1 Appendi A Review of Vectos This appendi is a summa of the mathematical aspects of vectos used in electicit and magnetism. Fo a moe detailed intoduction to vectos, see Chapte 1. A.1 DESCRIBING THE 3D WORLD: VECTORS Phsical phenomena take place in the 3D wold aound us. In ode to be able to make quantitative pedictions and give detailed, quantitative eplanations, we need tools fo descibing pecisel the positions and velocities of objects in 3D, and the changes in position and velocit due to inteactions. These tools ae mathematical entities called 3D vectos. 3D Coodinates We will use a 3D coodinate sstem to specif positions in space and othe vecto quantities. Usuall we will oient the aes of the coodinate sstem as shown in Figue A.1: + ais to the ight, + ais upwad, and + ais coming out of the page, towad ou. This is a ight-handed coodinate sstem: if ou hold the thumb, fist, and second finges of ou ight hand pependicula to each othe, and align ou thumb with the ais and ou fist finge with the ais, ou second finge points along the ais. (In some math and phsics tetbook discussions of 3D coodinate sstems, the ais points out, the ais points to the ight, and the ais points up, but we will also use a 2D coodinate sstem with up, so it makes sense alwas to have the ais point up.) Basic Popeties of Vectos: Magnitude and Diection A vecto is a quantit that has a magnitude and a diection. Fo eample, the velocit of a baseball is a vecto quantit. The magnitude of the baseball s velocit is the speed of the baseball, fo eample 20 metes/second. The diection of the baseball s velocit is the diection of its motion at a paticula instant, fo eample up o to the ight o west o in the + diection. A smbol denoting a vecto is witten with an aow ove it: Position v is a vecto. A position in space can also be consideed to be a vecto, called a position vecto, pointing fom an oigin to that location. Figue A.2 shows a position vecto that might epesent ou final position if ou stated at the oigin and walked 4 metes along the ais, then 2 metes paallel to the ais, then climbed a ladde so Figue A.1 Right-handed 3D coodinate sstem. = 3 m = 2 m = 4 m Figue A.2 A position vecto = 4, 3, 2 m and its,, and components. 1
2 2 Review of Vectos ou wee 3 metes above the gound. You new position elative to the oigin is a vecto that can be witten like this: = 4, 3, 2 m component =4m component =3m component =2m In thee dimensions a vecto is a tiple of numbes,,. Quantities like the position of an object and the velocit of an object can be epesented as vectos: = 4 m = 3 m = 2 m Figue A.3 The aow epesents the vecto = 4, 3, 2 m, dawn with its tail at location 0, 0, 2. Figue A.4 The position vecto 3, 1, 0, dawn at the oigin, in the plane. The components of the vecto specif the displacement fom the tail to the tip. The ais, which is not shown, comes out of the page, towad ou. Components of a Vecto =,, (a position vecto) 1 = 3.2, 9.2, 66.3 m (a position vecto) v = v,v,v (a velocit vecto) v 1 = 22.3, 0.4, 19.5 m/s (a velocit vecto) Each of the numbes in the tiple is efeed to as a component of the vecto. The component of the vecto v is the numbe v. The component of the vecto v 1 = 22.3, 0.4, 19.5 m/s is 19.5 m/s. A component such as v is not a vecto, since it is onl one numbe. It is impotant to note that the component of a vecto specifies the diffeence between the coodinate of the tail of the vecto and the coodinate of the tip of the vecto. It does not give an infomation about the location of the tail of the vecto (compae Figue A.2 and Figue A.3). Dawing Vectos In Figue A.2 we epesented ou position vecto elative to the oigin gaphicall b an aow whose tail is at the oigin and whose aowhead is at ou position. The length of the aow epesents the distance fom the oigin, and the diection of the aow epesents the diection of the vecto, which is the diection of a diect path fom the initial position to the final position (the displacement ; b walking and climbing ou displaced ouself fom the oigin to ou final position). Since it is difficult to daw a 3D diagam on pape, when woking on pape ou will usuall be asked to daw vectos which all lie in a single plane. Figue A.4 shows an aow in the plane epesenting the vecto 3, 1, 0. Vectos and Scalas A quantit which is epesented b a single numbe is called a scala. A scala quantit does not have a diection. Eamples include the mass of an object, such as 5 kg, o the tempeatue, such as 20C. Vectos and scalas ae ve diffeent entities; a vecto can neve be equal to a scala, and a scala cannot be added to a vecto. Scalas can be positive o negative: m =50kg T = 20 C Although a component of a vecto such as v is not a vecto, it s not a scala eithe, despite being onl one numbe. An impotant popet of a tue scala is that its value doesn t change if we oient the coodinate aes diffeentl. Rotating the aes doesn t change an object s mass, o the tempeatue, but it does change what we mean b the component of the velocit since the ais now points in a diffeent diection.
3 A.1. DESCRIBING THE 3D WORLD: VECTORS 3 Magnitude of a Vecto In Figue A.5 we again show the vecto fom Figue A.2, showing ou displacement fom the oigin. Using a 3D etension of the Pthagoean theoem fo ight tiangles (Figue A.6), the net distance ou have moved fom the stating point is (4 m)2 +(3m) 2 +(2m) 2 = 29 m = 5.39 m = 3 m = 4 m We sa that the magnitude of the position vecto is = 2 m =5.39 m The magnitude of a vecto is witten eithe with absolute-value bas aound the vecto as, o simpl b witing the smbol fo the vecto without the little aow above it,. The magnitude of a vecto can be calculated b taking the squae oot of the sum of the squaes of its components (see Figue A.6). Figue A.5 A vecto epesenting a displacement fom the oigin. MAGNITUDE OF A VECTOR If the vecto =,, then = (a scala). ( ) + 2 The magnitude of a vecto is alwas a positive numbe. The magnitude of a vecto is a single numbe, not a tiple of numbes, and it is a scala, not a vecto. The magnitude of a vecto is a tue scala, because its value doesn t change if ou otate the coodinate aes. Rotating the aes changes the individual components, but the length of the aow epesenting the vecto doesn t change. Can a Vecto be Positive o Negative? QUESTION Conside the vecto v = , 0, m/s. Is this vecto positive? Negative? Zeo? ( ) Figue A.6 The magnitude of a vecto is the squae oot of the sum of the squaes of its components (3D vesion of the Pthagoean theoem). None of these desciptions is appopiate. The component of this vecto is positive, the component is eo, and the component is negative. Vectos aen t positive, o negative, o eo. Thei components can be positive o negative o eo, but these wods just don t mean anthing when used with the vecto as a whole. On the othe hand, the magnitude of a vecto such as v is alwas positive. Mathematical Opeations Involving Vectos Although the algeba of vectos is simila to the scala algeba with which ou ae ve familia, it is not identical. Thee ae some algebaic opeations that cannot be pefomed on vectos. Algebaic opeations that ae legal fo vectos include the following opeations, which we will discuss in this chapte: adding one vecto to anothe vecto: a + w subtacting one vecto fom anothe vecto: b d finding the magnitude of a vecto: finding a unit vecto (a vecto of magnitude 1): ˆ multipling (o dividing) a vecto b a scala: 3 v o w/2 finding the ate of change of a vecto: Δ /Δt o d /dt. In late chaptes we will also see that thee ae two moe was of combining two vectos:
4 4 Review of Vectos the vecto dot poduct, whose esult is a scala the vecto coss poduct, whose esult is a vecto p 3p 2p p 1 2 p Opeations that ae Not Legal fo Vectos Although vecto algeba is simila to the odina scala algeba ou have used up to now, thee ae cetain opeations that ae not legal (and not meaningful) fo vectos: A vecto cannot be set equal to a scala. A vecto cannot be added to o subtacted fom a scala. A vecto cannot occu in the denominato of an epession. (Although ou can t divide b a vecto, note that ou can legall divide b the magnitude of a vecto, which is a scala.) Multipling a Vecto b a Scala A vecto can be multiplied (o divided) b a scala. If a vecto is multiplied b a scala, each of the components of the vecto is multiplied b the scala: If =,, then a = a, a, a 3p 2p Figue A.7 Multipling a vecto b a scala changes the magnitude of the vecto. Multipling b a negative scala eveses the diection of the vecto. If v = v,v,v then v b = v b, v b, v b ( 1 ) 6, 20, 9 = 3, 10, Multiplication b a scala scales a vecto, keeping its diection the same but making its magnitude lage o smalle (Figue A.7). Multipling b a negative scala eveses the diection of a vecto. Magnitude of a Scala You ma wonde how to find the magnitude of a quantit like 3, which involves the poduct of a scala and a vecto. This epession can be factoed: 3 = 3 The magnitude of a scala is its absolute value, so: 3 = 3 = Diection of a Vecto: Unit Vectos One wa to descibe the diection of a vecto is b specifing a unit vecto. A unit vecto is a vecto of magnitude 1, pointing in some diection. A unit vecto is witten with a hat (caet) ove it instead of an aow. The unit vecto â is called a-hat. QUESTION Is the vecto 1, 1, 1 a unit vecto? The magnitude of 1, 1, 1 is =1.73, so this is not a unit vecto. The vecto 1/ 3, 1/ 3, 1/ 3 is a unit vecto, since its magnitude is 1: ( 1 ) 2 +( 1 ) 2 +( 1 ) 2 = Note that eve component of a unit vecto must be less than o equal to 1.
5 A.1. DESCRIBING THE 3D WORLD: VECTORS 5 In ou 3D Catesian coodinate sstem, thee ae thee special unit vectos, oiented along the thee aes. The ae called i-hat, j-hat, and k-hat, and the point along the,, and aes, espectivel (Figue A.8): î= 1, 0, 0 ĵ= 0, 1, 0 ˆk = 0, 0, 1 One wa to epess a vecto is in tems of these special unit vectos: 0.02, 1.7, 30.0 =0.02î+( 1.7)ĵ+30.0ˆk We will usuall use the,, fom athe than the îĵˆk fom in this book, because the familia,, notation, used in man calculus tetbooks, emphasies that a vecto is a single entit. Not all unit vectos point along an ais, as shown in Figue A.9. Fo eample, the vectos ĝ = , , and ˆF = 0.424, 0.566, ae both unit vectos, since the magnitude of each is equal to 1. Note that eve component of a unit vecto is less than o equal to 1. k Figue A.8 The unit vectos î, ĵ, ˆk. Calculating Unit Vectos An vecto ma be factoed into the poduct of a unit vecto in the diection of the vecto, multiplied b a scala equal to the magnitude of the vecto. v = 1.5, 1.5, 0Ò m/s w = w ŵ Fo eample, a vecto of magnitude 5, aligned with the ais, could be witten as: 0, 5, 0 =5 0, 1, 0 v = 2, 2, 0Ò 2 2 Figue A.9 The unit vecto ˆv has the same diection as the vecto v, but its magnitude is 1, and it has no phsical units. Theefoe, to find a unit vecto in the diection of a paticula vecto, we just divide the vecto b its magnitude: CALCULATING A UNIT VECTOR ˆ = ( ), ˆ = =,, ( ) ( ), ( ) EXAMPLE Unit Vecto If v = 22.3, 0.4, 19.5 m/s, then ˆv = v v = 22.3, 0.4, 19.5 m/s = 0.753, , ( 22.3)2 +(0.4) 2 +( 19.5) 2 m/s Remembe that to divide a vecto b a scala, ou divide each component of the vecto b the scala. The esult is a new vecto. Note also that a unit vecto has no phsical units (such as metes pe second), because the units in the numeato and denominato cancel.
6 6 Review of Vectos Equalit of Vectos EQUALITY OF VECTORS A vecto is equal to anothe vecto if and onl if all the components of the vectos ae equal. w = means that w = and w = and w = The magnitudes and diections of two equal vectos ae the same: w = and ŵ =ˆ EXAMPLE Equal Vectos = 4, 3, 2 = ( )=5.39 ˆ = 4, 3, = 0.742, 0.557, If w = w = 4, 3, 2 w =5.39 ŵ = 0.742, 0.557, B Vecto Addition A ADDING VECTORS The sum of two vectos is anothe vecto, obtained b adding the components of the vectos. B A = A,A,A B = B,B,B A A + B = (A + B ), (A + B ), (A + B ) EXAMPLE Adding Vectos A + B B 1, 2, 3 + 4, 5, 6 = 3, 7, 9 A Figue A.10 The pocedue fo adding two vectos gaphicall: daw vectos tip to tail. To add A + B gaphicall, move B so the tail of B is at the tip of A then daw a new aow stating at the tail of A and ending at the tip of B. Waning: Don t Add Magnitudes! The magnitude of a vecto is not in geneal equal to the sum of the magnitudes of the two oiginal vectos! Fo eample, the magnitude of the vecto 3, 0, 0 is 3, and the magnitude of the vecto 2, 0, 0 is 2, but the magnitude of the vecto ( 3, 0, 0 + 2, 0, 0 ) is 1, not 5! Adding Vectos Gaphicall: Tip to Tail The sum of two vectos has a geometic intepetation. In Figue A.10 ou fist walk along displacement vecto A, followed b walking along displacement vecto B. What is ou net displacement vecto C = A + B? The component C
7 A.1. DESCRIBING THE 3D WORLD: VECTORS 7 of ou net displacement is the sum of A and B. Similal, the component C of ou net displacement is the sum of A and B. GRAPHICAL ADDITION OF VECTORS To add two vectos A and B gaphicall (Figue A.10): Daw the fist vecto A Move the second vecto B (without otating it) so its tail is located at the tip of the fist vecto Daw a new vecto fom the tail of vecto A to the tip of vecto B Vecto Subtaction The diffeence of two vectos will be ve impotant in this and subsequent chaptes. To subtact one vecto fom anothe, we subtact the components of the second fom the components of the fist: A B = (A B ), (A B ), (A B ) 1, 2, 3 4, 5, 6 = 5, 3, 3 Subtacting Vectos gaphicall: Tail to Tail To subtact one vecto B fom anothe vecto A gaphicall: Daw the fist vecto A Move the second vecto B (without otating it) so its tail is located at the tail of the fist vecto Daw a new vecto fom the tip of vecto B to the tip of vecto A Note that ou can check this algebaicall and gaphicall. As shown in Figue A.11, since the tail of A B is located at the tip of B, then the vecto A should be the sum of B and A B, as indeed it is: B +( A B)= A B A A B Figue A.11 The pocedue fo subtacting vectos gaphicall: daw vectos tail to tail; daw new vecto fom tip of second vecto to tip of fist vecto. Commutativit and Associativit Vecto addition is commutative: A + B = B + A Vecto subtaction is not commutative: A B B A The associative popet holds fo vecto addition and subtaction: The Zeo Vecto ( A + B) C = A +( B C) It is convenient to have a compact notation fo a vecto whose components ae all eo. We will use the smbol 0 to denote a eo vecto, in ode to distinguish it fom a scala quantit that has the value 0. 0 = 0, 0, 0 Fo eample, the sum of two vectos B +( B)= 0.
8 8 Review of Vectos 6 m Change in a Quantit: The Geek Lette Δ Fequentl we will want to calculate the change in a quantit. Fo eample, we ma want to know the change in a moving object s position o the change in its velocit duing some time inteval. The Geek lette Δ (capital delta suggesting d fo diffeence ) is used to denote the change in a quantit (eithe a scala o a vecto). We use the subscipt i to denote an initial value of a quantit, and the subscipt f to denote the final value of a quantit. If a vecto i denotes the initial position of an object elative to the oigin (its position at the beginning of a time inteval), and f denotes the final position of the object, then m Figue A.12 Relative position vecto. A = 1 A θ A Sting Figue A.13 A unit vecto whose diection is at a known angle fom the + ais. θ θ Figue A.14 A 3D unit vecto and its angles to the,, and aes. θ Δ = f i Δ means change of o f i (displacement) Δt means change of t ot f t i (time inteval) The smbol Δ (delta) alwas means final minus initial, not initial minus final. Fo eample, when a child s height changes fom 1.1 mto1.2m, the change is Δ =+0.1m, a positive numbe. If ou bank account dopped fom $150 to $130, what was the change in ou balance? Δ(bank account)= 20 dollas. Relative Position Vectos Vecto subtaction is used to calculate elative position vectos, vectos which epesent the position of an object elative to anothe object. In Figue A.12 object 1 is at location 1 and object 2 is at location 2. We want the components of a vecto that points fom object 1 to object 2. This is the vecto obtained b subtaction: 2 elative to 1 = 2 1. Note that the fom is alwas final minus initial in these calculations. Unit Vectos and Angles Suppose a taut sting is at an angle θ to the + ais, and we need a unit vecto in the diection of the sting. Figue A.13 shows a unit vecto Â pointing along the sting. What is the component of this unit vecto? Conside the tiangle whose base is A and whose hpotenuse is Â =1. Fom the definition of the cosine of an angle we have this: cos θ = adjacent hpotenuse = A 1, so A =cosθ In Figue A.13 the angle θ is shown in the fist quadant (θ less than 90 ), but this woks fo lage angles as well. Fo eample, in Figue?? the angle fom the + ais to a unit vecto ˆB is in the second quadant (θ geate than 90 ) and cos θ is negative, which coesponds to B being negative. What is tue fo is also tue fo and. Figue A.14 shows a 3D unit vecto ˆ and indicates the angles between the unit vecto and the,, and aes. Evidentl we can wite Vecto in plane ˆ = cos θ, cos θ, cos θ θ = 90º θ θ Figue A.15 If a vecto lies in the plane, cos θ =sinθ. These thee cosines of the angles between a vecto (o unit vecto) and the coodinate aes ae called the diection cosines of the vecto. The cosine function is neve geate than 1, just as no component of a unit vecto can be geate than 1. A common special case is that of a unit vecto ling in the plane, with eo component (Figue A.15). In this case θ + θ =90, so that cos θ = cos(90 θ )=sinθ, so that ou can epess the cosine of θ as the sine of θ,
9 A.1. DESCRIBING THE 3D WORLD: VECTORS 9 which is often convenient. Howeve, in the geneal 3D case shown in Figue A.14 thee is no such simple elationship among the diection angles, no among thei cosines. FINDING A UNIT VECTOR FROM ANGLES To find a unit vecto if angles ae given: Redaw the vecto of inteest with its tail at the oigin, and detemine the angles between this vecto and the aes. Imagine the vecto 1, 0, 0, which lies on the + ais. θ is the angle though which ou would otate the vecto 1, 0, 0 until its diection matched that of ou vecto. θ is positive, and θ 180. θ is the angle though which ou would otate the vecto 0, 1, 0 until its diection matched that of ou vecto. θ is positive, and θ 180. θ is the angle though which ou would otate the vecto 0, 0, 1 until its diection matched that of ou vecto. θ is positive, and θ 180. EXAMPLE Fom Angle to Unit Vecto A ope ling in the plane, pointing up and to the ight, suppots a climbe at an angle of 20 to the vetical (Figue A.16). What is the unit vecto pointing up along the ope? 20 Figue A.16 A climbe suppoted b a ope. Solution Follow the pocedue given above fo finding a unit vecto fom angles. In Figue A.17 we edaw the vecto with its tail at the oigin, and we detemine the angles between the vecto and the aes. If we otate the unit vecto 1, 0, 0 fom along the + ais to the vecto of inteest, we see that we have to otate though an angle θ =70. To otate the unit vecto 0, 1, 0 fom along the + ais to the vecto of inteest, we have to otate though an angle of θ =20. The angle fom the + ais to ou vecto is θ =90. Theefoe the unit vecto that points along the ope is this: θ = 90º θ = 20º θ = 70º cos 70, cos 20, cos 90 = 0.342, 0.940, 0 Figue A.17 Redaw the vecto with its tail at the oigin. Identif the angles between the positive aes and the vecto. In this eample the vecto lies in the plane. FURTHER DISCUSSION You ma have noticed that the component of the unit vecto can also be calculated as sin 70 =0.940, and it is often useful to ecognie that a vecto component can be obtained using sine instead of cosine. Thee is howeve some advantage alwas to calculate in tems of diection cosines. This is a method that alwas woks, including in 3D, and which avoids having to decide whethe to use a sine o a cosine. Just use the cosine of the angle fom the elevant positive ais to the vecto. EXAMPLE Fom Unit Vecto to Angles A vecto points fom the oigin to the location 600, 0, 300 m. What is the angle that this vecto makes to the ais? To the ais? To the ais?
10 10 Review of Vectos Solution 600, 0, 300 ˆ = = 0.894, 0, ( 600)2 +(0) 2 + (300) 2 m But we also know that ˆ = cos θ, cos θ, cos θ, so cos θ = 0.894, and the accosine gives θ = Similal, cos θ =0, θ =90 (which checks; no component) cos θ =0.447,θ =63.4 θ = 63.4º θ = 153.4º Figue A.18 Look down on the plane. The diffeence in the two angles is 90, as it should be. FURTHER DISCUSSION Looking down on the plane in Figue A.18, ou can see that the diffeence between θ = and θ =63.4 is 90, as it should be. A.2 VECTOR MULTIPLICATION Vectos can be added and subtacted, and the can be multiplied b a scala. Two vectos can also be multiplied, but two diffeent kinds of vecto multiplication ae defined: the dot poduct and the coss poduct. In the pevious volume the dot poduct was intoduced in the contet of wok, and the coss poduct was intoduced in the contet of angula momentum. The Dot Poduct The dot poduct is an opeation involving two vectos. This is encounteed in the epession fo wok in Chapte 6: W = F Δ =(F Δ + F Δ + F Δ) If F = 3, 2, 4 N and Δ = 2, 0, 5 m, then F Δ = ((3 2) + ( 2 0) + (4 5)) N m = 14 N m The esult of a dot poduct opeation is a scala (like the quantit wok). Note that the dot poduct of a vecto with itself gives the squae of the magnitude of the vecto:,,,, =( 2,2,2 )= 2 The magnitude of the dot poduct can also be calculated as: F Δ = F Δ cos θ = F Δ = F Δ whee θ is the angle between the two vectos, placed tail to tail. In the VPthon pogamming language, dot(vecto1,vecto2) gives the dot poduct of two vectos. The Coss Poduct k The coss poduct is discussed in detail in Chapte 18 in the contet of the Biot- Savat law fo finding the magnetic field of moving chages. In the VPthon pogamming language, coss(vecto1,vecto2) gives the coss poduct of two vectos. It is possible to evaluate the coss poduct in tems of unit vectos along the thee aes (Figue A.19). Fist, note that î î=0, ĵ ĵ=0, and ˆk ˆk =0, since when we coss a vecto with itself the angle between the two vectos is eo, and sin 0 =0. Second, î ĵ = ˆk, since the angle is 90 and the ight-hand ule gives a esult in the + diection (out of the page; Figue A.19). On the othe hand, ĵ î= ˆk, because the ight-hand ule gives a esult in the diection (into the Figue A.19 Coss poducts of unit vectos.
11 A.3. SUMMARY 11 page). Similal, ĵ ˆk =î, ˆk ĵ= î, ˆk î=ĵ, and î ˆk = ĵ. Putting this all togethe, we obtain the following geneal esult: A B =(A B A B )î+(a B A B )ĵ+(a B A B )ˆk o A B = (A B A B ), (A B A B ), (A B A B ) This appoach to calculating a coss poduct is paticulal useful in compute calculations. Note the cclic natue of the subscipts:,,. Common Eos in Vecto Multiplication (1) A dot poduct of two vectos esults in a scala, not anothe vecto. (2) A coss poduct of two vectos esults in anothe vecto, not a scala. Technicall, although a component of a vecto is a single numbe, it is not a scala. If ou otate ou coodinate aes, the,, and components of a vecto change, but a tue scala such as m =5kg doesn t change. A.3 SUMMARY Vectos A3Dvecto is a quantit with magnitude and a diection, which can be epessed as a tiple,,. A vecto is indicated b an aow:. A scala is a single numbe. Legal mathematical opeations involving vectos include: adding one vecto to anothe vecto subtacting one vecto fom anothe vecto multipling o dividing a vecto b a scala finding the magnitude of a vecto taking the deivative of a vecto Opeations that ae not legal with vectos include: A vecto cannot be added to a scala A vecto cannot be set equal to a scala A vecto cannot appea in the denominato (ou can t divide b a vecto) The smbol Δ denotes subtaction The smbol Δ (delta) means change of : Δt=t f t i, Δ = f i. Δ alwas means final minus initial.