jectives H P T E R 5 Vectors To nderstnd the concept of vector To ppl sic opertions to vectors To nderstnd the zero vector To se the nit vectors i nd j to represent vectors in two dimensions To se the fct tht, if nd re prllel, then = k for rel vle k. The converse of this lso holds. To se the nit vectors i, j nd k to represent vectors in three dimensions 5. Introdction to vectors Foreperiments in science or engineering some of the things which re mesred re completel determined their mgnitde. For emple, mss, length or time re determined nmer nd n pproprite nit of mesrement. e.g., length: cm is the length of the pge of prticlr ook time: s is the time for one thlete to rn m More is reqired to descrie velocit, displcement or force. The direction mst e recorded s well s the mgnitde. SMPLE e.g., displcement: km in direction north velocit: 6 km/h in direction soth est Qntities tht hve direction s well s mgnitde re represented rrows tht point in the direction of the ction nd whose lengths give the mgnitde of the qntit in terms of sitl chosen nit. rrows with the sme length nd direction re regrded s eqivlent. These rrows re directed line segments nd the sets of eqivlent segments re clled vectors. mridge Universit Press Uncorrected Smple Pges 978--52-6252-4 28 Evns, Lipson, Jones, ver, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird 9
hpter 5 Vectors 9 The five directed line segments shown ll hve the sme mgnitde nd direction. directed line segment from point to point is denoted. P For simplicit of lngge this is lso clled vector, i.e., the set of eqivlent segments cn e nmed throgh F H one memer of tht set. Note: = = P = EF = GH E G In hpter 8, vectors were introdced in the contet of trnsltions (in two dimensions). colmn of nmers ws introdced to represent the trnsltion nd it ws clled vector. This is consistent with the pproch here s the colmn 2 nits of nmers corresponds to set of eqivlent directed line segments. nits The colmn corresponds to the directed line segment 2 which goes cross nd 2 p. This nottion will e sed to represent directed line segment in the first section of this chpter. The nottion is widel sed to represent vectors t not to lrge etent in Victorin schools nd so the nottion, lthogh sefl, will e ndoned in the ltter sections of the chpter. Vectors re often denoted single old fce romn letter. The vector from to cn e denoted or single v. Tht is, v =. When vector is hndwritten the nottion is v. ddition of vectors Twovectors nd v cn e dded geometricll drwing line segment representing from to nd then line segment from to representing v. The sm + v is the vector from to. Tht is, + v =. The sme reslt is chieved if the order is reversed. This is represented in the following digrm SMPLE i.e. + v = nd + v = v + + v v + v v v mridge Universit Press Uncorrected Smple Pges 978--52-6252-4 28 Evns, Lipson, Jones, ver, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
92 Essentil dvnced Generl Mthemtics The ddition cn lso e chieved with the colmn vector nottion. 4 e.g., if = nd v = 4 then + v = + = 4 Sclr mltipliction Mltipliction rel nmer (sclr) chnges the length of the vector. e.g., 2 = + nd 2 + 2 = 2 is twice the length of nd is hlf the length of. 2 The vector k, k R +, with, hs the sme direction s, t its length is mltiplied fctor of k. When vector is mltiplied 2 the vector s direction is reversed nd its length doled. When vector is mltiplied the vector s direction is reversed nd the length remins the sme. 6 6 If =, =, 2 = nd 2 = 2 2 4 4 If = then = =. The directed line segment goes from to. Zero vector 4 + v SMPLE The zero vector is denoted nd represents line segment of zero length. The zero vector hs no direction. Strction of vectors In order to strct v from, dd v to. Foremple v v v v 4 2 2 v 2 mridge Universit Press Uncorrected Smple Pges 978--52-6252-4 28 Evns, Lipson, Jones, ver, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
hpter 5 Vectors 9 Emple rw the directed line segment defined. 2 is the vector cross to the right nd 2 down 2 Note: Here the vector strts t (, ) nd goes to (4, ). It cn strt t n point. Emple 2 The vector is defined the directed line segment from (2, 6) to (, ). If =, find nd. 2 The vector = = 6 5 Hence = nd = 5 Emple 2 If the vector = nd the vector v =, find 2 + v. 2 2 2 + v = 2 + 2 6 6 = + 2 6 = 4 SMPLE Polgons of vectors Fortwo vectors nd, + = 2 (2, 6) (, ) 4 mridge Universit Press Uncorrected Smple Pges 978--52-6252-4 28 Evns, Lipson, Jones, ver, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
94 Essentil dvnced Generl Mthemtics Forpolgon EF, + + + E + EF + F = Emple 4 Illstrte the vector sm + +, where,, nd re points in the plne. Prllel vectors + + = The non-zero [ vectors ] nd [ v re ] sid to e prllel if there eists k R\{} sch tht = kv. 2 6 If = nd v = then vector is prllel to v s v =. 9 Position vectors point, the origin, cn e sed s strting point for vector to indicte the position of point in spce reltive to tht point. In this chpter, is the origin for crtesin plne (three dimensionl work is considered riefl). Forpoint the position vector is. Liner comintions of non-prllel vectors If nd re non-zero, non-prllel vectors, then m + n = p + q implies m = p nd n = q n rgment is s follows: SMPLE m p = q n (m p) = (q n) = q n m p or = m p q n t nd re not prllel nd not zero F E q = n nd m = p mridge Universit Press Uncorrected Smple Pges 978--52-6252-4 28 Evns, Lipson, Jones, ver, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
hpter 5 Vectors 95 Emple 5 Let, nd e the vertices of tringle. Let e the midpoint of. Let = nd =. Find the following in terms of nd. c d e = = 2 2 (sme direction nd hlf the length) = = 2 (eqivlent vectors) c = + = + d = + = + 2 e = = ( + ) ( + = ) Emple 6 In the figre, = kp where k R\{}. Epress p in terms of k, q nd r. Epress FE in terms of k nd p to show FE is prllel to. c If FE = 4, find the vle of k. p = = + + = q + kp r ( k)p = q r nd hence p = (q r) k FE = 2q + p + 2r = 2(r q) + p t r q = kp p = (k )p... from FE = 2kp 2p + p = (2k )p Eercise 5 c F q q p If FE = 4 (2k )p = 4p 2k = 4 SMPLE k = 5 2 r r E Emple n the sme grph, drw the rrows which represent the following vectors. 4 c d 5 2 2 mridge Universit Press Uncorrected Smple Pges 978--52-6252-4 28 Evns, Lipson, Jones, ver, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
96 Essentil dvnced Generl Mthemtics Emple 2 Emple 4 Emple 2 The vector is defined the directed line segment from (, 5) to (6, 6). If =, find nd. The [ vector ] v is defined the directed line segment from (, 5) to (2, ). If v =, find nd. 4 = (, 2), = (, ), = (2, ) nd is the origin. Epress the following vectors in the form. c d e 5 = (2, ), = (4, ), = (, 4) nd is the origin. Sketch the following vectors. c d e 6 n grph pper, sketch the vectors joining the following pirs of points in the direction indicted. (, ) (2, ) (, 4) (, ) c (, ) (, 4) d (2, 4) (4, ) e ( 2, 2) (5, ) f (, ) (, ) 7 Identif vectors from 6 which re prllel to ech other. 8 Plot the points (, ), (, 4), (4, ), (2, ) on set of coordinte es. Sketch the vectors,,, nd. c Show tht i = ii = d escrie the shpe of the qdrilterl. 2 9 Let =, = nd c =. 2 Find i + ii 2c iii + c Show tht + is prllel to c. 2 9 Find the vles of m nd n sch tht m + n = 4 6 SMPLE In the figre,,, re the vertices of prllelogrm. = nd = M, N re the midpoints of nd respectivel. Epress the following in terms of nd. i M ii MN Find the reltionship etween MN nd. N M mridge Universit Press Uncorrected Smple Pges 978--52-6252-4 28 Evns, Lipson, Jones, ver, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
hpter 5 Vectors 97 Emple 5 Emple 6 2 The figre represents the tringle with = nd =. M, N re midpoints of nd respectivel. Epress nd MN in terms of nd. Hence descrie the reltion etween the two vectors (or directed line segments). The figre represents the reglr hegon EF with vectors F = nd =. Epress the following vectors in terms of nd. E c E d F e F f F g FE 4 In prllelogrm, = nd =. Epress ech of the following vectors in terms of nd. c d e 5 In tringle, = nd =. P is point on sch tht P = 2 P nd Q is point sch tht P = PQ. Epress ech of the following in terms of nd. P c P d PQ e Q 6 PQRS is qdrilterl in which PQ =, QR = v, RS = w. Epress the following vectors in terms of, v nd w. PR QS c PS 7 is prllelogrm. =, = v. M is the midpoint of. Epress nd M in terms of nd v. Epress M in terms of nd v. c If P is point on M nd P = 2 M, epress P in terms of nd v. d Find P nd hence show tht P lies on. e Find the rtio P : P. SMPLE 5.2 omponents of vectors The vector illstrted opposite cn e descried the colmn vector.from the digrm it is 4 possile to see tht cn e epressed s the sm of two vectors X nd X, i.e., = X + X. N M F (5, 7) (2, ) X E mridge Universit Press Uncorrected Smple Pges 978--52-6252-4 28 Evns, Lipson, Jones, ver, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
98 Essentil dvnced Generl Mthemtics In colmn vector nottion = + 4 4 This sggests the introdction of two importnt vectors. Let i e the vector of nit length with direction the positive direction of the is. Let j e the vector of nit length with direction the positive direction of the is. Note tht sing the colmn nottion, i = nd j =. For the emple ove, X = i nd X = 4j Therefore = i + 4j. It is possile to descrie n two-dimensionl vectors in this w. Forvector =, = i + j. Itissid tht is the sm of two components i nd j. The mgnitde of vector = i + j is denoted nd = 2 + 2 pertions with vectors now look more like sic lger. (i + j) + (mi + nj) = ( + m) i + ( + n) j k (i + j) = ki + kj For = i + j nd = mi + nj, = if = m nd = n Emple 7 Find if = i nd = 2i j Find 2i j. = + = i + (2i j) = i j 2i j = 4 + 9 = SMPLE Emple 8 j i i j, re points on the crtesin plne sch tht = 2i + j nd = i j. Find nd. mridge Universit Press Uncorrected Smple Pges 978--52-6252-4 28 Evns, Lipson, Jones, ver, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
hpter 5 Vectors 99 Emple 7 Emple 8 Unit vectors = + = + = (2i + j) + i j = i 4j = + 6 = 7 nit vector is vector of length one nit (i nd j re nit vectors). The nit vector in the direction of is â (prononced ht ). Emple 9 â = so â = â = or Let = i + 4j. Find, the mgnitde of, nd hence the nit vector in the direction of. = i + 4j so = 2 + 2 â = So â = (i + 4j) 5 Eercise 5 = 2 + 4 2 = 5 SMPLE, re points on the crtesin plne sch tht = i + 2j nd = i 5j. Find. 2 P is rectngle in which the vector = 5i nd the vector = 6j. Epress the following vectors in terms of i nd j. P c etermine the mgnitde of the following vectors. 5i 2j c i + 4j d 5i + 2j 4 The vectors nd v re given = 7i + 8j nd v = 2i 4j. Find v. Find constnts nd sch tht + v = 44j. mridge Universit Press Uncorrected Smple Pges 978--52-6252-4 28 Evns, Lipson, Jones, ver, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
4 Essentil dvnced Generl Mthemtics 5 In the tringle, = i nd = 4i + 5j. IfM is the midpoint of, find M in terms of i nd j. 6 In the rectngle PQ, P = 2i nd Q = j. M is the point on P sch tht M = 5 P. N is the point on MQ sch tht MN = 6 MQ. Find the following vectors in terms of i nd j. i M ii MQ iii MN iv N v i Hence show tht N is on the digonl. ii Stte the rtio of the lengths N : N. 7 The position vectors of nd re given = nd 5 =. Find the distnce etween nd. 8 Find the pronmerls in the following eqtions. 2i + j = 2 (li + kj) ( ) i + j = 5i + ( 4) j c ( + ) i + ( ) j = 6i d k (i + j) = i 2j + l (2i j) 9 Let = (2, ) nd = (5, ).Find Let = i, = i + 4j nd = i + j. Find c Let = (5, ), = (, 4) nd = (, ). Find sch tht = F sch tht F = c G sch tht = 2G 2 Let = i + 4j nd = 2i + 2j., nd re points sch tht =, =, = 2 nd is the origin. Find the coordintes of, nd.,, nd re the vertices of prllelogrm nd is the origin. = (2, ), = ( 5, 4) nd = (, 7). Find i ii iii iv v Hence find the coordintes of. SMPLE 4 The digrm shows prllelogrm PQR. The points P nd Q hve coordintes (2, 5) nd (8, ) respectivel. Find P nd PQ RQ nd R 5 (, 6), (, ), (, 5) re the vertices of tringle Find i ii iii Hence show tht is right-ngled tringle. R P Q mridge Universit Press Uncorrected Smple Pges 978--52-6252-4 28 Evns, Lipson, Jones, ver, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
hpter 5 Vectors 4 Emple 9 6 (4, 4), (, ) nd (7, ) re the vertices of the tringle. c Find the vector i ii iii Find i ii iii Hence show tht tringle is n isosceles right-ngled tringle. 7 (, 2), (, 7) re points on the crtesin plne. is the origin. M is the midpoint of. Find i ii iii iv M Hence find the coordintes of M. (Hint: M = + M) 8 Find the nit vector in the direction of ech of the following vectors. = i + 4j = i j c c = i + j d d = i j e e = 2 i + j f f = 6i 4j 5. Vectors in three dimensions Points in three dimensions (-) re represented in es s shown. Vectors in - re of the form = = i + j + zk where z i =, j = nd k = i, j nd k re represented in the figre. = i + j + zk Pthgors theorem 2 = 2 + 2 nd 2 = 2 + 2 = 2 + 2 + z 2 = 2 + 2 + z 2 Emple i k i z z z j zk (,, z) SMPLE j Let = i + j k nd = i + 7k.Find + c mridge Universit Press Uncorrected Smple Pges 978--52-6252-4 28 Evns, Lipson, Jones, ver, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
42 Essentil dvnced Generl Mthemtics Emple + = i + j k + i + 7k = 2i + j + 6k = i + 7k (i + j k) = 2i j + k c = Emple EFG is coid. = j, = k, = i. Epress ech of the following in terms of i, j, k. i E ii F iii GF iv G M, N re the midpoints of nd GF respectivel. Find MN. i E = + E = j + i ( E = ) ii F = E + EF = j + k + i ( EF = ) iii GF = = j iv G = = + = i + j MN = M + N = M + G + GN = 2 i + k + 2 j MN = 4 + + 9 4 4 = 2 Emple 2 If = i + 2j + 2k, find â. = 9 + 4 + 4 = 7 â = (i + 2j + 2k) 7 SMPLE Eercise 5 = i + 2 j + 2 k 7 7 7 Let = i + j + 2k, = 2i j + k, c = i + k. Find 2 + c c d + c e ( ) + 2c G F E Emple 2 2 If = i + j k find i â ii 2â find the vector in the direction of sch tht =5. mridge Universit Press Uncorrected Smple Pges 978--52-6252-4 28 Evns, Lipson, Jones, ver, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
Emple hpter 5 Vectors 4 If = i j + 5k nd = 2i j k find the vector c in the direction of sch tht c =. 4 P nd Q re points defined the position vectors i + 2j k nd 2i j k respectivel. M is the midpoint of PQ.Find PQ PQ c M 5 EFG is coid. = 2j, = 2k, = i G F Epress the following vectors in terms of i, j nd k. E c G d F e E f EG g E h 6 EFG is coid. = j, = 2k, = i M is sch tht M = E. N is the midpoint of F.Find MN MN 5.4 pplictions Emple Three points P, Q, nd R hve position vectors p, q, nd k(2p + q) respectivel, reltive to fied origin., P nd Q re not colliner. Find the vle of k if QR is prllel to p PR is prllel to q c P, Q nd R re colliner. QR = Q + R = q + k(2p + q) = q + 2kp + kq SMPLE If QR is prllel to p, there eists R\{} sch tht (k )q + 2kp = p k = nd 2k = k = PR = P + R = p + k(2p + q) = (2k )p + kq G E F E mridge Universit Press Uncorrected Smple Pges 978--52-6252-4 28 Evns, Lipson, Jones, ver, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
44 Essentil dvnced Generl Mthemtics If PR is prllel to q there eists m R\{} sch tht Emple c (2k )p + kq = mq k = 2 If P, Q nd R re colliner there eists n R\{} sch tht n PQ = QR n( p + q) = (k )q + 2kp k = n nd 2k = n which implies k = i.e., k = Eercise 5 In the digrm R = 4 5 P, P = p, Q = q nd PS : SQ = :4. R p Epress ech of the following in terms of p nd q. i R ii RP iii PQ iv PS v RS P S Wht cn e sid ot line segments RS nd Q? c Wht tpe of qdrilterl is RSQ? d The re of tringle PRS is5cm 2. Wht is the re of RSQ? 2 The position vectors of three points, nd reltive to n origin re, nd k respectivel. The point P lies on nd is sch tht P = 2P. The point Q lies on nd is sch tht Q = 6Q. c Find in terms of nd i the position vector of P ii the position vector of Q Given tht PQ is stright line, find i the vle of k ii the rtio P PQ SMPLE The position vector of point R is 7. Show tht PR is prllel to. The position vectors of two points nd reltive to n origin re i +.5j nd 6i.5j respectivel. i Given tht = nd E =, write down the position vectors of 4 nd E. ii Hence find E. q Q mridge Universit Press Uncorrected Smple Pges 978--52-6252-4 28 Evns, Lipson, Jones, ver, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
hpter 5 Vectors 45 Given tht E nd intersect t X nd tht X = pe nd X = q, find the position vector of X in terms of i p ii q c Hence determine the vles of p nd q. 4 The position vectors of P, Q with reference to n origin re p nd q nd M is the point on PQ sch tht PM = MQ Prove tht the position vector of M is m where p + q m = + The vector p = k nd the vector q = l where k nd l re positive rel nmers nd nd re nit vectors. i Prove tht the position vector of n point on the internl isector of PQ hs the form ( + ) ii If M is the point where the internl isector of PQ meets PQ, show tht = k l 5 RST is prllelogrm. U is the midpoint of RS nd V is the midpoint of ST. Reltive to the origin, r, s, t, nd v re the position vectors of R, S, T, U nd V respectivel. Epress s in terms of r nd t. Epress v in terms of s nd t. c Hence or otherwise show tht 4 ( + v) = (r + s + t) SMPLE mridge Universit Press Uncorrected Smple Pges 978--52-6252-4 28 Evns, Lipson, Jones, ver, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
46 Essentil dvnced Generl Mthemtics Review hpter smmr vector is set of eqivlent directed line segments. directed line segment from point to point is denoted. 2 In two dimensions, vector cn e represented colmn of nmers, e.g. is the vector 2 cross nd p. The sm + v cn e shown digrmmticll v + v = [ v + ] c + c If = nd v = then + v = d + d + v The vector k, k R + nd, hs the sme direction s t its length is mltiplied fctor k. The vector v is in the opposite direction to v t it hs the sme length. v = + ( v) Two non-zero vectors nd v re sid to e prllel if there eists k R\{} sch tht = kv. Forpoint, the position vector of is where is the origin. Ever vector cn e epressed s the sm of two vectors i nd j,where i is the nit vector in the positive direction of the is nd j is the nit vector in the positive direction of the is. j SMPLE The mgnitde of vector = i + j is denoted nd = 2 + 2 The nit vector in the direction of is. This vector is denoted â. In three dimensions vector cn e written s = i + j + zk, where i, j nd k re nit vectors s shown. z z (,, z) k j i i If = i + j + zk, = 2 + 2 + z 2 mridge Universit Press Uncorrected Smple Pges 978--52-6252-4 28 Evns, Lipson, Jones, ver, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
hpter 5 Vectors 47 Mltiple-choice qestions The vector v is defined the directed line segment from (, ) to (, 5). If v = i + j then = nd = 5 = 2nd = 4 = 2 nd = 4 = 2 nd = E = 4 nd = 2 2 If vector = nd vector = v then vector is eql to + v [ v ] v [ ] v E v + 2 If vector = nd vector = then + = 2 E 5 5 4 If vector = nd vector = then 2 = 2 9 9 9 E 7 7 5 5 PQRS is prllelogrm. If PQ = p nd QR = q, then in terms of p nd q, SQ eqls p + q p q q p 2q E 2p 6 i 5j = 2 4 4 8 E 6 7 If = 2i + j nd = i 2j then eqls i 5j i + 5j i j i + j E i + j 8 If = 2i + j nd = i 2j then eqls 6 26 26 24 E 6 9 If = 2i + j then the nit vector prllel to is 2i + j (2i + j) 5 (2i + j) (2i + j) E (2i + j) SMPLE If = i + j + k then â is 7 ( i + j + k) 7 ( i + j + k) ( i + j + k) E i + j + k 9 Short-nswer qestions (technolog-free) ( i + j + k) 9 Given tht = 7i + 6j nd = 2i + j, find the vles of for which is prllel to nd hve the sme mgnitde. 2 is prllelogrm where = 2i j, = i + 4j nd = 2i + 5j. Find the coordintes of the for vertices of the prllelogrm. Review mridge Universit Press Uncorrected Smple Pges 978--52-6252-4 28 Evns, Lipson, Jones, ver, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
48 Essentil dvnced Generl Mthemtics Review If = 2i j + k, = 2i 4j + 5k nd c = i 4j + 2k, find the vles of p nd q if + p + qc is prllel to the is. 4 The position vectors of P nd Q re 2i 2j + 4k nd i 7j + 2k respectivel. Find PQ nit vector prllel to PQ. 5 The position vectors of, nd re 2j + 2k, 4i + j + 8k nd i + 4j + 26k respectivel. Find if, nd re colliner. 6 = 4i + j nd is point on sch tht = 6 5. Find the nit vector in the direction of. Hence find. 7 In the digrm, ST = 2TQ, PQ =, SR = 2 nd SP =. Find ech of the following in terms of nd. i SQ ii TQ iii RQ iv PT v TR Q Show tht P, T nd R re colliner. 8 If = 5i sj + 2k nd = ti + 2j + k re eql vectors, find i s ii t iii 9 The vector p hs mgnitde 7 nits nd ering 5 nd the vector q hs mgnitde 2 nits nd ering 7. (These re compss erings on the horizontl plne.) rw digrm (not to scle) showing p, q nd p + q. lclte the mgnitde of p + q. If = 5i + 2j + k nd = i 2j + k, find + 2 c â d, nd re the points (, ), (, 4) nd (4, 6) respectivel. SMPLE is the point sch tht = +. Find the coordintes of. is the point (, 24) nd = h + k. Find the vles of h nd k. 2 Given tht p = i + 7j nd q = 2i 5j, find the vles of m nd n sch tht mp + nq = 8i + 9j. The points, nd hve position vectors, nd c reltive to n origin. Write down n eqtion connecting, nd c for ech of the following cses. is prllelogrm. divides in the rtio : 2, i.e., : = :2. P T R 2 S mridge Universit Press Uncorrected Smple Pges 978--52-6252-4 28 Evns, Lipson, Jones, ver, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird
hpter 5 Vectors 49 Etended-response qestions Let represent displcement km de est nd represent displcement km de north. The digrm shows 2 circle of rdis 25 km with centre t (, ). lighthose entirel srronded se is locted t. 2 The lighthose is not visile from points otside the circle. P 2 The ship is initill t P,kmwest nd 2 km soth of the lighthose. Write down the vector P. 4 The ship is trvelling prllel to vector = with speed 2 km/h. n hor fter leving P the ship is t R. Show tht the vector PR 6 =, nd hence find the vector R. 2 c Show tht when the ship reches R, the lighthose first ecomes visile. 2 Given tht p = i + j nd q = 2i + 4j find p q p q c r, sch tht p + 2q + r = 2 7 26 Let =, = 7, c = 9 nd d = 2 2 7 2 c 2 Find the vle of the sclr k sch tht + 2 c = kd Find the sclrs nd sch tht + = d Use or nswers to nd to find sclrs p, q nd r (not ll zero) sch tht p + q rc = 4 The qdrilterl PQRS is prllelogrm. The point P hs coordintes (5, 8), the point R hs coordintes (2, 7) nd the vector PQ is given PQ 2 =. 5 Find the coordintes of Q, nd write down the vector QR. Write down the vector RS, nd show tht the coordintes of S re (2, 2). 5 The digrm shows the pth of light [ em ] from its sorce t in the direction of the vector r =.tp the em P θ is reflected n djstle mirror nd meets the is t M. The position of M vries, depending on the djstment r of the mirror t P. M Given tht P = 4r, find the coordintes of P. The point M hs coordintes (k, ). Find in terms of k n epression for the vector PM. c Find the mgnitdes of vectors P, M nd PM, nd hence find the vle of k for which is eql to 9. d Find the vle for which M hs coordintes (9, ). SMPLE Review mridge Universit Press Uncorrected Smple Pges 978--52-6252-4 28 Evns, Lipson, Jones, ver, TI-Nspire & sio lsspd mteril prepred in collortion with Jn Honnens & vid Hird