Vectors. Chapter14. Syllabus reference: 4.1, 4.2, 4.5 Contents:

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hpter Vetors Syllus referene:.,.,.5 ontents: D E F G H I J K Vetors nd slrs Geometri opertions with vetors Vetors in the plne The mgnitude of vetor Opertions with plne vetors The vetor etween two points Vetors in spe Opertions with vetors in spe Prllelism The slr produt of two vetors The vetor produt of two vetors

8 VETORS (hpter ) OPENING PROLEM n eroplne in lm onditions is flying t 8 km h due est. old wind suddenly lows from the south-west t 5 km h, pushing the eroplne slightly off ourse. Things to think out: How n we illustrte the plne s movement nd the wind using sle digrm? Wht opertion do we need to perform to find the effet of the wind on the eroplne? n you use sle digrm to determine the resulting speed nd diretion of the eroplne? VETORS ND SLRS In the Opening Prolem, the effet of the wind on the eroplne is determined y oth its speed nd its diretion. The effet would e different if the wind ws lowing ginst the eroplne rther thn from ehind it. Quntities whih hve only mgnitude re lled slrs. Quntities whih hve oth mgnitude nd diretion re lled vetors. The speed of the plne is slr. It desries its size or strength. The veloity of the plne is vetor. It inludes oth its speed nd lso its diretion. Other exmples of vetor quntities re: ² elertion ² fore ² displement ² momentum For exmple, frmer Giles needs to remove fene post. He strts y pushing on the post sidewys to loosen the ground. Giles hs hoie of how hrd to push the post nd in whih diretion. The fore he pplies is therefore vetor. DIRETED LINE SEGMENT REPRESENTTION We n represent vetor quntity using direted line segment or rrow. The length of the rrow represents the size or mgnitude of the quntity, nd the rrowhed shows its diretion. For exmple, if frmer Giles pushes the post with fore of 5 Newtons (N) to the north-est, we n drw sle digrm of the fore reltive to the north line. N 5 5N Sle: m represents 5 N

VETORS (hpter ) 85 Exmple Drw sle digrm to represent fore of Newtons in north-esterly diretion. N Sle: m N 5 N E EXERISE. Using sle of m represents units, sketh vetor to represent: Newtons in south-esterly diretion 5 ms in northerly diretion n exvtor digging tunnel t rte of m min t n ngle of ± to the ground d n eroplne tking off t n ngle of ± to the runwy with speed of 5 ms. If represents veloity of 5 ms due est, drw direted line segment representing veloity of: ms due west 75 ms north-est. Drw sle digrm to represent the following vetors: fore of Newtons in the NW diretion veloity of 6 ms vertilly downwrds displement of units t n ngle of 5 ± to the positive x-xis d n eroplne tking off t n ngle of 8 ± to the runwy t speed of 5 km h. VETOR NOTTION onsider the vetor from the origin O to the point. We ll this the position vetor of point. ² This position vetor ould e represented y O or or ~ or O old used in text ooks used y students ² The mgnitude or length ould e represented y j O j or O or j j or j ~ j or j j For we sy tht nd tht is the vetor whih origintes t nd termintes t, is the position vetor of reltive to.

86 VETORS (hpter ) GEOMETRI VETOR EQULITY Two vetors re equl if they hve the sme mgnitude nd diretion. Equl vetors re prllel nd in the sme diretion, nd re equl in length. The rrows tht represent them re trnsltions of one nother. We n drw vetor with given mgnitude nd diretion from ny point, so we onsider vetors to e free. They re sometimes referred to s free vetors. GEOMETRI NEGTIVE VETORS nd re prllel nd equl in length, ut opposite in diretion. nd hve the sme length, ut they hve opposite diretions. We sy tht is the negtive of nd write =. - Exmple Q P S R PQRS is prllelogrm in whih PQ = QR =. Find vetor expressions for: QP RQ SR d SP nd QP = fthe negtive vetor of PQg RQ = fthe negtive vetor of QRg SR = fprllel to nd the sme length s PQg d SP = fprllel to nd the sme length s RQg EXERISE. Stte the vetors whih re: equl in mgnitude prllel in the sme diretion d equl e negtives of one nother. p q r s t

VETORS (hpter ) 87 The figure longside onsists of two equilterl tringles.,, nd lie on stright line. = p, E = q, nd D = r. E D Whih of the following sttements re true? E = r j p j = j q j q r = r d D = q e ED = p f p = q p DEF is regulr hexgon. Write down the vetor whih: i origintes t nd termintes t ii F is equl to. Write down ll vetors whih: E D i re the negtive of EF ii hve the sme length s ED. Write down vetor whih is prllel to nd twie its length. DISUSSION ² ould we hve zero vetor? ² Wht would its length e? ² Wht would its diretion e? GEOMETRI OPERTIONS WITH VETORS In previous yers we hve often used vetors for prolems involving distnes nd diretions. The vetors in this se re displements. typil prolem ould e: km runner runs est for km nd then south for km. N km How fr is she from her strting point nd in wht diretion? x km W E In prolems like these we use trigonometry nd Pythgors theorem to find the unknown lengths nd ngles. S GEOMETRI VETOR DDITION Suppose we hve three towns P, Q, nd R. trip from P to Q followed y trip from Q to R hs the sme origin nd destintion s trip from P to R. This n e expressed in vetor form s the sum PQ + QR = PR. P Q R

88 VETORS (hpter ) This tringulr digrm ould tke ll sorts of shpes, ut in eh se the sum will e true. For exmple: P R P Q Q R R P Q fter onsidering digrms like those ove, we n now define vetor ddition geometrilly: To onstrut + : Step : Drw. Step : t the rrowhed end of, drw. Step : Join the eginning of to the rrowhed end of. This is vetor +. DEMO Exmple Given nd s shown, onstrut +. + THE ZERO VETOR Hving defined vetor ddition, we re now le to stte tht: The zero vetor is vetor of length. For ny vetor : + = + = +( ) =( ) + =. When we write the zero vetor y hnd, we usully write. Exmple Find single vetor whih is equl to: + + E + E + + d + + D + DE E D + = fs showng + E + E = + + = = d + + D + DE = E E D

EXERISE. Use the given vetors p nd q to onstrut p + q: VETORS (hpter ) 89 p q p q q p d e f p q p q p q Find single vetor whih is equl to: + + D d + + D e + + D + f + + Given p nd q use vetor digrms to find: i p + q ii q + p. For ny two vetors p nd q, is p + q = q + p? onsider: Q One wy of finding PS is: R PS = PR + RS =( + )+. Use the digrm to show tht ( + ) + = +( + ). P S 5 nswer the Opening Prolem on pge 8. GEOMETRI VETOR SUTRTION To sutrt one vetor from nother, we simply dd its negtive. = +( ) For exmple, given nd then - -

9 VETORS (hpter ) Exmple 5 For r, s, nd t shown, find geometrilly: r s s t r r s t r-s r -s s s-- t r s -r -t Exmple 6 For points,,, nd D, simplify the following vetor expressions: D = + = fs = g D = + + D = D D EXERISE. For the following vetors p nd q, show how to onstrut p q: d p q p q q p q p

VETORS (hpter ) 9 For the vetors illustrted, show how to onstrut: p q r p + q r p q r r q p For points,,, nd D, simplify the following vetor expressions: + D D d + + D e + f + D VETOR EQUTIONS Whenever we hve vetors whih form losed polygon, we n write vetor eqution whih reltes the vriles. The vetor eqution n usully e written in severl wys, ut they re ll equivlent. Exmple 7 onstrut vetor equtions for: r s p q g d e t r f t = r + s r = p + q f = g + d + e We selet ny vetor for the LHS nd then tke nother pth from its strting point to its finishing point. EXERISE. onstrut vetor equtions for: r s r s s p q t t r d e f p q p q p q r s r t s r u t s

9 VETORS (hpter ) Exmple 8 Find, in terms of r, s, nd t: RS SR ST r R s S O t T RS = RO + OS = OR + OS = r + s = s r SR = SO + OR = OS + OR = s + r = r s ST = SO + OT = OS + OT = s + t = t s Find, in terms of r, s, nd t: i O ii iii O Find, in terms of p, q, nd r: i D ii iii s p r t q O D r GEOMETRI SLR MULTIPLITION slr is non-vetor quntity. It hs size ut no diretion. We n multiply vetors y slrs suh s nd, or in ft ny k R. If is vetor, we define = + nd = + + so =( ) =( )+( )+( ). If is then - - - - So, is in the sme diretion s ut is twie s long s is in the sme diretion s ut is three times longer thn hs the opposite diretion to nd is three times longer thn.

VETORS (hpter ) 9 If is vetor nd k is slr, then k is lso vetor nd we re performing slr multiplition. VETOR SLR MULTIPLITION If k>, k nd hve the sme diretion. If k<, k nd hve opposite diretions. If k =, k =, the zero vetor. Exmple 9 Given vetors r nd s, onstrut geometrilly: r + s r s -s r s -s r r + s r - s -s r Exmple Sketh vetors p nd q if: p =q p = q Suppose q is: q q p =q q p = q EXERISE. Given vetors r nd s, onstrut geometrilly: r s r d s e r s f r +s g r +s h (r +s) Sketh vetors p nd q if: p = q p = q p =q d p = q e p = q

9 VETORS (hpter ) opy this digrm nd on it mrk the points: i X suh tht MX = MN + MP ii Y suh tht MY = MN MP iii Z suh tht PZ = PM Wht type of figure is MNYZ? P M N p D is squre. Its digonls [] nd [D] interset t M. If = p nd = q, find in terms of p nd q: q D M d M D 5 P Q PQRSTU is regulr hexgon. If PQ = nd QR =, find in terms of nd : PX PS QX d RS U R X T S VETORS IN THE PLNE When we plot points in the rtesin plne, we move first in the x-diretion nd then in the y-diretion. For exmple, to plot the point P(, 5), we strt t the y origin, move units in the x-diretion, nd then 5 units P (, 5) in the y-diretion. 5 x In trnsformtion geometry, trnslting point units in the x-diretion nd units in the y-diretion n e hieved using the trnsltion vetor. i nd j re exmples of unit vetors So, the vetor from O to P is OP =. 5 Suppose tht i = is trnsltion unit in the positive x-diretion nd tht j = is trnsltion unit in the positive y-diretion. euse they hve length.

VETORS (hpter ) 95 We n see tht moving from O to P is equivlent to two lots of i plus 5 lots of j. OP =i +5j ) µ 5 = µ +5 y OP = 5 i i P (, 5) j j j j j x The point P(x, y) hs position vetor OP = x = xi + yj y i = j = omponent form is the se unit vetor in the x-diretion. is the se unit vetor in the y-diretion. The set of vetors fi, jg = oordinte system. ½, ¾ unit vetor form is the stndrd sis for the -dimensionl (x, y) ll vetors in the plne n e desried in terms of the se unit vetors i nd j. For exmple: =i j = i +j i i i -j j j j -i -i -i -i Two vetors re equl if their omponents re equl. Exmple y x Write O nd in omponent form nd in unit vetor form. omment on your nswers in. O = =i + j = The vetors O nd re equl. =i + j

96 VETORS (hpter ) EXERISE Write the illustrted vetors in omponent form nd in unit vetor form: d e f Write eh vetor in unit vetor form, nd illustrte it using n rrow digrm: d 5 Find in omponent form nd in unit vetor form: D d e f D D Write in omponent form nd illustrte using direted line segment: i +j i +j 5j d i j 5 Write the zero vetor in omponent form. D onsider vetor v = =i +j. THE MGNITUDE OF VETOR The mgnitude or length of v is represented y jvj. y Pythgors, jvj = + =+9= ) jvj = p units fsine jvj > g If v = µ v v v = v i + v j, the mgnitude or length of v is jvj = p v + v.

VETORS (hpter ) 97 Exmple If p = 5 p = 5 nd q =i 5j find: jpj jqj s i 5j =, 5 ) jpj = p +( 5) = p units jqj = p +( 5) = p 9 units. UNIT VETORS unit vetor is ny vetor whih hs length of one unit. i = nd j = re the se unit vetors in the positive x nd y-diretions respetively. Exmple Find k given tht k is unit vetor. Sine k is unit vetor, q ( ) + k = q ) 9 + k = ) 9 + k = fsquring oth sidesg ) k = 8 9 ) k = p 8 EXERISE D Find the mgnitude of: d e Find the length of: i + j 5i j i +j d i e kj Whih of the following re unit vetors? p p d 5 5 e 7 5 7

98 VETORS (hpter ) Find k for the unit vetors: k k 5 Given v = 8 p k d k k nd jvj = p 7 units, find the possile vlues of p. e µ k E OPERTIONS WITH PLNE VETORS LGERI VETOR DDITION onsider dding vetors = nd = Notie tht: ² the horizontl step for + is + ² the vertil step for + is +. If = µ nd = µ then + = µ. +. + + + + Exmple If = nd =, find +. hek your nswer grphilly. 7 + = + 7 + = +7 5 = Grphil hek: + LGERI NEGTIVE VETORS In the digrm we see the vetor = nd its negtive = If = µ µ. then =. - - -

LGERI VETOR SUTRTION To sutrt one vetor from nother, we simply dd its negtive. So, if = nd = then = +( ) = + = If = µ nd = µ, then = VETORS (hpter ) 99. Exmple 5 Given p =, q =, nd r =, find: 5 q p p q r q p = = + = 6 p q r µ = + = +5 = 5 LGERI SLR MULTIPLITION We hve lredy seen geometri pproh for integer slr multiplition: onsider = Notie tht: ² ( )v =. = + = µ = + + = If k is ny slr nd v = + = = 6 + = = 9 + µ v v, then kv = kv. kv ( ) v v () v = = v ² ()v = = ( ) v v () v =

VETORS (hpter ) Exmple 6 For p =, q = find: q p +q p q q = 6 = 9 p +q = + + () = +( ) 8 = 5 = = p µ q µ () () () ( ) = 9 Exmple 7 If p =i 5j nd q = i j, find jp qj. p q =i 5j ( i j) =i 5j +i +j =5i j VETOR RE GME ) jp qj = p 5 +( ) = p 6 units EXERISE E µ If =, =, nd = 5 find: + + + d + e + f + g + h + + Given p =, q =, nd r = find: 5 p q q r p + q r d p q r e q r p f r + q p onsider =. Use vetor ddition to show tht + =. Use vetor sutrtion to show tht =. For p =, q =, nd r = find: 5 p q p + q d p q e p r f p +r g q r h p q + r

5 onsider p = nd q = VETORS (hpter ). Find geometrilly nd then omment on the results: p + p + q + q + q p + q + p + q + q q + p + q + p + q 6 For r = nd s = find: j r j j s j j r + s j d j r s j e j s r j 7 If p = nd q = find: j p j j p j j p j d j p j e j pj f j q j g j q j h j qj i q j q 8 Suppose x = x = k. µ x x nd = µ. Show y equting omponents, tht if kx = then 9 From your nswers in 7, you should hve notied tht j kv j = j k jjvj. So, (the length of kv) = (the modulus of k) (the length of v). v y letting v =, prove tht j kv j = j k jjvj. v The modulus of k is its size. F THE VETOR ETWEEN TWO POINTS y ( _, ) ( _, ) x In the digrm, point hs position vetor O = nd point hs position vetor O =. ) = O + O = O + O = O O = = µ µ, The position vetor of reltive to is = O O =.

VETORS (hpter ) We n lso oserve this in terms of trnsformtions. In trnslting point to point in the digrm, the trnsltion vetor is. y (, ) (, ) - - x In generl, for two points nd with position vetors nd respetively, we oserve = + = = nd = + = = y x Exmple 8 Given points (, ), (, ), nd (, 5), find the position vetor of: from O from from The position vetor of reltive to O is The position vetor of reltive to is The position vetor of reltive to is O = = = =. µ = µ 5 = 5 7.. Exmple 9 [] is the dimeter of irle with entre (, ). If is (, ), find: the oordintes of. (-, ) (, ) = = If hs oordintes (, ), then = ( ) = +

VETORS (hpter ) ut =, so ) is ( 5, ). + = ) += nd = ) = 5 nd = EXERISE F Find given: (, ) nd (, 7) (, ) nd (, ) (, 7) nd (, ) d (, ) nd (, 5) e (6, ) nd (, ) f (, ) nd (, ) onsider the point (, ). Find the oordintes of: given = [PQ] is the dimeter of irle with entre. given =. Find P. Hene find the oordintes of Q. P (-, ) (, ) Q (, ) (6, 5) D is prllelogrm. Find. Find D. Hene find the oordintes of D. D 5 (, ) nd (, k) re two points whih re 5 units prt. Find nd j j. Hene, find the two possile vlues of k. (, -) Show, y illustrtion, why k should hve two possile vlues. j j is the mgnitude of. (, 5) 6 Find nd (, ) (, -). Explin why = +. Hene find. d hek your nswer to y diret evlution.

VETORS (hpter ) 7 Given = Given = Given PQ =, nd µ = nd µ = RQ =, find., find., nd RS =, find SP. 8 ('\\ 6) Find the oordintes of M. Find vetors, M, nd. Verify tht M = + M. (-'\\ ) (-'\\ ) G To speify points in -dimensionl spe we need point of referene O, lled the origin. Through O we drw mutully perpendiulr lines nd ll them the X, Y, nd Z-xes. We often think of the YZ-plne s the plne of the pge, with the X-xis oming diretly out of the pge. However, we nnot of ourse drw this. In the digrm longside the oordinte plnes divide spe into 8 regions, with eh pir of plnes interseting on the xes. The positive diretion of eh xis is solid line wheres the negtive diretion is dshed. X VETORS IN SPE Z Y ny point P in spe n e speified y n ordered triple of numers (x, y, z) where x, y, nd z re the steps in the X, Y, nd Z diretions from the origin O, to P. The position vetor of P is OP = @ x y = xi + yj + zk z where i = @, j = @, nd k = @ X x Z y P (x, y, z) z Y re the se unit vetors in the X, Y, nd Z diretions respetively.

8 < The set of vetors fi, j, kg = @, @, @ 9 = : ; is the stndrd sis for the -dimensionl (x, y, z) oordinte system. VETORS (hpter ) 5 To help us visulise the -D position of point on our -D pper, it is useful to omplete retngulr prism or ox with the origin O s one vertex, the xes s sides djent to it, nd P eing the vertex opposite O. -D POINT PLOTTER X x Z y P (x, y, z) z Y Exmple Illustrte the points: (,, ) (,, ) (,, ) Z Z Z X Y X Y X - Y THE MGNITUDE OF VETOR X Z P (,, ) Y Tringle O is right ngled t ) O = +... () fpythgorsg Tringle OP is right ngled t ) OP = O + fpythgorsg ) OP = + + p fusing ()g ) OP = + + The mgnitude or length of the vetor v = @ v p v is j v j = v + v + v. v

6 VETORS (hpter ) THE VETOR ETWEEN TWO POINTS If (x, y, z ) nd (x, y, z ) re two points in spe then: = O O = @ x x y y x-step y-step z z z-step is lled the vetor or the position vetor of reltive to. The mgnitude of is = p (x x ) +(y y ) +(z z ) whih is the distne etween the points nd. Exmple If P is (,, ) nd Q is (,, ), find: OP PQ j PQ j OP = @ PQ = @ ( ) = @ p j PQ j = +( ) + = p units VETOR EQULITY Two vetors re equl if they hve the sme mgnitude nd diretion. If = @ nd = @, then =, =, =, =. If nd do not oinide, then they re opposite sides of prllelogrm, nd lie in the sme plne. Exmple D is prllelogrm. is (,, ), is (,, ), nd D is (,, ). Find the oordintes of. Let e (,, ). [] is prllel to [D], nd they hve the sme length, so D = ) @ = @ (-),, D (,, ) (,, -) (,, )

VETORS (hpter ) 7 ) =, ) =6, =, =, nd = nd = So, is (6,, ). ³ + hek: Midpoint of [D] is, +, + whih is 5,, : ³ +6 Midpoint of [] is, +, + whih is 5,,. The midpoints re the sme, so the digonls of the prllelogrm iset. X EXERISE G onsider the point T(,, ). Drw digrm to lote the position of T in spe. Find OT. How fr is it from O to T? Illustrte P nd find its distne from the origin O: P(,, ) P(,, ) P(,, ) d P(,, ) Given (,, ) nd (,, ) find: nd the lengths j j nd j j. Given (,, ) nd (,, ) find O, 5 Given M(,, ) nd N(,, ) find: O, nd. the position vetor of M reltive to N the position vetor of N reltive to M the distne etween M nd N. 6 onsider (,, 5), (,, ), nd (,, ). Find the position vetor O nd its length O. Find the position vetor nd its length. Find the position vetor nd its length. d Find the position vetor nd its length. e Hene lssify tringle. 7 Find the shortest distne from Q(,, ) to: the Y -xis the origin the Y OZ plne. 8 Show tht P(,, ), Q(, 6, 5), nd R(,, ) re verties of n isoseles tringle. 9 Use side lengths to lssify tringle given the oordintes: (,, ), (, 8, ), nd ( 9, 6, 8) (,, ), (,, ), nd (, 6, 6). The verties of tringle re (5, 6, ), (6,, 9), nd (,, ). Use distnes to show tht the tringle is right ngled. Hene find the re of the tringle. sphere hs entre (,, ) nd dimeter [] where is (,, ). Find the oordintes of nd the rdius of the sphere.

8 VETORS (hpter ) Stte the oordintes of ny generl point on the Y -xis. Use nd the digrm opposite to find the oordintes of two points on the Y -xis whih re p units from (,, ). ~` ` X - Z - Y Find,, nd if: @ = @ + @ 5 = @ + 5 Find k given the unit vetor: @ k k @ unit vetor hs length. 5 (,, ), (, 5, ), (,, ), nd D(r, s, t) re four points in spe. Find r, s, nd t if: = D = D 6 qudrilterl hs verties (,, ), (,, ), (7,, 5), nd D(5,, 6). Find nd D. Wht n e dedued out the qudrilterl D? 7 PQRS is prllelogrm. P is (,, ), Q is (,, 5), nd R is (,, ). Use vetors to find the oordintes of S. Use midpoints of digonls to hek your nswer. H OPERTIONS WITH VETORS IN SPE The rules for lger with vetors redily extend from -D to -D: If = @ nd = @ then + = @ + +, = @, + nd k = @ k k for ny slr k. k

VETORS (hpter ) 9 INVESTIGTION PROPERTIES OF VETORS IN SPE There re severl properties or rules whih re vlid for the ddition of rel numers. For exmple, we know tht + = +. Our tsk is to identify some similr properties of vetors. Wht to do: Use generl vetors = @, = @, nd = @ to find: + nd + + +( ) nd ( + ) d ( + ) + nd +( + ) Summrise your oservtions from. Do they mth the rules for rel numers? Prove tht for slr k nd vetors = @ nd = @ : jkj = jkjjj k( + ) =k + k From the Investigtion you should hve found tht for vetors,, nd k R : ² + = + fommuttive propertyg ² ( + )+ = +( + ) fssoitive propertyg ² + = + = fdditive identityg ² +( ) =( )+ = fdditive inverseg ² jk j {z} = {z} j k j {z} j j where k is prllel to length of k length of modulus of k ² k( + ) = k + k fdistriutive propertyg The rules for solving vetor equtions re similr to those for solving rel numer equtions. However, there is no suh thing s dividing vetor y slr. Insted, we multiply y the reiprol slr. For exmple, if x = then x = nd not. hs no mening in vetor lger. Two useful rules re: ² if x + = then x = ² if kx = then x = k (k 6= ) Proof: ² If x + = then x + +( ) = +( ) ) x + = ) x = ² If kx = then k (kx) = k ) x = k ) x = k

VETORS (hpter ) Exmple Solve for x: x r = s x = d x r = s ) x = s + r ) x = (s + r) x = d ) d =x ( d) =x ) Exmple If = @, find j j. j j =jj = p ( ) + + = p +9+ = p units Exmple 5 Find the oordintes of nd D: (-,-5, ) (-,-, ) D = ( ) @ ( 5) = @ O = O + = O + = @ 5 + @ 6 ) is (,, ) = @ OD = O + D = O + = @ 5 + @ 9 = @ ) Dis (,, ) EXERISE H Solve the following vetor equtions for x: x = q x = n x = p d q +x = r e s 5x = t f m x = n Suppose = @ nd = @. Find x if: + x = x = x =

If O = @ nd O = @ VETORS (hpter ), find nd hene the distne from to. For (,, ) nd (,, ) find: in terms of i, j, nd k the mgnitude of 5 If = @ nd = @, find: j j j j j j d j j 6 If e j j f j j g j + j h j j = i j + k nd = i + j k find in terms of i, j, nd k. 7 onsider the points (,, ), (,, ), (,, ), nd D(,, ). Dedue tht D =. 8 Find the oordintes of, D, nd E. (-'\\ 5' ) ('\ ' -) D E 9 Use vetors to determine whether D is prllelogrm: (, ), (, ), (, ), nd D(, ) (5,, ), (,, ), (,, 6), nd D(, 5, 5) (,, ), (,, ), (, 6, ), nd D(,, ). Use vetor methods to find the remining vertex of: (, ) (,-) P (-,, ) Q (-, 5, ) W (-, 5, 8) X D (8,-) S (,, 7) R Z (,, 6) Y (,-,-) D In the given figure [D] is prllel to [O] nd hlf its length. Find, in terms of nd, vetor expressions for: D d OD e D f D O If = @, = @, nd D = @, find: D D

VETORS (hpter ) For = @, = @, nd = @, find: + + d e f + If = @, = @, nd = @, find: j j j j j + j d j j e j j f j j 5 Find slrs,, nd : @ = @ @ + @ + @ = @ @ + @ 7 = @ 7 9 I PRLLELISM Qe re prllel vetors of different length. Two non-zero vetors re prllel if nd only if one is slr multiple of the other. Given ny non-zero vetor v nd non-zero slr k, the vetor kv is prllel to v. ² If is prllel to, then there exists slr k suh tht = k. ² If = k for some slr k, then I is prllel to, nd I j j = j k jj j. j k j is the modulus of k, wheres j j is the length of vetor.

VETORS (hpter ) Exmple 6 Find r nd s given tht = @ r is prllel to = @ s. Sine nd re prllel, = k for some slr k. ) @ = k @ s r ) =ks, =k nd r = k onsequently, k = nd ) = s nd r = ( ) ) r = nd s = UNIT VETORS Given non-zero vetor v, its mgnitude j v j is slr quntity. If we multiply v y the slr The length of this vetor is j v j, we otin the prllel vetor j v j v. j v j j v j = j v j j v j =, so v is unit vetor in the diretion of v. j v j ² unit vetor in the diretion of v is j v j v. ² vetor of length k in the sme diretion s is = k j j. ² vetor of length k whih is prllel to ould e = k j j. Exmple 7 If =i j find: unit vetor in the diretion of vetor of length units in the diretion of vetors of length units whih re prllel to. j j = p +( ) = p 9+ = p units ) the unit vetor is p (i j) = p i p j vetor of length units in the diretion of is p (i j) The vetors of length units whih re prllel to re p i p j nd p i + p j. = p i p j

VETORS (hpter ) Exmple 8 Find vetor of length 7 in the opposite diretion to the vetor = @. The unit vetor in the diretion of is j j = p @ = p6 @. ++ We multiply this unit vetor y 7. The negtive reverses the diretion nd the 7 gives the required length. Thus = p 7 @ 6. hek tht j j =7. OLLINER POINTS Three or more points re sid to e olliner if they lie on the sme stright line.,, nd re olliner if = k for some slr k. Exmple 9 Prove tht (,, ), (,, ), nd (,, 9) re olliner. = @ 5 nd = @ = @ 5 8 ) = ) [] is prllel to []. Sine is ommon to oth,,, nd re olliner. EXERISE I = @ nd = @ 6 r re prllel. Find r nd s. s Find slrs nd given tht @ nd @ re prllel. Wht n e dedued from the following? = D RS = KL =

VETORS (hpter ) 5 The position vetors of P, Q, R, nd S re @, @, @, nd @ respetively. Dedue tht [PR] nd [QS] re prllel. Wht is the reltionship etween the lengths of [PR] nd [QS]? 5 If =, write down the vetor: in the sme diretion s nd twie its length in the opposite diretion to nd hlf its length. 6 Find the unit vetor in the diretion of: i +j i k i j + k 7 Find vetor v whih hs: the sme diretion s nd length units the opposite diretion to nd length units. 8 is(, ) nd point is units from in the diretion Find. Find O using O = O +. Hene dedue the oordintes of.. 9 Find vetors of length unit whih re prllel to = @. Find vetors of length units whih re prllel to = @. Find vetor in: the sme diretion s @ nd with length 6 units the opposite diretion to @ nd with length 5 units. Prove tht (,, ), (,, ), nd (9, 8, ) re olliner. Prove tht P(,, ), Q(5, 5, ), nd R(, 7, ) re olliner. (,, ), (, 9, 7), nd (,, ) re olliner. Find nd. d K(,, ), L(,, 7), nd M(,, ) re olliner. Find nd.

6 VETORS (hpter ) The tringle inequlity sttes tht: In ny tringle, the sum of ny two sides must lwys e greter thn the third side. Prove tht j + j 6 j j + j j using geometril rgument. Hint: onsider: ² is not prllel to nd use the tringle inequlity ² nd prllel ² ny other ses. J THE SLR PRODUT OF TWO VETORS For ordinry numers nd we n write the produt of nd s or. There is only one interprettion for this produt, so we n use power nottion =, =, nd so on s shorthnd. However, there re two different types of produt involving two vetors. These re: I I The slr produt of vetors, whih results in slr nswer nd hs the nottion v ² w (red v dot w ). The vetor produt of vetors, whih results in vetor nswer nd hs the nottion v w (red v ross w ). onsequently, for vetor v, v or (v) vetor produts it would refer to. hs no mening nd is not used, s it is not ler whih of the SLR PRODUT The slr produt of two vetors is lso known s the dot produt or inner produt. If v = @ v nd w = @ w, the slr produt of v nd w is defined s v v w w v ² w = v w + v w + v w. NGLE ETWEEN VETORS onsider the vetors v = @ v nd w = @ w v w. We trnslte one of the vetors so tht they oth originte from the sme point. v v w w v w This vetor is v + w = w v nd hs length j w v j. Using the osine rule, j w v j = j v j + j w j j v jjwjos µ ut w v = @ w w @ v v = @ w v w v w v w v

VETORS (hpter ) 7 ) (w v ) +(w v ) +(w v ) = v + v + v + w + w + w j v jjwjos µ ) v w + v w + v w = j v jjwjos µ ) v ² w = j v jjwjos µ The ngle µ etween two vetors v nd w n e found using os µ = v ² w j v jjw j LGERI PROPERTIES OF THE SLR PRODUT The slr produt hs the following lgeri properties for oth -D nd -D vetors: I v ² w = w ² v I v ² v = j v j I I These properties re proven y using generl vetors suh s: v ² (w + x) =v ² w + v ² x (v + w) ² (x + y) =v ² x + v ² y + w ² x + w ² y v = @ v nd w = @ w. v v e reful not to onfuse the slr produt, whih is the produt of two vetors to give slr nswer, with slr multiplition, whih is the produt of slr nd vetor to give prllel vetor. They re quite different. w w GEOMETRI PROPERTIES OF THE SLR PRODUT I For non-zero vetors v nd w: v ² w =, v nd w re perpendiulr or orthogonl. I jv ² wj = j v jjwj, v nd w re non-zero prllel vetors. I If µ is the ngle etween vetors v nd w then: v ² w = j v jjwjos µ If µ is ute, os µ> nd so v ² w > If µ is otuse, os µ< nd so v ² w <. The ngle etween two vetors is lwys tken s the ngle µ suh tht ± 6 µ 6 8 ±, rther thn reflex ngle. v w The first two of these results n e demonstrted s follows: If v is perpendiulr to w then µ =9 ±. ) v ² w = j v jjw j os µ = j v jjw j os 9 ± = If v is prllel to w then µ = ± or 8 ±. ) v ² w = j v jjwjos µ = j v jjwjos ± or j v jjwjos 8 ± = jvjjwj ) j v ² w j = j v jjwj To formlly prove these results we must lso show tht their onverses re true.

8 VETORS (hpter ) DISUSSION v Eline hs drwn vetor v on plne whih is sheet of pper. It is therefore -dimensionl vetor. i How mny vetors n she drw whih re perpendiulr to v? ii re ll of these vetors prllel? Edwrd is thinking out vetors in spe. These re -dimensionl vetors. He is holding his pen vertilly on his desk to represent vetor w. i How mny vetors re there whih re perpendiulr to w? ii If Edwrd ws to drw vetor (in penil) on his desk, would it e perpendiulr to w? iii re ll of the vetors whih re perpendiulr to w, prllel to one nother? Exmple If p = @ nd q = @, find: p ² q the ngle etween p nd q. Sine p ² q < the ngle is otuse. p ² q = @ ² @ =( ) + () + ( ) = + = p ² q = j p jjqjos µ ) os µ = p ² q j p jjqj = p p +9+ ++ = p 7 ) µ = ros ³ p 7 ¼ 9 ± EXERISE J For p = µ, q =, nd r = 5, find: q ² p q ² r q ² (p + r) d r ² q e p ² p f i ² p g q ² j h i ² i For = @, = @, nd = @, find: ² ² j j d ² e ² ( + ) f ² + ²

If p = @ VETORS (hpter ) 9 nd q = @, find: p ² q the ngle etween p nd q. Find the ngle etween m nd n if: m = @ nd n = @ m =j k nd n = i +k 5 Find: (i + j k) ² (j + k) i ² i i ² j 6 Find p ² q if: j p j =, j q j =5, µ =6 ± j p j =6, j q j =, µ = ± 7 Suppose j v j = nd j w j =. Stte the possile vlues of v ² w if v nd w re: i prllel ii t 6 ± to eh other. Suppose ² = nd is unit vetor. i ii Explin why nd re not perpendiulr. Find j j if nd re prllel. Suppose j j = j d j = p 5. Wht n e dedued out nd d if: i ² d =5 ii ² d = 5? 8 In the given figure: Stte the oordintes of P. Find P nd P. Find P ² P using. d Wht property of semi-irle hs een x dedued in? (-, ) (, ) 9 Use = @, = @, nd = @ to prove tht ² ( + ) = ² + ². Hene, prove tht ( + ) ² ( + d) = ² + ² d + ² + ² d. y P Exmple Find t suh tht = 5 nd = t re perpendiulr. Sine nd re perpendiulr, ² = ) ² = 5 t ) ( )() + 5t = ) +5t = ) 5t = ) t = 5 If two vetors re perpendiulr then their slr produt is zero.

VETORS (hpter ) Find t if the given pir of vetors re: i perpendiulr ii prllel. t p = nd q = r = t t + nd s = t t = nd = t + t Show tht @ nd @ re perpendiulr. 5 Show tht = @, = @, nd = @ 5 re mutully perpendiulr. Find t if the following vetors re perpendiulr: i = @ nd = @ t ii @ t nd @ t. t Vetors re mutully perpendiulr if eh one is perpendiulr to ll the others. Exmple onsider the points (, ), (6, ), nd (5, ). Use slr produt to hek if tringle is right ngled. If it is, stte the right ngle. =, =, nd =. ² =( ) + ( ) ( ) = +=. )? nd so tringle is right ngled t. We do not need to hek ² nd ² euse tringle nnot hve more thn one right ngle. Use slr produt to hek if tringle is right ngled. If it is, stte the right ngle. (, ), (, 5), nd (, ) (, 7), (, ), nd (, 6) (, ), (5, 7), nd (, ) d (, ), (5, ), nd (7, ) onsider tringle in whih is (5,, ), is (6,, ), nd is (,, ). Using slr produt only, show tht the tringle is right ngled. (,, ), (,, ), (,, 6), nd D(, 5, 5) re verties of qudrilterl. Prove tht D is prllelogrm. Find j j nd j j. Wht n e sid out D? Find ² D. Desrie wht property of D you hve shown.

VETORS (hpter ) Exmple Find the form of ll vetors whih re perpendiulr to ² = + = ) is vetor perpendiulr to The required vetors hve the form k µ. re ll of the vetors prllel?., k 6=. ll of these vetors re prllel. 5 Find the form of ll vetors whih re perpendiulr to: 5 d e 6 Find ny two vetors whih re not prllel, ut whih re oth perpendiulr to 7 Find the ngle of tringle for (,, ), (,, ), nd (,, ). Hint: To find the ngle t, use nd. Wht ngle is found if nd re used? @. Exmple Use vetor methods to determine the mesure of. m m m The vetors used must oth e wy from (or oth towrds ). If this is not done you will e finding the exterior ngle t. Pling the oordinte xes s illustrted, is(,, ), is (,, ), nd is (,, ). ) = @ nd = @ Z ² os ( ) = j j j j () + ( )() + ( )( ) = p p +6+9 ++9 X Y = p 9 ³ ) = ros p 9 ¼ 9:8 ±

VETORS (hpter ) 8 The ue longside hs sides of length m. Find, using vetor methods, the mesure of: S R P P S P Q S R D 9 [KL], [LM], nd [LX] re 8, 5, nd units long respetively. P is the midpoint of [KL]. Find, using vetor methods, the mesure of: Y NX Y NP W K P Z N X L Y M onsider tetrhedron D. Find the oordintes of M. Find the mesure of D M. (,, ) D (,, ) M (,, ) (,, ) Find t if i + tj +(t )k nd ti +j + tk re perpendiulr. Find r, s, nd t if = @, = @, nd = @ s t re mutully perpendiulr. r Find the ngle mde y: i nd @ j nd @. Find three vetors,, nd suh tht 6= nd ² = ² ut 6=. Show, using j x j = x ² x, tht: j + j + j j =j j +j j j + j j j = ² 5 nd re the position vetors of two distint points nd, neither of whih is the origin. Show tht if j + j = j j then is perpendiulr to, using: vetor lgeri method geometri rgument. 6 If j j = nd j j =, find ( + ) ² ( ). 7 Explin why ² ² is meningless. K THE VETOR PRODUT OF TWO VETORS We hve seen how the slr produt of two vetors results in slr. The seond form of produt etween two vetors is the vetor produt or vetor ross produt, nd this results in vetor. The vetor produt rises when we ttempt to find vetor whih is perpendiulr to two other known vetors.

Suppose x = @ x y is perpendiulr to oth = @ nd = @. z ½ x + y + z = ) fs dot produts re zerog x + y + z = ½ x + y = z... () ) x + y = z... () VETORS (hpter ) We will now try to solve these two equtions to get expressions for x nd y in terms of z. To eliminte x, we multiply () y nd () y. x y = z f ()g x + y = z f ()g dding these gives ( )y =( )z ) y =( )t nd z =( )t for ny non-zero t. Sustituting into (), x + ( )t = ( )t ) x =( + + )t ) x = ( )t ) x =( )t The simplest vetor perpendiulr to oth nd is otined y letting t =. x = @, whih we ll the vetor ross produt of nd. In this se we find The vetor ross produt of vetors = @ nd = @ is = @. Rther thn rememering this formul, mthemtiins ommonly write the vetor ross produt s: The signs lternte +,, +. i j k = = i j + k {z } {z } {z } {z } From this form: over up the top row nd i olumn. over up the top row nd j olumn. over up the top row nd k olumn. where =( ) is the produt of the min (red) digonl minus the produt of the other digonl (green). e reful to get the sign of the middle term orret! We find tht =( )i ( )j +( )k =( )i +( )j +( )k

VETORS (hpter ) LGERI PROPERTIES OF THE VETOR ROSS PRODUT The vetor ross produt hs the following lgeri properties for non-zero -dimensionl vetors,,, nd d: I is vetor whih is perpendiulr to oth nd. I = for ll. I = for ll nd. This mens tht nd hve the sme length ut opposite diretion. I ² ( ) is lled the slr triple produt. I I ( + ) =( )+( ) ( + ) ( + d) =( )+( d)+( )+( d). You will prove or verify these results in the next Exerise. Exmple 5 If = @ nd = @ fter finding,, find. hek tht your nswer is perpendiulr to oth nd. i j k = = i j + k =( ( ) )i ( ( ) ( ))j +( ( ))k = @ 7 7 Exmple 6 For = @, = @, nd = @, find: ² ( ) i j k = ² ( ) = @ ² @ 8 = i j + k =6++ = = @ 8

VETORS (hpter ) 5 EXERISE K. lulte: @ @ @ @ (i + j k) (i k) d (i k) (j +k) Suppose = @ nd = @. Find. Hene determine ² ( ) nd ² ( ). Explin your results. i, j, nd k re the unit vetors prllel to the oordinte xes. Find i i, j j, nd k k. omment on your results. Find: i i j nd j i ii j k nd k j iii i k nd k i. omment on your results. Using = @ nd = @, prove tht: = for ll -dimensionl vetors = ( ) for ll -dimensionl vetors nd. 5 Suppose = @, = @, nd = @. Find: ² ( ) 6 Suppose = i +k, = j + k, nd =i k. Find: ( )+( ) d ( + ) 7 Prove tht ( + ) = + using = @, = @, nd = @. hek tht ( + ) = ( ) +( ) for other vetors,, nd of your hoosing. 8 Use ( + ) =( )+( ) to prove tht ( + ) ( + d) =( )+( d)+( )+( d). Note tht sine x y = (y x), the order of the vetors must e mintined. 9 Use the properties of vetor ross produt to simplify: ( + ) ( + ) ( + ) ( + ) ( ) d ² ( )

6 VETORS (hpter ) Exmple 7 Find ll vetors perpendiulr to oth = @ nd = @. i j k = = i j + k = 6i +j k = (i j + k) The vetors hve the form k @ where k is ny non-zero rel numer. Find ll vetors perpendiulr to oth: @ nd @ @ nd @ 5 i + j nd i j k d i j k nd i +j k. Find ll vetors perpendiulr to oth = @ nd = @. Hene find two vetors of length 5 units whih re perpendiulr to oth nd. Exmple 8 Find vetor whih is perpendiulr to the plne pssing through the points (,, ), (,, ), nd (,, ). v = @ nd = @ 5 i j k ) v = 5 = 5 i 5 j + k = i +j +k = (i 7j 5k) The vetor v must e perpendiulr to oth nd. ny non-zero multiple of @ 7 will e perpendiulr to the plne. 5

Find vetor whih is perpendiulr to the plne pssing through the points: (,, ), (,, 5), nd (,, ) P(,, ), Q(,, ), nd R(,, ). VETORS (hpter ) 7 DIRETION OF Z Z We hve lredy oserved tht = ( ), so nd re in opposite diretions. However, wht is the diretion of eh? In the lst Exerise, we sw tht i j = k nd j i = k. X i k j Y X i k j Y In generl, the diretion of x y is determined y the right hnd rule: x x y y If the fingers on your right hnd turn from x to y, then your thum points in the diretion of x y. THE LENGTH OF Using = @, j j = p ( ) +( ) +( ) However, nother very useful form of the length of exists. This is: j j = j jjjsin µ where µ is the ngle etween nd. Proof: j j j j sin µ = j j j j ( os µ) = j j j j jj j j os µ = j j j j ( ² ) =( + + )( + + ) ( + + ) whih on expnding nd then ftorising gives =( ) +( ) +( ) = j j ) j j = j jj j sin µ fs sin µ>g The immedite onsequenes of this result re: ² If u is unit vetor in the diretion of then = j jjjsin µ u. In some texts this is the geometri definition of. ² If nd re non-zero vetors, then =, is prllel to.

8 VETORS (hpter ) EXERISE K. Find i k nd k i using the originl definition of. hek tht the right hnd rule orretly gives the diretion of i k nd k i. k hek tht = j jjjsin µ u, where u is the unit vetor i Y j in the diretion of, is true for i k nd k i. onsider = @ nd = @ X. Find ² nd. Find os µ using ² = j jjjos µ. Find sin µ using sin µ + os µ =. d Find sin µ using j j = j jj j sin µ. Prove the property: If nd re non-zero vetors then =, is prllel to. onsider the points (,, ) nd (,, ). Find: i O nd O REVIEW SET ii O O iii j O O j Explin why the re of tringle O is j O O j. 5,, nd re distint points with non-zero position vetors,, nd respetively. If =, wht n e dedued out O nd? If + + =, wht reltionship exists etween nd? If 6= nd =, prove tht + = k for some slr k. NON-LULTOR Using sle of m represents units, sketh vetor to represent: n eroplne tking off t n ngle of 8 ± to runwy with speed of 6 ms displement of 5 m in north-esterly diretion. Simplify: + D. onstrut vetor equtions for: p r If PQ =, RQ = q, nd RS =, find SP. 5 O p [] is prllel to [O] nd is twie its length. Find, in terms of p nd q, vetor expressions for: q M OM. k O j l Z (,, -) (-),, n m

VETORS (hpter ) 9 6 Find m nd n if @ m nd @ re prllel vetors. n 7 If = @ 7 nd = @ 6, find. 8 If p =, q =, nd r =, find: p ² q q ² (p r) 5 9 onsider points X(, 5), Y(, ), W(, ), nd Z(, ). Use vetors to show tht WYZX is prllelogrm. onsider points (, ), (, ), nd (, k). Find k if is right ngle. Explin why: ² ² is meningless the expression ² does not need rkets. Find ll vetors whih re perpendiulr to the vetor. 5 In this question you my not ssume ny digonl properties of prllelogrms. O is prllelogrm with O = p nd O = q. M is the midpoint of []. Find in terms of p nd q: i O ii OM Hene show tht O, M, nd re olliner, nd tht M is the midpoint of [O]. Find the vlues of k suh tht the following re unit vetors: O p q 7 k M k k 5 Suppose j j =, j j =, nd j j =5. Find: ² ² ² 6 6 Find nd if J(,, ), K(,, ), nd L(,, ) re olliner. 7 Given j u j =, j v j =5, nd u v = @, find the possile vlues of u ² v. 8 [] nd [D] re dimeters of irle with entre O. If O = q nd O = r, find: i D in terms of q nd r ii in terms of q nd r. Wht n e dedued out [D] nd []?

VETORS (hpter ) 9 Find t given tht @ t nd @ t re perpendiulr. t t + Show tht K(,, ), L(,, ), nd M(,, ) re verties of right ngled tringle. REVIEW SET opy the given vetors nd find geometrilly: x + y y x x LULTOR y Show tht (,, ), (,, ), nd (,, ) re verties of n isoseles tringle. If r = nd s = find: j s j j r + s j j s r j Find slrs r nd s suh tht r + s =. 5 Given P(,, ) nd Q(,, ), find: PQ the distne etween P nd Q the midpoint of [PQ]. 6 If (,, ), (, 5, ), (,, ) re verties of tringle whih is right ngled t, find the vlue of. 7 Suppose = @ nd = @. Find x given x =. 8 Find the ngle etween the vetors =i + j k nd =i +5j + k. 9 Find two points on the Z-xis whih re 6 units from P(,, 5). t Determine ll possile vlues of t if nd + t re perpendiulr. t Prove tht P( 6, 8, ), Q(, 6, 8), nd R(9,, 7) re olliner. If u = @ nd v = @, find: u ² v the ngle etween u nd v. [P] nd [Q] re ltitudes of tringle. Let O = p, O = q, nd O = r. Find vetor expressions for nd in terms of p, q, nd r. Using the property ² ( ) = ² ², dedue tht q ² r = p ² q = p ² r. Hene prove tht [O] is perpendiulr to []. Q O P

VETORS (hpter ) Find two vetors of length units whih re prllel to i j + k. 5 Find the mesure of D M. D (,-, ) (,-, ) 6 Find ll vetors perpendiulr to oth 7 Find k given tht @ k p k (-,, -) @ nd @. is unit vetor. M (, 5, -) Find the vetor whih is 5 units long nd hs the opposite diretion to 8 Find the ngle etween @ nd @. @. 9 Determine the mesure of Q DM given tht M is the midpoint of [PS]. Q P M R S 7 m m D m REVIEW SET Find single vetor whih is equl to: PR + RQ PS + SQ + QR For m = @ 6, n = @, nd p = @, find: 6 m n + p n p j m + p j Wht geometril fts n e dedued from the equtions: = D =? Given P(, 5, 6) nd Q(, 7, 9), find: the position vetor of Q reltive to P the distne from P to Q the distne from P to the X-xis.

VETORS (hpter ) 5 P Q In the figure longside, OP = p, OR = r, nd M RQ = q. M nd N re the midpoints of [PQ] nd p O N q [QR] respetively. Find, in terms of p, q, nd r: OQ PQ ON d MN r R 6 Suppose p = @, q = @, nd r = @. Find x if: p x = q x = r 7 Suppose j v j = nd j w j =. If v is prllel to w, wht vlues might v ² w tke? 8 Find unit vetor whih is prllel to i + rj +k nd perpendiulr to i +j k. 9 Find t if @ t + nd @ t +t re perpendiulr vetors. t Find ll ngles of the tringle with verties K(,, ), L(,, ), nd M(,, ). µ 5 Find k if the following re unit vetors: @ k k k k Use vetor methods to find the mesure of G in the retngulr ox longside. F E G H 5 m Using p =, q =, nd r =, verify tht: 5 p ² (q r) =p ² q p ² r. P(,, ) nd Q(,, ) re two points in spe. Find: PQ the ngle tht PQ mkes with the X-xis. 5 MP =, 5 Suppose OM =, D 8 m MP ² PT =, nd j MP j = j PT j. m Write down the two possile position vetors OT. 6 Given p =i j +k nd q = i j +k, find the ngle etween p nd q. 7 Suppose u =i + j, v =j, nd µ is the ute ngle etween u nd v. Find the ext vlue of sin µ. 8 Find two vetors of length units whih re perpendiulr to oth i +k nd i j + k.