MST124 Essentil mthemtics 1 Unit 5 Coordinte geometry nd vectors
Contents Contents Introduction 4 1 Distnce 5 1.1 The distnce etween two points in the plne 5 1.2 Midpoints nd perpendiculr isectors 7 2 Circles 11 2.1 The eqution of circle 11 2.2 The circle through three points 19 3 Points of intersection 23 3.1 Points of intersection of line nd curve 23 3.2 Points of intersection of two circles 26 3.3 Using the computer for coordinte geometry 28 4 Working in three dimensions 29 4.1 Three-dimensionl coordintes 29 4.2 The distnce etween two points in three dimensions 32 4.3 The eqution of sphere 34 5 Vectors 37 5.1 Wht is vector? 37 5.2 Vector lger 48 5.3 Using vectors 50 6 Component form of vector 57 6.1 Representing vectors using components 57 6.2 Vector lger using components 61 6.3 Position vectors 68 6.4 Converting from component form to mgnitude nd direction, nd vice vers 73 6.5 Using vectors in component form 81 7 Sclr product of two vectors 84 7.1 Clculting sclr products 84 7.2 Finding the ngle etween two vectors 89 Lerning outcomes 91 Solutions to ctivities 92 Acknowledgements 107 Index 108 3
5 Vectors 5 Vectors This section mrks the strt of the second prt of this unit, in which you ll lern out type of mthemticl oject clled vector. Vectors ply n importnt role in the study nd nlysis of phenomen in physics nd engineering. 5.1 Wht is vector? Some mthemticl quntities cn t e specified just y stting their size insted you need to stte size nd direction. For exmple, to fully descrie the motion of ship on the ocen t point in time during its voyge, it s not enough to specify how fst the ship is moving you lso need to descrie its direction of motion. You sw other exmples of this in Unit 2, when you considered ojects moving long stright lines. To specify the position of point P on stright line reltive to some other point, sy O, on the line, you first choose one direction long the line to e the positive direction; then you stte the distnce etween the two points, nd ttch plus or minus sign to indicte the direction. The resulting quntity is clled the displcement of P from O. For exmple, in Figure 28, if the positive direction is tken to e to the right, then the displcement of A from O is 2cm, nd the displcement of B from O is 4cm. Third prty copyright imge not ville in this we version You cn move only in one dimension long tightrope (unless you fll off!) A O B 2 cm 4 cm Figure 28 Positions long stright line Similrly, if n oject is moving long stright line, then you cn descrie its motion y giving its speed, nd ttching plus or minus sign to indicte its direction. The resulting quntity is clled the velocity of the oject. Plus nd minus signs provide convenient wy to specify direction when you re deling with movement long stright line tht is, in one dimension. Exmples of movement of this type include the motion of cr long stright rod, or tht of tightrope wlker long tightrope. However, we often need to del with movement in two or three dimensions. For exmple, someone stnding in flt field cn move cross the field in two dimensions, nd person in spce cn move in three dimensions. In generl, displcement is the position of one point reltive to nother, whether in one, two or three dimensions. To specify displcement, you need to give oth distnce nd direction. For exmple, consider the points O, P nd Q in Figure 29. You cn specify the displcement of P You cn move in two dimensions cross flt field You cn move in three dimensions in spce 37
Unit 5 Coordinte geometry nd vectors from O y sying tht it is 1km north-west of O. Similrly, you cn specify the displcement of Q from O y sying tht it is 1km north-est of O. P O Q N 1 km Figure 29 Points in plne Just s distnce together with direction is clled displcement, so speed together with direction is clled velocity. For exmple, if you sy tht someone is wlking t speed of 5kmh 1 south, then you re specifying velocity. Quntities, such s displcement nd velocity, tht hve oth size nd direction re clled vectors, or vector quntities. (In Ltin, the word vector mens crrier.) Another exmple of vector quntity is force. The size of vector is usully clled its mgnitude. In contrst to vectors, quntities tht hve size ut no direction re clled sclrs, or sclr quntities. Exmples of sclrs include distnce, speed, time, temperture nd volume. So sclr is numer, possily with unit. Notice tht the mgnitude of the displcement of one point from nother is the distnce etween the two points, nd the mgnitude of the velocity of n oject is its speed. In everydy English the words speed nd velocity re often used interchngely, ut in scientific nd mthemticl terminology there is n importnt difference: speed is sclr nd velocity is vector. The concept of vectors evolved over long time. Isc Newton (1642 1727) delt extensively with vector quntities, ut never formlised them. The first exposition of wht we would tody know s vectors ws y Josih Willrd Gis in 1881, in his Elements of vector nlysis. This work ws derived from erlier ides of Willim Rown Hmilton (1805 1865). Josih Willrd Gis (1839 1903) 38
5 Vectors Any vector with non-zero mgnitude cn e represented y n rrow, which is line segment with n ssocited direction, like the one in Figure 30. The length of the rrow represents the mgnitude of the vector, ccording to some chosen scle, nd the direction of the rrow represents the direction of the vector. Two-dimensionl vectors re represented y rrows in plne, nd three-dimensionl vectors re represented y rrows in three-dimensionl spce. For exmple, the rrow in Figure 30 might represent displcement of 30km north-west, if you re using scle of 1cm to represent 10km. Alterntively, the sme rrow might represent velocity of 30ms 1 north-west, if you re using scle of 1cm to represent 10ms 1. N Figure 30 An rrow tht represents vector Once you ve chosen scle, ny two rrows with the sme length nd the sme direction represent the sme vector. For exmple, ll the rrows in Figure 31 represent the sme vector. 39
Unit 5 Coordinte geometry nd vectors Figure 31 Severl rrows representing the sme vector Vectors re often denoted y lower-cse letters. We distinguish them from sclrs y using old typefce in typed text, nd y underlining them in hndwritten text. For exmple, the vector in Figure 30 might e denoted y v in print, or hndwritten s v. These conventions prevent reders from confusing vector nd sclr quntities. Rememer to underline hndwritten vectors (nd mke typed ones old) in your own work. Vectors tht represent displcements re sometimes clled displcement vectors. There is useful lterntive nottion for such vectors. If P nd Q re ny two points, then the vector tht specifies the displcement from P to Q (illustrted in Figure 32) is denoted y PQ. P Figure 32 The vector PQ The mgnitude of vector v is sclr quntity. It is denoted y v, which is red s the mgnitude of v, the modulus of v, or simply mod v. For exmple, if the vector v represents velocity of 30ms 1 north-west, then v = 30ms 1. Similrly, if the vector PQ represents displcement of 3m south-est, then PQ = 3m. Rememer tht the distnce etween the points P nd Q cn lso e denoted y PQ, so PQ = PQ. Notice tht the nottion for the mgnitude of vector is the sme s the nottion for the mgnitude of sclr tht you met in Unit 3. For exmple, you sw there tht 3 = 3. So this nottion cn e pplied to either vectors or sclrs. Q 40
5 Vectors When we re working with vectors, it s often convenient, for simplicity, not to distinguish etween vectors nd the rrows tht represent them. For exmple, we might sy the vector shown in the digrm rther thn the vector represented y the rrow shown in the digrm. This convention is used throughout the rest of this unit. Over the next few pges you ll lern the sics of working with vectors. Equl vectors As you d expect, two vectors re equl if they hve the sme mgnitude nd the sme direction. Activity 23 Identifying equl vectors The following digrm shows severl displcement vectors. () Which vector is equl to the vector? () Which vector is equl to the vector PQ? Q E D P e A R S c F C f H G d B The zero vector The zero vector is defined s follows. Zero vector The zero vector, denoted y 0 (old zero), is the vector whose mgnitude is zero. It hs no direction. The zero vector is hndwritten s 0 (zero underlined). For exmple, the displcement of prticulr point from itself is the zero vector, s is the velocity of n oject tht is not moving. 41
Unit 5 Coordinte geometry nd vectors Addition of vectors To understnd how to dd two vectors, it s helpful to think out displcement vectors. For exmple, consider the sitution shown in Figure 33. Suppose tht n oject is positioned t point P nd you first move it to the point Q, then you move it gin to the point R. The two displcements re the vectors PQ nd QR, respectively, nd the overll, comined displcement is the vector PR. This method of comining two vectors to produce nother vector is clled vector ddition. We write PQ+ QR = PR. R P R * QR * P P Q * Q Figure 33 The result of dding two displcement vectors Vectors re lwys dded in this wy. The generl rule is clled the tringle lw for vector ddition, nd it cn e stted s follows. Tringle lw for vector ddition To find the sum of two vectors nd, plce the til of t the tip of. Then + is the vector from the til of to the tip of. + The sum of two vectors is lso clled their resultnt or resultnt vector. You cn dd two vectors in either order, nd you get the sme result either wy. This is illustrted in Figure 34. Digrms () nd () show how the vectors + nd + re found using the tringle lw for vector ddition. When you plce these two digrms together, s shown in digrm (c), the two resultnt vectors coincide, ecuse they lie long the digonl of the prllelogrm formed y the two copies of nd. 42
5 Vectors + + + () () (c) Figure 34 The vectors + nd + re equl In fct, Figure 34(c) gives n lterntive wy to dd two vectors, the prllelogrm lw for vector ddition, which cn e stted s follows. Prllelogrm lw for vector ddition To find the sum of two vectors nd, plce their tils together, nd complete the resulting figure to form prllelogrm. Then + is the vector formed y the digonl of the prllelogrm, strting from the point where the tils of nd meet. + The prllelogrm lw is convenient in some contexts, nd you ll use it in Unit 12. In this unit we ll lwys use the tringle lw, s it s simpler in the sorts of situtions tht we ll del with here. You cn dd more thn two vectors together. To dd severl vectors, you plce them ll tip to til, one fter nother; then their sum is the vector from the til of the first vector to the tip of the lst vector. For exmple, Figure 35 illustrtes how three vectors, nd c re dded. c + + c Figure 35 The sum of three vectors, nd c The order in which you dd the vectors doesn t mtter you lwys get the sme resultnt vector. 43
Unit 5 Coordinte geometry nd vectors Activity 24 Adding vectors The digrm elow shows three vectors u, v nd w drwn on grid. u v w Drw rrows representing the following vector sums. (Use squred pper or sketch grid.) () u+v () u+w (c) v+w (d) u+v+w (e) u+u As you d expect, dding the zero vector to ny vector leves it unchnged. Tht is, for ny vector, +0 =. Note tht you cn t dd vector to sclr. Expressions such s v+x, where v is vector nd x is sclr, re meningless. Negtive of vector The negtive of vector is denoted y, nd is defined s follows. Negtive of vector The negtive of vector, denoted y, is the vector with the sme mgnitude s, ut the opposite direction. For ny points P nd Q, the position vectors PQ nd QP hve the property tht PQ = QP, since PQ nd QP hve opposite directions. If you dd ny vector to its negtive, y plcing the two vectors tip to til in the usul wy, then you get the zero vector. In other words, for ny vector, +( ) = 0, s you d expect. The negtive of the zero vector is the zero vector; tht is, 0 = 0. 44
5 Vectors Sutrction of vectors To see how vector sutrction is defined, first consider the sutrction of numers. Sutrcting numer is the sme s dding the negtive of the numer. In other words, if nd re numers, then mens the sme s +( ). We use this ide to define vector sutrction, s follows. Vector sutrction To sutrct from, dd to. Tht is, = +( ). Activity 25 Sutrcting vectors The digrm elow shows three vectors u, v nd w drwn on grid. u v w Drw rrows representing the following negtives nd differences of vectors. (Use squred pper or sketch grid.) () v () w (c) u v (d) v w (e) u+v w You hve lredy seen tht for ny vector, we hve +( ) = 0. Tht is, for ny vector, we hve = 0, s you would expect. 45
Unit 5 Coordinte geometry nd vectors Multipliction of vectors y sclrs You cn multiply vectors y sclrs. To understnd wht this mens, first consider the effect of dding vector to itself, s illustrted in Figure 36. Figure 36 A vector dded to itself The resultnt vector + hs the sme direction s, ut twice the mgnitude. We denote it y 2. We sy tht this vector is sclr multiple of the vector, since 2 is sclr quntity. In generl, sclr multipliction of vectors is defined s elow. Note tht in this ox the nottion m mens the mgnitude of the sclr m. Sclr multiple of vector Suppose tht is vector. Then, for ny non-zero rel numer m, the sclr multiple m of is the vector whose mgnitude is m times the mgnitude of tht hs the sme direction s if m is positive, nd the opposite direction if m is negtive. Also, 0 = 0. (Tht is, the numer zero times the vector is the zero vector.) Rememer tht sclr multiple of vector is vector. Vrious sclr multiples of vector re shown in Figure 37. 3 2 1 2 Figure 37 Sclr multiples of vector By the definition ove, if is ny vector, then ( 1) is the vector with the sme mgnitude s ut the opposite direction. In other words, s 46
5 Vectors you would expect, ( 1) =. Activity 26 Multiplying vectors y sclrs The digrm elow shows three vectors u, v nd w drwn on grid. u v w Drw rrows representing the following vectors. (Use squred pper or sketch grid.) () 3u () 2v (c) 1 2v (d) 3u 2v (e) 2v+w The next exmple illustrtes how you cn use sclr multiples of vectors to represent quntities in prcticl situtions. Exmple 10 Scling velocities Suppose tht the vector u represents the velocity of cr trvelling with speed 50kmh 1 long stright rod heding north. Write down, in terms of u, the velocity of second cr tht is trvelling in the sme direction s the first with speed of 75kmh 1. Solution The velocity of the second cr hs the sme direction s u, nd hence is sclr multiple of it. The speed of the second cr is 1.5 times the speed of the first. The velocity of the second cr is 1.5u. The ctivity elow involves winds mesured in knots. A knot is unit of speed often used in meteorology, nd in ir nd mritime nvigtion. Its usul revition is kn, nd 1kn = 1.852kmh 1. Conventionlly, the direction of wind is usully given s the direction from which it lows, rther thn the direction tht it lows towrds. So, for exmple, southerly wind is one lowing from the south, towrds the north. 47
Unit 5 Coordinte geometry nd vectors Activity 27 Scling velocities Suppose tht the vector v represents the velocity of wind of 35 knots lowing from the north-est. Express the following vectors in terms of v. () The velocity of wind of 70 knots lowing from the north-est. () The velocity of wind of 35 knots lowing from the south-west. 5.2 Vector lger In Susection 5.1, you met some properties of the ddition, sutrction nd sclr multipliction of vectors. For exmple, you sw tht for ny vectors nd, + = +, +0 =, = 0 nd + = 2. All the properties tht you met cn e deduced from the eight sic lgeric properties of vectors listed elow. Properties of vector lger The following properties hold for ll vectors, nd c, nd ll sclrs m nd n. 1. + = + 2. (+)+c = +(+c) 3. +0 = 4. +( ) = 0 5. m(+) = m+m 6. (m+n) = m+n 7. m(n) = (mn) 8. 1 = These properties re similr to properties tht hold for ddition, sutrction nd multipliction of rel numers. Similr properties lso hold for mny different systems of mthemticl ojects. You ll meet further exmples of such systems lter in the module. Property 1 sys tht the order in which two vectors re dded does not mtter. This property cn e descried y sying tht vector ddition is commuttive. Similrly, ddition of rel numers is commuttive, ecuse + = + for ll rel numers nd, nd multipliction of rel numers is commuttive, ecuse = for ll rel numers nd. Sutrction of rel numers is not commuttive, ecuse it is not true tht = for ll rel numers nd. 48
5 Vectors Property 2 sys tht finding + nd then dding c to the result gives the sme finl nswer s finding +c nd then dding to the result. You might like to check this for prticulr cse, y drwing the vectors s rrows. This property is descried y sying tht vector ddition is ssocitive. It llows us to write the expression ++c without there eing ny miguity in wht is ment you cn interpret it s either (+)+c or +(+c), ecuse oth men the sme. Addition nd multipliction of rel numers re lso ssocitive opertions. Property 5 sys tht dding two vectors nd then multiplying the result y sclr gives the sme finl nswer s multiplying ech of the two vectors individully y the sclr nd then dding the two resulting vectors. This property is descried y sying tht sclr multipliction is distriutive over the ddition of vectors. Similrly, property 6 sys tht sclr multipliction is distriutive over the ddition of sclrs. You will notice tht nothing hs een sid out whether vectors cn e multiplied or divided y other vectors. There is useful wy to define multipliction of two vectors two different wys, in fct! You will meet one of these wys in Section 7. Division of vector y nother vector is not possile. The properties in the ox ove llow you to perform some opertions on vector expressions in similr wy to rel numers, s illustrted in the following exmple. Exmple 11 Simplifying vector expression Simplify the vector expression 2(+)+3(+c) 5(+ c). Solution Expnd the rckets, using property 5 ove. 2(+)+3(+c) 5(+ c) = 2+2+3+3c 5 5+5c Collect like terms, using property 6 ove. = 2 5+2+3 5+3c+5c = 8c 3. The properties in the ox ove lso llow you to mnipulte equtions contining vectors, which re known s vector equtions, in similr wy to ordinry equtions. For exmple, you cn dd or sutrct vectors on oth sides of such n eqution, nd you cn multiply or divide oth 49
Unit 5 Coordinte geometry nd vectors sides y non-zero sclr. You cn use these methods to rerrnge vector eqution to mke prticulr vector the suject, or to solve vector eqution for n unknown vector. Activity 28 Mnipulting vector expressions nd equtions () Simplify the vector expression 4( c)+3(c )+2(2 3c). () Rerrnge ech of the following vector equtions to express x in terms of nd. (i) 2+4x = 7 (ii) 3( )+5x = 2( ) 5.3 Using vectors In this susection you ll see some exmples of how you cn use two-dimensionl vectors in prcticl situtions. When you use vector to represent rel-world quntity, you need mens of expressing its direction. For two-dimensionl vector, one wy to do this is to stte the ngle mesured from some chosen reference direction to the direction of the vector. You hve to mke it cler whether the ngle is mesured clockwise or nticlockwise. If the vector represents the displcement or velocity of n oject such s ship or n ircrft, then its direction is often given s compss ering. There re vrious different types of compss erings, ut in this module we will use the following type. A ering is n ngle etween 0 nd 360, mesured clockwise in degrees from north to the direction of interest. For exmple, Figure 38 shows vector v with ering of 150. N W 150 º v E A nvigtionl compss Figure 38 A vector with ering of 150 S 50
5 Vectors When erings re used in prctice, there re vrious possiilities for the mening of north. It cn men mgnetic north (the direction in which compss points), true north (the direction to the North Pole) or grid north (the direction mrked s north on prticulr mp). We ll ssume tht one of these hs een chosen in ny prticulr sitution. Notice tht the rottionl direction in which erings re mesured is opposite to tht in which ngles re usully mesured in mthemtics. Berings re mesured clockwise (from north), wheres in Unit 4 you sw ngles mesured nticlockwise (from the positive direction of the x-xis). Activity 29 Working with erings () Write down the erings tht specify the directions of the following vectors. (The cute ngle etween ech vector nd the gridlines is 45.) N c () Drw rrows to represent vectors (of ny mgnitude) with directions given y the following erings. (i) 90 (ii) 135 (iii) 270 When you work with the directions of vectors expressed using ngles, you often need to use trigonometry, s illustrted in the next exmple. 51
Unit 5 Coordinte geometry nd vectors Exmple 12 Adding two perpendiculr vectors An explorer wlks for 3km on ering of 90, then turns nd wlks for 4km on ering of 0. Find the mgnitude nd ering of his resultnt displcement, giving the ering to the nerest degree. Solution Represent the first prt of the wlk y the vector, nd the second prt y the vector. Then the resultnt displcement is +. Drw digrm showing, nd +. Since nd re perpendiculr, you otin right-ngled tringle. N + Use Pythgors theorem to find the mgnitude of +. Since = 3km, = 4km nd the tringle is right-ngled, + = 2 + 2 = 3 2 +4 2 = 25 = 5km. Use sic trigonometry to find one of the cute ngles in the tringle. From the digrm, tnθ = = 4 3, so θ = tn 1( 4 3) = 53 (to the nerest degree). Hence find the ering of +. The ering of + is 90 θ = 37 (to the nerest degree). Stte conclusion, rememering to include units. So the resultnt displcement hs mgnitude 5km nd ering of pproximtely 37. 52
5 Vectors Activity 30 Adding two perpendiculr vectors A ycht sils on ering of 60 for 5.3km, then turns through 90 nd sils on ering of 150 for further 2.1km. Find the mgnitude nd ering of the ycht s resultnt displcement. Give the mgnitude of the displcement in km to one deciml plce, nd the ering to the nerest degree. The vectors tht were dded in Exmple 12 nd Activity 30 were perpendiculr, so only sic trigonometry ws needed. In the next ctivity, you re sked to dd two displcement vectors tht ren t perpendiculr. You need to drw cler digrm nd use the sine nd cosine rules to find the required lengths nd ngles. Activity 31 Adding two non-perpendiculr vectors The gr of rootic rm moves 40cm from its strting point on ering of 90 to pick up n oject, nd then moves the oject 20cm on ering of 315. Find the resultnt displcement of the gr, giving the mgnitude to the nerest centimetre, nd the ering to the nerest degree. In some exmples involving vectors, it cn e quite complicted to work out the ngles tht you need to know from the informtion tht you hve. You often need to use the following geometric properties. Opposite, corresponding nd lternte ngles Where two lines intersect: opposite ngles re equl (for exmple, θ = φ). Where line intersects prllel lines: corresponding ngles re equl (for exmple, α = β); lternte ngles re equl (for exmple, α = γ). 53
Unit 5 Coordinte geometry nd vectors B A ABC C Figure 39 ABC The next exmple illustrtes how to use some of these geometric properties. You should find the tutoril clip for this exmple prticulrly helpful. The exmple uses the stndrd nottion ABC (red s ngle ABC ) for the cute ngle t the point B etween the line segments AB nd BC. This is illustrted in Figure 39. Exmple 13 Adding two non-perpendiculr vectors The displcement from Exeter to Belfst is 460km with ering of 340, nd the displcement from Belfst to Glsgow is 173km with ering of 36. Use this informtion to find the mgnitude (to the nerest kilometre) nd direction (s ering, to the nerest degree) of the displcement from Exeter to Glsgow. Solution Denote Exeter y E, Belfst y B, nd Glsgow y G. Drw digrm showing the displcement vectors EB, BG nd their resultnt EG. Mrk the ngles tht you know. Mrk or stte ny mgnitudes tht you know. B G N 36 º 340 º E We know tht EB = 460km nd BG = 173km. To enle you to clculte the mgnitude nd ering of EG, you need to find n ngle in tringle BEG. Use geometric properties to find EBG. Since the ering of EB is 340, the cute ngle t E etween EB nd north is 360 340 = 20, s shown in the digrm elow. Hence, since lternte ngles re equl, the cute ngle t B etween EB nd south is lso 20. So EBG = 180 36 20 = 124. 54
5 Vectors 36 º G N B 124 º 20 º 20 º µ 340 º E Now use the cosine rule to clculte EG. Applying the cosine rule in tringle EBG gives so EG 2 = EB 2 +BG 2 2 EB BG cos124, EG = 460 2 +173 2 2 460 173 cos124 = 574.91... = 575km (to the nerest km). To find the ering of EG, first find BEG. Let BEG = θ, s mrked in the digrm. Then, y the sine rule, BG sinθ = EG sin124 sinθ = BGsin124 EG = 173sin124 574.91.... Now ( ) 173sin124 sin 1 574.91... so = 14.44..., θ = 14.44... or θ = 180 14.44... = 165.55.... If θ = 165.55..., then the sum of θ nd EBG (two of the ngles in tringle EBG) is greter thn 180, which is impossile. So θ = 14.44.... Hence the ering of EG is 340 +θ = 340 +14.44... = 354.44.... Stte conclusion. The displcement of Glsgow from Exeter is 575km (to the nerest km) on ering of 354 (to the nerest degree). 55
Unit 5 Coordinte geometry nd vectors As mentioned in Susection 5.1, velocity is vector quntity, since it is the speed with which n oject is moving together with its direction of motion. So the methods tht you hve seen for dding displcements cn lso e pplied to velocities. It my t first seem strnge to dd velocities, ut consider the following sitution. Suppose tht oy is running cross the deck of ship. If the ship is motionless in hrour, then the oy s velocity reltive to the se ed is the sme s his velocity reltive to the ship. However, if the ship is moving, then the oy s velocity reltive to the se ed is comintion of his velocity reltive to the ship nd the ship s velocity reltive to the se ed. In fct, the oy s resultnt velocity reltive to the se ed is the vector sum of the two individul velocities. Activity 32 Adding velocities A ship is steming t speed of 10.0 ms 1 on ering of 30 in still wter. A oy runs cross the deck of the ship from the port side to the strord side, perpendiculr to the direction of motion of the ship, with speed of 4.0ms 1 reltive to the ship. (The port nd strord sides of ship re the sides on the left nd right, respectively, of person on ord fcing the front.) Find the resultnt velocity of the oy, giving the speed in ms 1 to one deciml plce nd the ering to the nerest degree. When ship sils in current, or n ircrft flies through wind, its ctul velocity is the resultnt of the velocity tht it would hve if the wter or ir were still, nd the velocity of the current or wind. In prticulr, the direction in which the ship or ircrft is pointing this is clled its heding, when it is given s ering my e different from the direction in which it is ctully moving, which is clled its course. This is ecuse the current or wind my cuse it to continuously drift to one side. 56
6 Component form of vector Activity 33 Finding the resultnt velocity of ship in current A ship hs speed in still wter of 5.7ms 1 nd is siling on heding of 230. However, there is current in the wter of speed 2.5ms 1 flowing on ering of 330. Find the resultnt velocity of the ship, giving the speed in ms 1 to one deciml plce nd the ering to the nerest degree. 57