd b 155
Mths in Motion Multipling b Number You lred cme cross the multipliction of vector b number, when ou were decomposing vector into its components. There, ou wrote the component of vector s the product of number (the element) nd the unit vector i. In generl, if s is number nd vector, it pplies tht: s s s s s So ech of the elements is multiplied b the number s. This keeps the direction of s equl to tht of. However, the mgnitude becomes s times s lrge. Adding nd Subtrcting In element nottion, dding two vectors nd b ppers s follows: + b b + b b + b + b + b When subtrcting, minus signs replce the plus signs. The rest of the procedure is ver similr to tht of dding up. Adding up more thn two vectors is simpl n etension of the bove. The resultnt consists of elements tht re the sum of the corresponding elements of the seprte vectors. So if ou hve to clculte the resultnt of number of vectors, it is importnt to first find the elements of these vectors b decomposing them into their components. After dding up the elements per dimension (, nd ), ou cn clculte the mgnitude nd direction of the resultnt s described in the previous section. 156
Chpter 6: Vectors Emple 6.4.1 Suppose tht two forces, represented b the vectors 1 nd, re cting on point (see figure). Both forces lie in the plne. 1 is pointing stright down (long the negtive is) nd hs mgnitude of 10 N. is mking n ngle of 60 o with the positive is nd hs 0 N mgnitude. is 60 o is 1 In order to be ble to determine the mgnitude nd direction of the resultnt, ou first hve to find the elements of the seprte vectors. or tht ou hve to decompose those vectors into their components. You cn clculte the elements with the help of the direction cosines: o 1 cos 60 0 10 o 1 cos 30 0 3 10 3 Although ou cn write down directl 1 nd 1 (since 1 is running long the is), it is instructive to lso clculte the elements of this vector with the help of the direction cosines: o 1 1 cos 90 10 0 0 o 1 1 cos 180 10 1 10 You cn now dd up the elements per dimension: 1 + 0 + 10 10 1 + 10 + 10 3 7.3 157
Mths in Motion Now tht ou hve the elements of the resultnt ou cn determine its mgnitude nd direction. + (10) + (7.3) 1.4 N cosα α rccos 10 rccos 1.4 36. o cos β β rccos 7.3 o rccos 53.8 1.4 is β α 1 + is 1 Eercises 6.4 In the eercises below, severl forces re cting on point. Clculte the resultnt in ech sitution (its elements, mgnitude nd direction). 1. orce 1 is 5 N, lies in the plne nd mkes n ngle of 45 o with both the positive nd is. orce is 10 N, lso lies in the plne nd mkes 60 o ngle with the positive is nd 150 o ngle with the positive is.. orce 1: 1 4 N; α1 60 o ; β1 60 o ; γ1 45 o orce : 10 N; α 45 o ; β 45 o ; γ 90 o 158
Chpter 6: Vectors 3. See the figure below for the detils. is 15 N 45 o 30 ο 3 0 N is 110 N 4. See the figure below for the detils. orce vector 1 lies in the plne, in the plne. is 1 3 N is is 30 o 5 o 4 N 159
Mths in Motion 6.5 Vector unctions Until now, the vectors ou hve come cross hd fied elements. However, this does not lws hve to be the cse. Elements of vector cn lso vr s function of certin vrible. If this is the sitution, the whole vector vries s function of tht certin vrible. An emple is force vector tht vries s function of time: ( t) ( t) ( t) ( t) Check tht when the elements do not vr to the sme etent (so when the elements differ in the w the depend on time) the force vector will chnge both in mgnitude nd in direction. In Humn Movement Sciences vector functions come in prticulrl hnd when describing vring forces, velocities or ccelertions. You m dd vring positions to this list, if ou specif the position of n object b using so-clled position vector. This is vector tht strts in the origin nd ends t certin point with coordintes (,, ). As consequence, the elements of position vector r re equl to the coordintes of the point towrds which the vector is pointing: r r r r Plese note tht contrr to norml vectors, for position vectors it is certinl importnt wht their position in spce is. Emple 6.5.1 Suppose tht the vector function describing n object's trjector through spce ppers s follows: ( t) t r ( t) ( t) 4t ( t) 10 5t With this ou cn clculte the object's position for n prticulr time t. or instnce, t t 0 the object is in (0, 0, 10) nd t t 1 it is in (, 4, 5). 160
Chpter 6: Vectors 6.5.1 The Derivtive of Vector unction Just s for n other function, ou cn determine the derivtive for vector function c(t). This comes down to differentiting element-wise: c c ( t) c c ( t) ( t) ( t) dc dt dc dc dc / dt / dt / dt The derivtive of vector function is gin vector function. You cn esil see tht this is correct b writing c(t) s the sum of its components nd subsequentl differentiting this epression: c ( t) c ( t) i + c ( t) j c ( t) k + With the product rule nd the sum rule (see Section.3) this becomes: dc dt dc dt i + c di dt dc + dt j + c dj dc + dt dt k + c dk dt When the unit vectors i, j nd k do not depend on t, so when ou re deling with n invrible coordinte sstem, the derivtives of the unit vectors re equl to ero. This mens tht the bove epression reduces to: dc dt dc dc dc i + j + dt dt dt k In the following emples the focus is on the vector functions for position nd velocit, where the ltter (velocit) is the derivtive of the former (position). Emple 6.5.1.1 Suppose tht n object is following trjector s indicted in the figure on the net pge. At certin time t the object is in point A. Some time t lter it hs covered distnce s to rrive t point B. Plese check tht ou cn determine the vector r (drwn in the figure) with: r r( t + t) r( t) 161
Mths in Motion This difference vector r is clled the displcement nd indictes the etent to which the position vector r hs chnged, both in mgnitude nd direction. is r(t) A r s B r(t+ t) is is B dividing the displcement r b the time intervl in which this displcement took plce ( t), ou will get the vector quntit v tht represents the verge velocit during the time intervl: v r t To end up t the momentr velocit v(t) (i.e. the velocit t prticulr moment), ou hve to mke the time intervl infinitesimll smll, which leds to the derivtive (see lso Section.): dr v ( t) dt Suppose, it is given tht: Then, the velocit vector is: ( t) t r ( t) ( t) 4t m ( t) 10 5t d / dt dr v ( t) d / dt 4 m/s dt d / dt 10t 16
Chpter 6: Vectors 6.6 Vector Multiplictions While division hs not been defined for vectors, multipliction comes in two flvours: the dot product nd the cross product. Both products pper to be of use in Humn Movement Sciences, prticulrl in biomechnicl contet. 6.6.1 The Dot Product The dot product of two vectors nd b (nottion: product), is defined s: b, hence the nme dot b b + b + b This mens tht for ech dimension (, nd ) ou hve to multipl the corresponding elements nd dd up the results to one single number. Becuse of this numericl outcome, nother nme for the dot product ou m find in literture is sclr product. It cn be demonstrted 1 tht ou cn lso write the definition of the dot product in the following w: b b cosφ Here, φ is the (smllest) ngle between the vectors nd b with mgnitudes nd b respectivel. B combining both definitions ou get: b + b + b b cosφ This reltion comes in hnd when ou wnt to find the ngle between two vectors for which ou know the elements. 1 The proof, tht mkes use of the cosine rule, is beond the scope of this book. 163
Mths in Motion Emple 6.6.1.1 ind the (smllest) ngle between the vectors 1 nd b 0 3 4 Use the reltion: b b + b + b cosφ b b + b (. 0) + (. 3) + (1. 4) 0 + 6 + 4 10 b + + + + + b b b 0 + + 3 + 1 + 4 9 3 5 5 10 10 o So: 10 (3)(5) cosφ cosφ φ rccos 48. 15 15 Emple 6.6.1. In mechnics the concept work (W) is defined s the mgnitude of force times the mgnitude of the displcement r of the force's point of ppliction in the direction of the force. So if the force nd the displcement re not pointing in the sme direction, s in the figure, then onl the mgnitude of the component r, the component tht is pointing in the direction of the displcement, is contributing to the work. φ r r You find the mgnitude of tht component b projecting the force vector perpendiculrl on the line of displcement: r cosφ With this, the epression for work becomes: W cosφ r r cos( φ) r The sclr quntit work W cn thus be written s the dot product of the force vector nd the displcement vector r. 164
Chpter 6: Vectors Eercises 6.6.1 1. Is the dot product commuttive, in other words: is b b?. Wht does it men if b 0? 3. 1 Are the vectors u 1 nd v 4 perpendiculr to ech other? 1 4. Given re force nd the displcement r of the force's point of ppliction. Clculte the work W nd the ngle φ between the force nd the displcement. 1 if N nd r 3 6 m 6 b. 1 3 if N nd r 6 m 6 5. Given re two vectors u nd v in the plne; u hs mgnitude 6 nd o o o α β 45, v hs mgnitude 15, α 15 nd β 75 (α nd β re the ngles between the vector nd the positive is nd the positive is respectivel). Clculte u v. 6.6. The Cross Product The cross product of two vectors nd b (nottion: product) is defined s: b, hence the nme cross b b b b b b b Becuse the outcome is vector, the cross product is lso referred to s the vector product. 165
Mths in Motion Just s for the dot product there is lso n lterntive definition for the cross product: The result of the cross product mgnitude is given b: b is vector c of which the c bsinφ Here, φ is the (smllest) ngle between nd b. The direction of c is perpendiculr to the surfce tht is spnned b nd b. c φ b This definition does not et determine the direction of the resulting vector unmbiguousl. In the sitution s sketched in the figure, it cn be directed upwrds (s drwn), but lso downwrds. To find the right direction ou hve to ppl the so-clled corkscrew rule: rotte n (imginr) corkscrew from to b over the smllest ngle. The direction in which the corkscrew will move, is the direction of c. Plese note tht this definition is pplicble in right-hnded oriented coordinte sstem onl. Emple 6.6..1 If: 1 3 nd b 4 5 6 then: c b 6 3 5 3 3 4 1 6 6 1 5 4 3 166
Chpter 6: Vectors Emple 6.6.. Suppose ou wnt to find vector tht is perpendiculr to plne in which lie points A(, 0, 0), B(5, 4, 1) nd C(0,, 1). You cn solve this problem when ou know two vectors tht lie in the plne. You then simpl hve to determine the cross product of these two vectors. One of the vectors tht is ling in the plne, is the vector tht runs from A to B. Check tht ou cn find this vector b subtrcting the position vector to A from the position vector to B (see figure). ra A O rab rb ra rb B r AB r B r A 5 3 4 0 4 1 0 1 Another vector in the plne is for emple the one tht runs from B to C. You cn construct this one with: r BC r C r B 0 5 5 4 1 1 0 The cross product of the two vectors found is: r AB r BC 3 5 (4 0) (1 ) 4 (1 5) (3 0) 5 1 0 (3 ) (4 5) 14 This vector is perpendiculr to the plne. If ou hd strted with two other vectors in the plne, ou might hve found different result. However, ll possible vectors ou m find, hve the sme direction. The onl difference is their mgnitude. In other words: their respective elements re the sme ecept for multipliction fctor. 167
Mths in Motion In biomechnics ou use the cross product in problems involving ngulr velocit. The net emple serves to illustrte this. Emple 6.6..3 If n object is following circulr pth with rdius R, then the distnce s trvelled long the circumference is ssocited with certin ngle φ: s R φ v s R φ B differentiting this epression ou will find the speed (i.e. the mgnitude of the velocit) long the circulr pth: v ds dt dφ R dt The quntit dφ/dt, lso referred to s ω, is clled the ngulr speed (unit: rdins per second). So ou cn write v s: v ω R Here, speed nd ngulr speed re sclr quntities (numbers). In order to be ble to del with three-dimensionl problems, ou need to chnge over to vector nottion. You lred know the velocit vector v with its tngentil direction (see Emple 6.5.1.). Although it m seem strnge t first sight, ou cn lso regrd the ngulr speed s vector: the ngulr velocit ω. This is vector tht is perpendiculr to the plne in which the circulr motion tkes plce. In the figure on the net pge the sitution hs been drwn for n object tht rottes counterclockwise round the is. 168
Chpter 6: Vectors is ω R v θ r is is In the figure ou see tht: R r sinθ Becuse v ω R, it follows tht: v ω r sinθ Check tht v is perpendiculr to the plne spnned b ω nd r nd tht ou cn therefore write tht: v ω r In this cse, the corkscrew rule requires the ngulr velocit to be directed upwrd. Check tht if ou were to direct the ngulr velocit downwrd, the object would rotte the other w round. 0 Suppose tht t certin moment: ω 0 rd/s nd 5 1 r 4 m. 3 You cn now clculte tht t tht moment: ω ω 0 3 5 4 0 v ω r ω ω 5 1 0 3 5 m/s ω ω 0 4 0 1 0 169
Mths in Motion Eercises 6.6. 1. Is the cross product commuttive, in other words: is b b?. Wht does it men if b 0? 3. Clculte the cross product b for the following instnces:. 1 3 nd b 4 5 0 b. 1 0 0 nd b 1 0 0 c. 1 3 nd b 6 4 8 4. In 3D spce lies surfce with vertices A(1,, 3), B(, 3, 5), C(3,, 4) nd D(, 1, ). Determine n epression for vectors tht re perpendiculr to this surfce. 5. Suppose tht ou re studing the movements of kicking infnt. or tht purpose ou hve plced mrkers on the hip (H), knee (K) nd foot () of the right leg. With the help of 3D video nlsis ou subsequentl register the sptil positions of these mrkers. At certin moment H (13, 1, 1), K (6, 15, 10) nd (, 11, 4). Determine, for tht moment, vector tht is perpendiculr to the plne defined b H, K nd. 6. An object is moving long circulr pth of which the is of rottion is running through the origin. Given is tht ω i + k [rd/s] nd tht the point (0, 4, 0) [m] lies on the circulr pth.. Determine the velocit vector v in tht point. b. Wht is the mgnitude of v? c. Wht ngle is v mking with the positive is? 170