UNCORRECTED PAGE PROOFS

Transcription

1 8 Kik off with CAS 8 Introdution to vetors 8 Operations on vetors Vetors 8 Magnitude, diretion and omponents of vetors 85 i, j notation 86 Appliations of vetors 87 Review 8

2 8 Kik off with CAS Eploring the lengths of vetors with CAS A vetor in two dimensions from the origin to the terminal point P (, ) is represented as OP = r = i j We often need to alulate the length of the vetor Using CAS this an e done using the norm ommand and representing a vetor as a matri: norm ([ ]) Use CAS to find the length of eah of the following vetors a i +j i j i j d i +j e 8i 6j f 5i +j Using CAS, define the following vetors: a = i +j, Then alulate: a a a 5a d a + e a f a + 5 Please refer to the Resoures ta in the Prelims setion of our eookplus for a omprehensive step--step guide on how to use our CAS tehnolog

3 8 Units & AOS Topi Conept Introdution to vetors Conept summar Pratie questions Introdution to vetors A salar quantit is one that is speified size, or magnitude, onl Distane is an eample of a salar quantit; it needs onl a numer to speif its size or magnitude Time, length, volume, temperature and mass are salars A vetor quantit is speified oth magnitude and diretion Displaement measures the final position ompared to the starting position and requires oth a magnitude (eg distane 800 m) and a diretion (eg 0 T) Displaement is an eample of a vetor quantit Fore, veloit and aeleration are also vetors The all require a size and a diretion to e speified ompletel Representation of vetors Vetors an e represented direted line segments S N For eample, if north is straight up the page and a sale of m = 0 m is used, then a displaement of 00 m south is W represented a 5 m line straight down the page We plae an arrow S on the line to indiate the diretion of the vetor, as shown at right The start and end points of a vetor an e laelled with apital letters 00 m For eample, the vetor shown at right an have the starting point, or tail, laelled S and the end point, or head, laelled F This vetor an then e referred to as SF F m = 0 m The vetor an also e represented a lower-ase letter over a tilde, for eample, s Representing a vetor as an ordered pair (a, ) A vetor in the plane an e desried an ordered pair (a, ) The values a and are alled omponents; a gives the hange of position relative to the positive -ais and gives the hange of position relative to the positive -ais of the end of the vetor ompared to the start For eample (, ) represents a hange of position of units in the positive -diretion and units in the positive -diretion Note that the vetor represented (, ) doesn t neessaril start at the origin It an e in an position on the Cartesian plane Representing a vetor as a olumn matri a An vetor an e written as a olumn matri, whih is a matri onsisting of a single olumn with two elements For eample, the vetor represented the direted line segments desried in the previous setion an e written as the olumn matri The top numer gives the displaement relative to the positive diretion on the -ais and the ottom numer gives the displaement relative to the positive diretion on the -ais E 0 7 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

4 WoRKED EXAMPLE Position vetors A position vetor desries a point in the Cartesian plane Position vetors start at the origin O (0, 0) For eample, for A (, ) the position vetor OA is shown at right Note we an also use (, ) to desrie an vetor that travels three units aross and one up, ut it is onl a position vetor if it starts at (0, 0) Write the following vetors in the form (a, ) and a a OC DA a From O to C, we travel + units in the positive -diretion and + unit in the positive -diretion From D to A, we travel 5 units in the positive -diretion and + units in the positive -diretion WoRKED EXAMPLE A O WritE a OC = (, ) and OC = DA = ( 5, ) and DA = 5 C D If we started at (5, ), where would we end up after a displaement of (, )? WritE Write (5, ) + (, ) (5, ) + (, ) We start at (5, ) and move + units in the positive -diretion and + units in the positive -diretion = (8, 0) Write the answer We would end up at the point (8, 0) WoRKED EXAMPLE Draw d, the position vetor of (, ), on a set of aes A position vetor must start at (0, 0) and end at the point speified Make sure the arrow is pointing awa from the origin Lael the vetor DrAW (, ) d 0 A 0 O Topi 8 VECToRS 75

5 WoRKED EXAMPLE EXErCisE 8 PrACtisE Equalit of vetors Two vetors are equal if the are: equal in magnitude parallel, and point in the same diretion Whih of the vetors shown at right are equal? Vetors a and e are of equal length, parallel and point in the same diretion Vetors and g are of equal length, parallel and point in the same diretion WoRKED EXAMPLE 5 Introdution to vetors WE Eamine the diagram at right Represent eah of the following vetors as an ordered pair (a, ) a A AF e D AC d C WritE a = e = g An airraft flies 00 km north, then 00 km east Draw a vetor diagram to represent the path taken the airraft and also the displaement of the airraft from its starting point to its finishing point Take north as vertiall up the page and east to the right Draw a short vertial direted line segment to represent a displaement of 00 km north DrAW N W E S Draw a horizontal direted line segment with its tail joined to the head of the first This represents a displaement of 00 km east 00 km Draw a direted line segment from the tail of the north vetor (point S) to the head of the east vetor (point F) This represents the displaement of the airraft from its starting point to its finishing point 00 km S f 00 km a 00 km 00 km g D F e 0 E F d A C 76 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

6 Eamine the diagram in question Represent eah of the following vetors as a olumn matri a a CD CA ED d EF e FE WE If we started at the point (, ) where would we end up after a displaement of (, 6)? If we started at the point (, 5), where would we end up after eah of these displaements? a (, ) (, 5) (0, ) d (, 5) 5 WE Draw d, the position vetor (, ), on a set of aes 6 Draw the position vetor for eah of the following points on the same set of aes A (, ), (, ), C (0, ), D (, ) 7 WE Whih of the vetors shown in the diagram elow are equal? g a e h d f 8 Whih of the vetors shown in the diagram elow are equal? a 5 d e WE5 An aeroplane flies 000 km north from airport A to airport It then travels to airport C, whih is 00 km north-east of Draw a vetor diagram to represent the path taken the aeroplane and the displaement of the finishing point from the starting point 0 A oat travels 0 km north and then 0 km west Draw a vetor diagram showing the path of the oat and the displaement of the finishing point from the starting point Topi 8 Vetors 77

7 Consolidate Master Eamine the diagram elow Represent the hange of position of eah of the vetors shown in the form (a, ) g 5 a f d h e Represent the hange of position of eah of the vetors shown in question in the form a Represent eah of the following vetors on separate diagrams a The position vetor of (, ) The position vetor of (0, 5) The position vetor of (, ) Represent eah of the following vetors on separate diagrams a A displaement of (, 8) starting from the point (, ) A displaement of (, 5) starting from the point (, 6) A displaement of (0, ) starting from the point (, 5) d The position vetor of (, ) followed (, 5) 5 A vetor that starts at the point (, ) and finishes at the point (, ) is represented a displaement of: A (, 5) (5, ) C (, ) D ( 5, ) E (, ) 6 Draw two direted line segments represented the vetor 5 7 The direted line segment shown in the diagram represents the vetor d Find a and if d = a 8 Sketh the following vetors on separate aes, if A = (, ), = (0, ), C = (, ) and O is the origin a OA A AC d C 9 Epress eah of the vetors from question 8 in the form a 0 Marus les 0 km in an easterl diretion and then travels 0 km due south Draw a vetor diagram to represent Marus s path and the displaement of the finishing point from the starting point In questions and, draw vetor diagrams to represent the paths desried and the displaement of the finishing point from the starting point iana rows straight aross a river in whih a urrent is flowing at 5 km/h iana an row at 5 km/h An aeroplane takes off and flies at an angle of elevation of 5 for 5 km It then flies horizontall for 00 km (, ) 0 d (, ) 78 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

8 8 Units & AOS Topi Conept Operations on vetors Conept summar Pratie questions Operations on vetors Addition of vetors If we travel from A to and then from to C, the omined effet is to start from A and finish at C We write A + C = AC Notie that the tail of the seond vetor C is joined to the head of the first vetor A If the addition is reversed, so that the tail of the first vetor is C joined to the head of the seond vetor, the omined effet is also a vetor equal to AC So A + C = C + A This shows that hanging the order in whih vetors are added does not alter the omined effet of the vetors This method for adding two vetors is alled the triangle rule for vetors The addition of vetors a and an e shown forming a vetor from the tail of a to the head of Negative vetors Just as moving units on the -ais is opposite in diretion to moving units along the -ais, the negative of a given vetor is opposite in diretion to the original vetor The vetor has the same magnitude as ut is in the opposite diretion Sutration of vetors Sutration of vetors an e performed omining vetor addition and negative vetors a = a + ( ) For eample, if a and are vetors as shown at right, then we an find a : epressing it as an addition: a = a + ( ) reversing the arrow on vetor so that it eomes adding to a as shown to form a A a a + a a A C a a Topi 8 Vetors 79

9 WoRKED EXAMPLE 6 Using d, e and f as shown in the diagram, draw vetor diagrams to show: a d + e d + e + f e f a Draw the vetor d and join the tail of e to the head of d d + e is shown the vetor drawn from the tail of d to the head of e d + e + f is otained joining the head of d + e (from part a) with the tail of f d + e + f is shown the vetor drawn from the tail of d (or d + e) to the head of f Reverse the arrow on f to otain f and join the head of e to the tail of f e f is shown the vetor drawn from the tail of e to the head of f WoRKED EXAMPLE 7 DrAW a e Salar multipliation A displaement of (, ) followed another displaement of (, ) equals a displaement of (, 6) We ould write this as (, ) = (, 6) The vetor represented (, ) has een multiplied the numer to give the vetor represented (, 6) d d e f f e f f e e d e d d + e If a = (, ), = ( 5, ) and = (, ), find eah of the following: a a + a a + + a Add the orresponding omponents of eah vetor to give the answer for a + Sutrat the orresponding omponents of eah vetor to give the answer for a a + + ma e alulated adding the orresponding omponents of a and and WritE a a + = (, ) + ( 5, ) = (, 6) a = (, ) (, ) = (, ) d f e e f d + e + f a + + = (, ) + ( 5, ) + (, ) = ( 6, 9) 80 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

10 WoRKED EXAMPLE 8 This proess is alled multipliation a salar or salar multipliation Salar multipliation means that the vetor is made larger or smaller a sale fator In the ase aove, the salar is In general, we an sa that if k R: ka is a vetor k times as ig as a and in the same diretion as a for k > 0 ka is in the opposite diretion to a for k < 0 If a = (5, ) and = (, ), alulate: a a + ( a ) WritE a Multipl eah omponent of a to otain a a a = (5, ) = (0, 8) a a a a 5a Add the omponents of a and to otain a + a + = (0, 8) + (, ) = (7, 6) Sutrat the omponents of a from to otain a a = (, ) (5, ) = ( 8, 6) Multipl the omponents of a to otain ( a ) ( a ) = ( 8, 6) = (, 8) WoRKED EXAMPLE 9 AEF and CDE are parallelograms with A represented F E D a and AF represented The length of C is twie the length of A Epress the following vetors in terms of a and A a C a C AC D a C and A are in the same diretion and C is twie as ig as A WritE a C = A Replae A a = a AC = A + C using vetor addition AC = A + C Replae A and C a and a respetivel = a + a Simplif = a CD = AF eause opposite sides of a parallelogram are parallel and the same size CD = D = C + CD, using the triangle rule to add vetors D = C + CD Replae C and CD a and respetivel = a + Topi 8 VECToRS 8

11 WoRKED EXAMPLE 0 Simplif the epression A + C EC A + C represents a vetor from A to with the vetor from to C added on This is the same as the vetor from A to C EC is the same as vetor CE eause the negative of a vetor reverses the diretion AC + CE represents a vetor starting at A going to C and then from C to E This is the same as AE EXErCisE 8 PrACtisE ConsoliDAtE Operations on vetors WE6 Using vetors a, and as shown, sketh: a a d a + e a + f + Using a, and from question, sketh: a a + a + a d e a + + f a WE7 If m = (, ), n = (, 0) and p = (, 5), find eah of the following a m + n m + n + p Using m, n and p from question, find eah of the following a n p m n p 5 WE8 Using m, n and p from question, alulate the following a n p m + n p 6 Using m, n and p from question, alulate the following a (m + n ) p n 7 WE9 Refer to the ue shown on the right Let a = OA, = OC and d = OD Write, in terms of a, and d, the vetors representing: a DE AC e EA WritE A + C EC = AC EC = AC + CE O d AE 8 Using the ue from question 7, write the following vetors in terms of a, and d a EG DF OF d AG e D 9 WE0 Show that OA + A + C = OC 0 In simplest form, MN QP + NP + QR equals: A 0 MR C MQ D QN E NR Draw two vetors u and v suh that u + v = (0, 0) Draw two possile representations of u + v = (, 5) = AE G D a C O F E A 8 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

12 Master Draw two possile representations of u + v = (, ) Using the ue shown in question 7, write all the vetors that are equal to the following vetors a OA OC OD d GF e O f AD 5 ACDEF is a regular heagon with vetors OA and O C represented a and respetivel Write, in terms of a and, the following vetors: D a a DO DA AD d A O e C f AC g CD h ED i EA j DF 6 Show that EF + GH GF EH = 0 7 Epress in simplest form A + C + DE DC 8 In terms of vetors a and in the figure at right, the vetor joining O to D is given : A a + a + C a D a E none of these 9 In terms of vetors a and, the vetor joining E to O at right is: A a + a C a D a E none of these 0 The parallelogram ACD an e defined the two vetors and In terms of these vetors, find: a the vetor from A to D the vetor from C to D A the vetor from D to A retangular prism CDEFGHIJ an e defined J the vetors r, s and t, as shown on the right Epress F in terms of r, s and t: t a the vetor joining C to H G the vetor joining C to J s C the vetor joining G to D d the vetor joining F to I e the vetor joining H to E f the vetor joining D to J g the vetor joining C to I h the vetor joining J to C A ue PQRSTUVW an e defined the three vetors a, and as shown at right Epress in terms of a, and : a the vetor joining P to V the vetor joining P to W the vetor joining U to Q d the vetor joining S to W e the vetor joining Q to T a O a O D S P E E r U T D a H R Q F I A C E D W V Topi 8 Vetors 8

13 8 Units & AOS Topi Conept Magnitude of a vetor Conept summar Pratie questions WoRKED EXAMPLE Magnitude, diretion and omponents of vetors Magnitude The magnitude of a vetor an e alulated from the length of the line segment representing the vetor The magnitude of a vetor a is denoted a or a Diretion The diretion of a vetor an e found appling appropriate trigonometri ratios to find a relevant angle This angle is usuall the angle that the vetor makes with a given diretion suh as north, the positive -ais, or the horizontal or vertial Find the magnitude and diretion, relative to the positive -ais, of the vetor (, ) Draw a diagram of the vetor and denote it as a with the angle etween a and the positive -ais as θ The magnitude of a is the length of the line segment representing the vetor Use Pthagoras theorem to alulate this length DrAW/WritE 0 θ a a = + = Calulate the angle θ using trigonometr tan (θ) = θ = 7 orret to deimal plae 5 State the solution with the angle down from The vetor (, ) has a magnitude of units the positive -ais given as a negative and makes an angle of 7 with the positive -ais The angle that a vetor makes with the positive -ais an e (, ) found using trigonometr If the vetor points in the negative -diretion, then ou will need to add our found angle θ to 80 θ 90 or sutrat it from 80 to find the required angle See the θ diagram at right 0 Upward vetors are epressed as positive angles antilokwise from the positive -ais Downward vetors are epressed as negative angles lokwise from the positive -ais In general, if r = (a, ), then the diretion of r ompared to the positive -ais is found appropriatel adjusting θ where tan (θ) = a 8 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

14 WoRKED EXAMPLE Vetor omponents We have seen that two vetors ma e added to give one resultant vetor The reverse proess ma e used to epress one vetor as the sum of two other vetors This proess is alled reaking the vetor into two omponents A vetor an e roken into two perpendiular omponents suh as and or north and east It ma e onvenient to find the effet of a vetor in a partiular diretion We do this reaking the vetor into two omponents A fore F ating as shown will move an ojet to the right and upwards The fore F an e separated into two omponent parts; one in the horizontal diretion, H, and the other in the vertial diretion, V F = H + V The effet of the fore in the horizontal diretion is given entirel H and the effet in the vertial diretion is given V reaking F into omponent parts in two perpendiular diretions, we an analse the effet of the vetor in one or oth of these diretions Write the horizontal and vertial omponents of a vetor of magnitude 5 and angle of 0 with the positive -ais Represent the vetor on the Cartesian plane Construt a right-angled triangle with the vetor as the hpotenuse and the other sides H for horizontal and V for vertial Calulate the angle etween the vetor and the -ais and indiate it on the graph DrAW/WritE 5 V V 5 H H 0 Calulate V using the sine ratio sin (60 ) = V 5 V = 5 sin (60 ) = 5 (or ) F H Angle = 80 0 = 60 V Topi 8 VECToRS 85

15 5 Calulate H using the osine ratio os (60 ) = H 5 6 State the solution, adding negative signs where neessar WoRKED EXAMPLE EXErCisE 8 PrACtisE A ar travels km in a diretion N0 E From its starting point, how far has it travelled: a north a Draw a vetor diagram representing the motion of the ar Call the vetor a and its eastern and northern omponents e and n respetivel Calulate n (the magnitude of n ) using the osine ratio State the distane travelled the ar to the north Calulate e (the magnitude of e ) using the sine ratio State the distane travelled the ar to the east H = 5 os (60 ) = 5 (or 5) The vetor has a horizontal omponent of 5 and a vertial omponent of 5 east? DrAW/WritE a N e km n 0 a Magnitude, diretion and omponents of vetors 0 WE Calulate the eat magnitude and diretion, relative to the positive -ais, of the following displaements a (6, ) (, ) (, ) d (, ) E os (0 ) = n n = os (0 ) = = 6 The ar has travelled to approimatel 6 km north of its starting point sin (0 ) = e e = sin (0 ) = 6 The ar has travelled to 6 km east of its starting point 86 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

16 Consolidate Calulate the eat magnitude and diretion, relative to the positive -ais, of the following displaements a (, ) (, ) (, 0) d (, ) WE Write the horizontal and vertial omponents of these vetors Write our answers in eat form where possile a Magnitude, angle of 60 with the -ais Magnitude, angle of 50 with the -ais Magnitude 0, angle of 60 with the -ais d Magnitude, angle of 0 with the -ais e Magnitude 0, angle of 5 with the -ais f Magnitude, parallel to the -ais g Magnitude, parallel to the -ais Write the horizontal and vertial omponents of these vetors Write our answers in eat form where possile a A speed of 0 m/s for one seond vertiall downwards A move of magnitude 0 m at an angle of 0 antilokwise from the negative diretion of the -ais A move of magnitude 0 m at an angle of 0 antilokwise from the positive diretion of the -ais d A speed of 50 m/s horizontall to the right e A fore of 0 N at an angle of 0 to the horizontal f A fore of 98 N vertiall downwards g A fore of 50 N at an angle of 5 to the horizontal 5 WE A aht sails km in a diretion S5 E From its starting point how far has it travelled: a south east? 6 A ar travels 0 km in a diretion N5 W From its starting point, how far has it travelled: a north (to the nearest km) west (to deimal plae)? G F 7 Refer to the diagram of the ue shown If the sides of the ue are unit in length, write the magnitudes of these vetors C in eat form D E a OA A O d OD e AD f DF O A g OE h EF i OF j AG 8 A vetor has a horizontal omponent of ( > 0) and a vertial omponent of ( > 0) Write the magnitude and diretion from the positive -ais of the vetor 9 Find the magnitude and diretion of eah of the following vetors Epress the diretion relative to the positive -ais a (, ) a 0 d 0 (, ) Topi 8 Vetors 87

17 0 Write the horizontal and vertial omponents of a vetor of magnitude 0 on an angle of 0 with the positive -ais Give answers orret to deimal plae For eah of the following, find: i the magnitude of the vetor ii the diretion of eah vetor (Epress the diretion with respet to the positive -ais) a (6, 6) (, 7) v w 0 a (, 5) 0 d 0 0 (0, 0) Using the vetor shown at right, find: a the magnitude of u the diretion of u (epress the angle with respet to the 0 θ u positive -ais) (, 5) the true earing of u A vetor with a true earing of 60 degrees and a magnitude of 0 has: A -omponent =, -omponent = -omponent =, -omponent = C -omponent = 5, -omponent = 5 D -omponent = 5, -omponent = 5 E none of the aove Consider the vetor w, shown on the right, with a N magnitude of 00 and on a earing of 0 T Find the and omponents of w Epress answers as 0 W eat values Justine les 8 km in a northerl diretion She then w S travels 6 km in an easterl diretion Calulate the magnitude and diretion of her displaement 6 Epress the horizontal and vertial N omponents of a vetor to the nearest whole numer represented a ship that sails on a earing of for 5 km 9º 7 For the following pairs of vetors, alulate the magnitude and diretion of a + a a = 0 km north and = 6 km north-east a = 5 units east and = 0 units S0 W a = 0 units and = 8 units in the opposite diretion Master E 88 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

18 THINK 85 Units & AOS Topi Conept i, j notation and resolving a vtor Conept summar Pratie questions WORKED EXAMPLE 8 For the following pairs of vetors, alulate the magnitude and diretion of a + and a a a = km west and = km south a = 0 km and = 5 km in the same diretion a = 50 units in a diretion 00 T and = 0 units in a diretion 0 T i, j notation Unit vetors A unit vetor is an vetor with a magnitude or length of unit The vetor i is defined as the unit vetor in the positive -diretion The vetor j is defined as the unit vetor in the positive j -diretion O i For eample, a displaement of d = (, 5) represents a move of units in the positive -diretion and 5 units in the positive -diretion An alternative wa of representing this is: d = i + 5j An vetor in two dimensions an e represented as a omination of i and j vetors, the oeffiient of i representing the magnitude of the horizontal omponent and the oeffiient of j representing the magnitude of the vertial omponent In general we ma represent an two-dimensional vetor r as: r = i + j where, R This vetor, r, ma also e written as a olumn matri: r = i + j = = where i = 0 and j = 0 For eample, d = i + 5j = 5 = a Draw a vetor to represent a = i j Find the magnitude and diretion of the vetor a a Draw aes with i and j as unit vetors in the - and -diretions respetivel DRAW/WRITE a j O Represent i j as a vetor from 0 that is j units in the positive -diretion and unit in the θ negative -diretion, and mark the angle etween a and the -ais as θ O a i i O r Topi 8 VECTORS 89

19 The magnitude of a (that is, a ) ma e found using Pthagoras theorem WoRKED EXAMPLE 5 As we have seen, angles are usuall given with respet to the positive -diretion We ma generalise this proedure: For an vetor, r : r = i + j magnitude r, r = + the diretion from the positive -ais is given appropriatel adjusting θ where tan (θ) = Addition, sutration and multipliation a salar for a vetor in i, j form follow the rules of normal arithmeti, with eah omponent treated separatel If a = i + j and = i + j a + = ( + ) i + ( + ) j a = ( ) i + ( ) j ka = k i + k j If a = i + j and = i + 5j, epress in i, j form: a a + a a Add the i omponents and j omponents separatel a is alulated multipling the i and j omponents of a a is alulated sutrating the i and j omponents of respetivel from a a = + ( ) WritE = 0 Find the value of angle θ using the tangent ratio tan (θ) = θ 8 Give the diretion of vetor a relative to the Vetor a makes an angle of 8 from positive -ais the positive -ais O θ r a a + = (i + j ) + ( i + 5j ) = i i + j + 5j = i + 6j a = (i + j) = 6i + j a = 6i + j ( i + 5j) = 6i + j + i 5j = 8i j i j 90 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

20 WoRKED EXAMPLE 6 OA = i + j and O = i + j a Represent OA and O on a diagram Find, in terms of i and j, the vetor A If M is the midpoint of A, find the vetor OM in terms of i and j a Draw aes with i and j as unit vetors in the - and -diretions respetivel Represent OA as i + j and O as i + j on the aes A ma e epressed as AO + O using the triangle rule for adding vetors Multipling two vetors The dot produt is one method of multipling one vetor another vetor It is also alled a salar produt as the result of this multipliation is a salar (magnitude onl) The produt of two vetors u and v is denoted u v Consider the two vetors u and v, as shown DrAW/WritE a O definition, the dot produt u v is given : u v = u v os (θ) where θ is the angle etween (the positive diretions of) u and v Note: The vetors are not aligned as for addition or sutration; instead their two tails are joined A A = AO + O Change AO to negative OA = OA + O OA = i + j O = i + j Epress this in i, j form A = (i + j ) + ( i + j ) Simplif = i + j Mark the point M in the middle of A M A O Epress OM as the sum of OA + A OM = OA + AM OM = OA + A Epress this in i, j form = (i + j ) + ( i + j ) Simplif = i + 5j u θ v Topi 8 VECToRS 9

21 WoRKED EXAMPLE 7 Let u = i + j and v = 6i Find u v WritE/DrAW Find the magnitudes of u and v u = + = 5 v = 6 = 6 Draw a right-angled triangle showing the angle that u makes with the positive -ais sine v is u along the -ais 5 Find os (θ), knowing that u = 5 and the -omponent of u is Note: An easier method for finding the dot produt is to multipl the orresponding and omponents of the two vetors Perpendiular vetors If two vetors are perpendiular then the angle etween them is 90 0 θ os (θ) = 5 Find u v using the dot produt equation u v = u v os (θ) 5 Simplif = = 8 WoRKED EXAMPLE 8 Find u v if u = i + 5j and v = i + j WritE Write down u v u v = (i + 5j ) (i + j ) Multipl the orresponding omponents u v = + 5 Simplif = u v = u v os (90 ) = u v 0 sine os (90 ) = 0 = 0 If u v = 0, then u and v are perpendiular 9 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

22 WoRKED EXAMPLE 9 Find the onstant a if the vetors u = i + j and v = i aj are perpendiular WritE Find the dot produt u v = (i + j ) ( i aj ) Simplif = a Set u v equal to zero sine u and v are perpendiular u v = a = 0 Solve the equation for a a = WoRKED EXAMPLE 0 Parallel vetors If two vetors are parallel ( ) then the angle etween them is either 0 (if ating in the same diretion) or 80 (if ating in opposite diretions) u v Same diretion, θ = 0 For θ = 0 (same diretion) or u v = u v os (θ) u v Opposite diretions, θ = 80 = u v os (0 ) For θ = 80 (opposite diretions) eomes = u v as os (0 ) = u v = u v os (θ) = u v os (80 ) u = u v as os (80 ) = Let u = 5i + j Find a vetor parallel to u suh that the dot produt is 87 The dot produt is positive so the vetors are in the same diretion, with θ = 0 WritE u v = u v u v = 87 Find the magnitude of u u = 5 + = 5 + = 9 Sustitute u into the formula found in step u v = 87 v = 87 9 = 9 v Topi 8 VECToRS 9

23 The magnitude of v is times the magnitude of u, hene v is times larger than u v = u 5 Multipl u to find v (5i + j ) = 5i + 6j 6 Answer the question v = 5i + 6j WoRKED EXAMPLE EXErCisE 85 PrACtisE Finding the angle etween two vetors The dot produt formula an e used to find the angle etween two vetors u v = u v os (θ) re-arranging this formula, we get: os (θ) = u v u v θ = os u v u v Let u = i + j and v = i j Find the angle etween them to the nearest degree WritE Find the dot produt u v u v = (i + j ) (i j ) = + = Find the magnitude of u u = + = 5 i, j notation WE Draw a vetor to represent eah of the following a i + j i j i + j d i j e i + j f 5i g 6j h i = 5 Find the magnitude of v v = + ( ) Sustitute the results into the formula for the angle etween two vetors = θ = os u v u v = os 5 = os ( ) = Answer to the nearest degree = 9 i 8i 6j j 5i + j 9 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

24 Consolidate Calulate the magnitude and diretion of eah of the vetors in question WE5 If a = i + j, = i j and = j, find the following in i, j form a a a + a d e a + + f g a + + h i a j a If u = i j and v = i + j, find the following in eat form a u u + v v d u v 5 WE6 OA = i j and O = i + j a Represent OA and O on a diagram Find, in terms of i and j, the vetor A If M is the midpoint of A, find the vetor OM in terms of i and j 6 OAC is a retangle in whih the vetor OA = i and O = 6j Epress the following in terms of i and j a OC OM where M is the midpoint of OA AC d ON where N is the midpoint of O e A f MN 7 WE7 Find the dot produt of u = i + j and v = i + 5j 8 Let u = i j and v = i j Find the dot produt of u and v 9 WE8 Let u = i + 6j and v = i j Find u v 0 Find u v, when u = i + j and v = i + j WE9 Find the onstant a if the vetors u = i + 7j and v = 7i aj are perpendiular Find the onstant a, if the vetors v = ai + j and u = 6i j are perpendiular WE0 Let u = i j Find a vetor parallel to u suh that its dot produt with u is 0 Let u = i j Find a vetor parallel to u suh that their dot produt is 80 5 WE Let u = i + j and v = i 5j Find the angle etween them to the nearest degree 6 Find the angle, to the nearest degree, etween the vetors i 6j and i + j 7 Represent the following position vetors in the form i + j (, ) (, ) (, ) a h (5, ) d e f g ( 6, ) (, ) (, ) (6, ) Topi 8 Vetors 95

25 Master 8 The position of the points A, and C is defined : OA = i, O = 0i + j and OC = i + j a Find the vetors representing the three sides of the triangle AC (that is, find in i, j form the vetors A, AC and C) Calulate the magnitude of these three sides Leave answers in eat form What tpe of triangle is AC? 9 M, N and P are three points defined : OM = i + j, ON = i + j and OP = 5i + 0j a Find MN and NP Show that MN and NP are parallel vetors 0 a = i j and = i + j a Find a and a + Eplain wh a + is parallel to the -ais The magnitude of the vetor i + j is: A + C 6 D E + If a = i 5j and = i j, then a equals: A 9i j 9i + j C i j D i + j E i 9j The angle the vetor i j makes with the positive -ais is nearest to: A 7 5 C 5 D 7 E 7 Find the vetor a +, whih represents the planned shot of a pool plaer a = i + j = 6 i 0 j 5 Vetor m = i + j The magnitude of m is Find the value of 6 Find a vetor perpendiular to i 6j 7 Let u = i + j and v = i + j Find os (θ), where θ is the angle etween the two vetors Give answer in eat form 8 Consider the vetors u and v at right Their magnitudes are 7 and 8 respetivel Find u v (to the nearest whole numer) 86 Appliations of vetors Vetors have a wide range of appliations, suh as in orienteering, navigation, mehanis and engineering Vetors are applied whenever quantities speified oth magnitude and diretion are involved 8 v 50 u 96 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

26 WoRKED EXAMPLE When solving prolems involving vetors: Draw a vetor diagram depiting the situation desried Use the appropriate skills to answer the question eing asked A oat is eing rowed straight aross a river at a speed of 6 km/h The river is flowing at km/h If i is the unit vetor in the diretion that the river is flowing and j is the unit vetor in the diretion straight aross the river, represent the veloit of the oat in terms of i and j Hene, find the magnitude and diretion of the veloit of the oat, orret to deimal plae Draw a set of aes with i in the diretion of the positive -ais and j in the diretion of the positive -ais Indiate the veloit vetor of the oat, v, starting at O and finishing at the point (, 6) Represent the veloit of the oat in terms of i and j DrAW/WritE j i Note: The magnitude of veloit is referred to as speed j O i 6 O v = i + 6j The magnitude of v is + 6 v = Evaluate the magnitude orret to deimal plae 6 Draw a right-angled triangle with v as the hpotenuse and θ as the angle etween v and the i diretion 6 O v = 0 6 km/h 7 Epress θ using the tangent ratio tan (θ) = 6 = 8 Evaluate θ orret to deimal plae θ = 76 9 State the magnitude and diretion of the veloit of the oat v θ 6 The veloit of the oat has a magnitude of approimatel 6 km/h and is direted at approimatel 76 from the riverank Topi 8 VECToRS 97

27 WoRKED EXAMPLE a Epress a in terms of i and j An airraft is heading north with an airspeed of 500 km/h A wind of 80 km/h is lowing from the south-west Using i and j as unit vetors in the diretions east and north respetivel: a represent the airraft s air veloit in terms of i and j represent the airraft s eat ground veloit, v, in terms of i and j find the diretion in whih the airraft is heading and its ground speed, orret to deimal plae Draw a set of aes with i in the diretion of the positive -ais and j in the diretion of the positive -ais Indiate the vetor representing the airraft s airspeed, a, starting at O and finishing at the point (0, 500) Indiate the vetor representing the wind speed, w, plaing its tail at the head of the first vetor, direted in a diretion 5 from the north with a magnitude of 80, as the wind speed is 80 km/h from the south-west Represent the omined effet of the two speeds with a vetor, v, using the triangle rule 5 Epress w, eatl, in terms of i and j using asi trigonometr 6 Epress the airraft s ground veloit, v, as the sum of a and w WritE/DrAW a a = 500j N W S j i O a 80 w v E 0 w = 80 sin (5 ) i + 80 os (5 )j = 0 i + 0 j v = a + w 7 Epress v in terms of i and j = 0 i + ( ) j Indiate the angle etween v and the -ais as θ 0 Use the tangent ratio to evaluate θ to deimal w tan (θ) = plae The length of the horizontal omponent v of v is 0 The length of the vertial a 006 omponent of v is θ = 58 O Calulate the magnitude of v orret to deimal plae State the diretion and magnitude of the ground speed of the airraft θ v = (0 ) + ( ) 559 The airraft is fling with a ground speed of 559 km/h in a N58 E diretion 98 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

28 Units & AOS Topi Conept 5 Appliations of vetors Conept summar Pratie questions WoRKED EXAMPLE Statis When the vetor sum of the fores ating on a stationar partile is zero, the situation is said to e stati and the partile will remain stationar The partile is also said to e in equilirium In the ase of two fores, we have the situation shown elow In the ase of three fores, we have the situation shown in P F the diagram elow left Where the three fores are ating so that the partile is in equilirium, the lines representing the fores an e rearranged into a triangle of fores (as in the diagram elow right) eause their vetor sum is zero Hene, prolems an e solved using trigonometr (inluding the sine rule and osine rule) and sometimes Pthagoras theorem F F P P F + P + R = 0 Note: The three fores are still ating in the same diretion and have the same magnitudes (or lengths) as the did in the real situation R Three fores are ating on the partile P as shown in the diagram elow Fore A is vertiall up and has a magnitude of 0 N (0 newtons); fore is horizontall to the right and has a magnitude of 0 N If the partile is in equilirium, find the magnitude of fore C to the nearest tenth of a newton and give its diretion to the nearest tenth of a degree A C 0 N Draw the three fores as a triangle of fores Lael the angle etween fores A and C as θ P 0 N DrAW/WritE 0 C A θ 0 Calulate C using Pthagoras theorem C = A + = = = 000 C = 000 Evaluate C orret to deimal plae C = 7 newtons R Topi 8 VECToRS 99

29 5 Evaluate θ using the tangent ratio tan (θ) = 0 0 θ = tan () θ = 6 6 State the answer to the question The fore has a magnitude of 7 N and is ating downwards at an angle of 6 from the vertial WoRKED EXAMPLE 5 EXErCisE 86 PrACtisE Geometri proofs Vetors an also e used to prove a range of geometri theorems From earlier in the topi, ou will rememer that two vetors are equal if the are equal in magnitude, are parallel and point in the same diretion One important vetor propert that is useful in geometri proofs is that if a = k, where k R (k 0), then the two vetors, a and are parallel Show that the line joining the midpoint of two sides of a triangle is parallel to the third side and equal to half of its length Let side A represent vetor A and side C represent vetor C Use the smol a for vetor A, and for vetor C Let side AC represent vetor AC Epress AC in terms of a and Appliations of vetors WritE Let A = a and C = AC = A + C AC = a + Epress MN in terms of a and MN = M + N Simplif the epression taking out as a ommon fator = A + C = a + = (a + ) 5 Epress MN in terms of AC = AC 6 MN is parallel to AC sine AC is a multiple of MN Therefore, MN is parallel to AC and its length is half the length of AC WE A oat is eing rowed straight aross a river at a speed of 7 km/h The river is flowing at 5 km/h If i is the unit vetor in the diretion that the river is flowing and j is the unit vetor in the diretion straight aross the river, represent the veloit of the oat in terms of i and j Hene, find the magnitude and diretion of the veloit of the oat orret to deimal plae A M N C 00 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

30 Q P M R Consolidate A oat is eing rowed straight aross a river at a speed of 0 km/h The river is flowing at km/h Find the magnitude and diretion of the veloit of the oat WE An airraft is heading north with an airspeed of 650 km/h A wind of 60 km/h is lowing from the south-west Using i and j as unit vetors in the diretions east and north respetivel: a represent the airraft s airspeed represent the airraft s ground speed in terms of i and j find the diretion in whih the airraft is heading and its ground speed An airraft is heading south with an airspeed of 600 km/h A wind of 50 km/h is lowing in a S0 W diretion Find the diretion in whih the airraft is heading and the ground speed 5 WE Three oplanar fores are ating on the partile P A as shown Fore A is vertiall up and has magnitude of 6 N; fore is horizontall to the right and has a 6 N P magnitude of 8 N If the partile is in equilirium, 8 N find the magnitude of fore C to the nearest tenth of a C newton and give its diretion to the nearest tenth of a degree 6 Three oplanar fores are ating on the partile P as shown C at right Fore A has a magnitude of 5 N, and fore has a A magnitude of 0 N If the partile is in equilirium, find the magnitude of fore C to the nearest tenth of a newton and P 5 N 0º give its diretion to the nearest tenth of a degree 7 WE5 PQR is a triangle in whih M is the midpoint of QR as 0 N shown on the left Prove that PM = (PR QP) 8 Prove that if the midpoints E, F, G and H of a rhomus ACD are joined, then a parallelogram EFGH is formed (Etension: Show that the parallelogram is, in fat, a retangle) 9 Fores of i + j and i + j at simultaneousl on an ojet Find the magnitude and diretion of the resultant of the two fores 0 Fores of 5i j, i j and i + j at simultaneousl on an ojet Find the magnitude and diretion of the resultant of the three fores A hiker is loated at a position given (8, 6), where the oordinates represent the distanes in kilometres east and north of O respetivel If a ampsite is at a position given (, ), find the distane and diretion of the hiker from the ampsite A hiker is loated at a position given ( 5, ), where the oordinates represent the distanes in kilometres east and north of O respetivel If a ampsite is at a position given (, ), find the distane and diretion of the hiker from the ampsite Topi 8 Vetors 0

31 MAstEr The position vetors for an arrow and a moving target are shown at right, where t is the time in seonds sine the target egan to move, and h is the height of the target in metres If the arrow is to hit the target, when must this happen and what must the value of h e for this to our? Fores of i + j, i 5j, i + j and i j at on a partile that is in equilirium Find the values of and 5 A river flows through the jungle from west to east at a speed of km/h An eplorer wishes to ross the river oat, and attempts this travelling at 5 km/h due north Using i and j as unit vetors in the diretions east and north respetivel: a epress the veloit of the river and the veloit of the oat in terms of i and j draw vetors represented the veloit of the river and the veloit of the oat alulate the magnitude of the resultant vetor d find the earing of the oat s journe, orret to the nearest degree 6 oat A travels east at 0 km/h, while oat travels south from the same point at 5 km/h Find the veloit of oat A with respet to oat 7 A river flows west east at 5 m/s A swimmer, in still water, an swim m/s and tries to swim diretl aross the river from south to north a Draw a vetor diagram to illustrate this situation Find the resultant speed of the swimmer orret to deimal point Find the earing of the swimmer orret to the nearest degree d If it took the swimmer minutes to reah the opposite ank, how wide is the river? e How far downstream would the swimmer e arried? 8 In the drawing at right, AC is a triangle Point D is along C the line C suh that D = r C The vetors q, r and t are A as shown in the diagram Prove that: t = t (q + r) q D 9 A ushwalker starts walking at 800 am from a ampsite at (, 8), where the oordinates represent the distanes in kilometres east and north of O respetivel After hour she is at (, 65) Take i and j as unit vetors along OX and OY a Write, in terms of i and j, her position at the start and after hour Calulate the distane travelled in hour 0 S 0t i + (t 0t ) j (5t + 5)i + h j W km/h E N h 0 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

32 She then ontinues at the same rate and in the same diretion What is her position vetor after: i hours ii hours? d Show that her position t hours after 800 am is given : r = ( + t) i + (8 5t) j Another ushwalker ommenes walking from his ampsite, also at 800 am His position is given : r = (7 8t) i + ( + 05t) j e What are the oordinates of this ushwalker s ampsite? f What is his position after hours of walking? g equating i and j omponents, show that the two ushwalkers meet h Find the distane from eah ampsite that eah ushwalker has travelled when the meet 0 The i, j sstem ma e etended to three dimensions with a unit vetor k in the z-diretion Take i, j and k as unit vetors in the diretions east, north and vertiall up respetivel z k O j i Frank travels km in a diretion N0 E from O to a point A He then lims a 00 m high liff a Write the vetor OA in i, j form Calulate how far Frank has travelled to the north of his starting point If T represents the top of the liff, write down the vetors AT and OT using i, j, k omponents d Calulate the magnitude of OT Topi 8 Vetors 0

33 ONLINE ONLY 87 Review the Maths Quest review is availale in a ustomisale format for ou to demonstrate our knowledge of this topi the review ontains: Multiple-hoie questions providing ou with the opportunit to pratise answering questions using CAS tehnolog short-answer questions providing ou with the opportunit to demonstrate the skills ou have developed to effiientl answer questions using the most appropriate methods studon is an interative and highl visual online tool that helps ou to learl identif strengths and weaknesses prior to our eams You an then onfidentl target areas of greatest need, enaling ou to ahieve our est results wwwjaplusomau Etended-response questions providing ou with the opportunit to pratise eam-stle questions A summar of the ke points overed in this topi is also availale as a digital doument REVIEW QUESTIONS Download the Review questions doument from the links found in the Resoures setion of our eookplus Units & Vetors Sit topi test 0 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

34 8 Answers EXERCISE 8 a (, ) (, ) (, ) d (, ) e ( 6, ) 5 a d (, 7) e a (5, 7) (, 0) (, ) d (, 0) 5 6 A 0 D C 7 a = ; = f 8 a = d ; = km 00 km A 0 0 km F S 5 C W W 0 km N S N S a = (, ), = (0, ), = (, ), d = (, ), e = (, ), f = (, ), g = (, 0), h = (, ) E E a = a a d = g = 0 = 0 = e = f = 0 (, ) (, ) h = (6, ) (, ) (, 8) 5 (0, 5) (, ) (, 5) 5 6 (, 6) 0 Topi 8 Vetors 05

35 d 5 6 (, ) 8 (, 8) 7 (0, ) 6 5 (, 5) (7, ) (, 5) (, ) (, 5) 0 Note: Vetor an start at an point Started at (0, 0) and (, ) to draw different line segments for the vetor 5 7 = d a =, = 8 a 0 A 0 A d 9 a 0 S 0 km C 0 A C 0 W 0 km F 5 km/h 5 km/h 5 km 5 00 km EXERCISE 8 a a a N S E d d a + e a + a a a a + f d a + a + a 06 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

36 e a + + a a (, ) (, 8) f a a a (5, 5) ( 5, ) 5 a (, 5) (, ) 6 a (, 6) ( 6, 0) 7 a a a + a + d d e d f 8 a a + a + a + + d d a + + d e a + d 9 Teaher to hek student proofs 0 u or an two vetors equal in length ut opposite in diretion v 5 (, 5) v or an similar result u O v (, ) u O or an similar result a C, DE, GF A, DG, EF AE, CG, F d DE, C, OA e DF f G 5 a a a a d a + e a f a + g h a + i a + j a 6 Teaher to hek student proofs 7 AE 8 9 D 0 a a r + s s + t r s d r + s e t s f s + t r g r + s + t h s t a a + a + + a d a + e + a EXERCISE 8 Magnitude Diretion a d 5 Magnitude Diretion a Parallel to -ais d 5 a,, 5 5, 5 d, e 0, 0 f 0, g, 0 a 0, 0 5, 5 0, 0 d 50, 0 e 76, 7 f 0, 98 g 07, 5 5 a 9 5 km 6 a km km 7 a d e f g h i j 8 +, angle θ from the positive diretion of the -ais where tan (80 θ) = 9 a 5, 5 5, 5 0 9, a i 6 ii 5 i 65 ii 97 i 9 ii d i 50 ii 8 a 59 9 T C 50, km, N69 E 6 6, 09 7 a 86 km, N66 E 9 units, S09 E units in diretion of a 8 a 697 km, SW and 697 km, NW 5 km in diretion of a and 5 km in diretion of a 60 units, 87 T and 60 units, 6 T Topi 8 Vetors 07

37 EXERCISE 85 a e g i O O O O O d f h j O O O O 5 O 5 a 5, 69 5, 69, 5 d, 5 e 7, f 5, 0 g 6, 90 h, 80 i 0, j, 6 a 9i + 6j i + j i + j d i j e i j f i g i + j h 8j i i 0j j i 7j a 9 5 a 0 d 7 O i + j 5 A j i i + j 6 a i + 6j i 6j d j e i + 6j f i + j i 8j 6 5 i 8 5 j a = i + j, = i + j, = i + j, d = 6i j, e = i j, f = i j, g = 6i j, h = 5i + j 8 a 6i + j, j, 6i + j 0,, 0 Isoseles 9 a MN = i + j, NP = i + 6j Parallel sine NP = MN 0 a 8i 8j, j The i omponent is zero C A C 9i + 7j 5 = ±5 6 6i + j or 6i j EXERCISE 86 5i + 7j ; 7 km/h; 70 from the river ank 06 km/h, 7 from the river ank a 650j 0 i + ( ) j N5 E, 697 km/h S W, 68 km/h 5 N, 60 from the vertial 6 78 N, 5 from the vertial 7 Teaher to hek student proofs 8 Teaher to hek student proofs 9 78 units, 50 from the positive -ais 0 6 units, 8 from the positive -ais 6 km, N5 E or 5 T 9 km, N58 W s, m = 5, = 5 a i, 5j r (, 5) d N E W N S E 08 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

38 6 5 km/h on a earing of N5 E 7 a s 58 m/s d 60 metres r N59 E e 600 metres 8 Teaher to hek student proofs 9 a i + 8j, i + 65j 5 km i 5j d Chek with our teaher ii i + 5j e (7, ) f (8, ) h First 75 km, seond 56 km 0 a i + j km 0k, i + j + 0k d 00 Topi 8 Vetors 09