# 1.3 SCALARS AND VECTORS

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1 Bridge Course Phy I PUC SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd the Sun is finite. The study of speed of light involves the distne trveled y the ry of light nd time onsumed. ny thing tht is mesurle is termed s quntity. The quntities tht ome ross in physis is referred to s physil quntity. Exmple: Mss, length, time, temperture, et., Whenever we mesure physil quntity, the mesured vlue is lwys numer. This numer mkes sense only when the relevnt unit is speified. Thus, the result of mesurement hs numeril vlue nd unit of mesure. For exmple, the mss of ody is 3 kg. Here quntity hving numeril vlue 3 nd the unit of mesure kg re used. The numeril vlue together with the unit is lled the mgnitude. To desrie ertin physil quntities only mgnitude is required. prt from the mss of ody, distne to ny ple, time, temperture, height, the numer of osilltions of pendulum nd the numer of ooks in g re some exmples of suh numers. They hve no diretion ssoited with them. Quntities whih require only mgnitude for their omplete speifitions nd hving no diretion ssoited with them re lled slr quntities. To desrie ertin physil quntities like displement long with the mgnitude, the diretion is essentil. Consider ody moving from X to Y. XY is the displement. On the ontrry, if the ody moves from Y to X, the displement is YX. Quntities whih require oth mgnitude nd diretion for their omplete speifition re lled vetors. Exmple for vetor quntities: momentum, fore, torque, mgneti field et., Note: The physil quntity like eletri urrent possesses oth the mgnitude nd diretion, still they re not vetors, nd similrly ny form of energy is slr. Representtion of vetor: vetor n e onveniently represented y stright line with n rrow hed. The length of the vetor represents its mgnitude nd the rrow hed indites its diretion. Steps involved representing vetor: 1. By hoosing proper sle, drw line whose length is proportionl to the mgnitude of the vetor. 2. By following the stndrd onvention to show diretion, indite the diretion of the vetor y mrking n rrow hed t one end of the line. Exmple: 1) To represent the displement of ody long x-xis. N Sle: 1 m = 5 km W E 0 35 km S Fig -1 Grphil representtion of vetor The vetor represented y the direted line segment O in fig (1) is denoted y O (to e red s vetor O) or simple nottion s ( to e red s vetor ). For vetoro, O is the initil point nd is the terminl point.

2 The mgnitude of O (or ) is denoted y O, (red s modulus of vetor O) or ( or simply O or ) nd is lwys positive. Position vetor: To lote the position of ny point P in Y P(x,y) plne or spe, generlly fixed point of referene lled the origin O is tken. The vetor OP is lled the position O X vetor of P with respet to O s shown in fig (2). Fig 2 Note: i) Given point P, there is one nd only one position vetor for the point with respet to the origin O. ii) Position vetor of point P hnges if the position of the origin O is hnged. Kinds of vetors: 1) Unit vetor: vetor hving unit mgnitude is lled unit vetor. If is vetor hving mgnitude 0, then is unit vetor hving the sme diretion s r. It is represented s (red s p). = 1. Thus = or = =. The unit vetors long the x, y nd z-xis re usully denoted s i, j nd k respetively. 2) Zero vetor or Null vetor: vetor hving zero mgnitude is lled Null vetor or Zero vetor. It is represented s O. Note: ) Zero vetor hs no speifi diretion. ) The position vetor of origin is zero vetor. ) Zero vetors re only of mthemtil importne. 3) Equl vetors: Vetors re sid to e equl if oth vetors hve sme mgnitude nd diretion. Fig (3) = 4) Prllel vetors (Like vetors): Vetors re sid to e prllel if they hve the sme diretions. Fig (4) The vetors nd represent prllel vetors. Note: Two equl vetors re lwys prllel ut, two prllel vetors my or my not e equl vetors. Bridge Course Phy I PUC 25

3 5) nti prllel vetors (Unlike vetors): Vetors re sid to e nti prllel if they ts in opposite diretion. Fig (5) The vetors nd re nti prllel vetors. 6) Negtive vetor : The negtive vetor of ny vetor is vetor hving equl mgnitude ut ts in opposite diretion. Fig (6) = - OR = - 7) Conurrent vetors (Co initil vetors ): vetors hving the sme initil point re lled onurrent vetors or o initil vetors. O, nd re onurrent t point O. Fig (7) 8) Coplnr vetors: The vetors in the sme plne re lled oplnr vetors. O Fig (8) The vetor nd re oplnr vetors Fig (8) The vetors nd re onurrent oplnr vetors. 9) Orthogonl vetors: Two vetors re sid to e orthogonl to one nother if the ngle etween them is 90. The vetor nd re orthogonl to one nother. 0 Fig (9) VECTOR LGEBR: ddition of vetors: The ddition of slrs involves only the ddition of their mgnitudes. But, when vetor is dded with nother vetor we hve to onsider their diretion lso. vetor n e dded with nother vetor provided oth the vetors represents the sme physil quntity. For exmple, the ddition of vetor representing displement of ody with nother vetor representing veloity of the ody is meningless. Bridge Course Phy I PUC 26

4 METHODS OF VECTOR DDITION: I.Tringle method of vetor ddition OR Til to tip method of vetor ddition: C Illustrtion: + = + = B Fig (10) Explntion: To dd with, trnslte, y drwing prllel to itself so tht the origin or initil point of is t the tip of vetor. nd re two vetors represented y two sides of tringle tken in the sme sense (diretion). The vetor sum of nd (lso lled resultnt of nd ) is represented y the third side of the tringle tken in opposite sense( diretion). Sttement: Tringle lw of vetor ddition sttes tht if two vetors n e represented in mgnitude nd diretion y two sides of tringle tken in the sme order, then their resultnt is represented ompletely y the third side of the tringle tken in opposite order. II. Prllelogrm method of vetor ddition: To dd two vetors pled with ommon initil point, the prllelogrm method of vetor is used. Illustrtion: B C = θ θ θ O O O α + = + = Fig (11) Fig (11) Fig (11) Explntion: To dd vetor with inlined t n ngleθ, drw equl vetor of t the tip of. By lw of tringle method of vetor ddition = + (fig-11). Repet the proess, y drwing equl vetor of t the tip of (fig-11). gin y lw of tringle method of vetor ddition = +. Note tht + = +, tht is vetor ddition follows ommuttive rule., the digonl of the ompleted prllelogrm represents the vetor sum of nd ompletely oth in mgnitude nd diretion. Sttement of prllelogrm lw of vetor ddition: It sttes tht if two vetors ting t point n e represented oth in mgnitude nd diretion y the two djent sides of prllelogrm drwn from tht point, the resultnt is represented ompletely y the digonl of the prllelogrm pssing through tht point. In fig(11), if nd re two vetors nd θ is the ngle etween them, then their vetor sum is represented y the digonl. Bridge Course Phy I PUC 27

5 It n e shown tht the mgnitude of If α is the ngle mde y the diretion of with tht of, then sinθ tnα = + osθ is, = osθ Note: It is ommon mistke to drw the sum vetor s the digonl running etween the tips of the two vetors s shown in fig (12). + = Fig (12) Think: In ft, the digonl represents the differene etween the two vetors, not their sum! Note: 1) The dvntge of the prllelogrm method is tht one n get oth the sum nd the differene of two vetors if one knows how to identify the pproprite diretions. 2) The resultnt of two vetors does not depend on the order in whih the vetors re dded. This ft leds to ) Commuttive lw of vetor ddition: + = + ) ssoitive lw of vetor ddition: If, nd re three vetor,then +( + )= ( + ) + III. Lw of polygon of vetor ddition: Sttement: It sttes tht if numer of vetors re represented oth in mgnitude nd diretion y the sides of polygon tken in the sme order, then their sum (resultnt) is represented oth in mgnitude nd diretion y the losing side of the polygon tken in the opposite order. C Let the vetors,, nd d represent the sides d O, B, BC nd CD of the polygon OBCDO. D B Their resultnt R is represented y the losing side OD of the polygon tken in the opposite order. R O Fig (13) Sutrtion of two vetors: Sutrtion of one vetor from nother vetor n e relized using the definition of negtive of vetor s follows. - = + Fig (14) Bridge Course Phy I PUC 28

6 Tringle method: - = Fig (15) Prllelogrm method: - = Fig (16) Note: 1) Sutrtion of one vetor with nother vetor is regrded s the ddition of one vetor with negtive of nother vetor. 2) The knowledge of sutrtion of vetors is useful in understnding the onept of reltive veloity. RESOLUTION OF VECTORS: Resolution of vetor mens the proess of splitting of vetor into omponents. If vetor is resolved into two omponents long the two mutully perpendiulr diretions, they re lled retngulr omponents. Consider vetor R represented y OC oth in mgnitude nd diretion s shown in fig (17). Drw OX nd OY whih re mutully perpendiulr to eh other from O. Y Let the vetor R mkes n ngle θ with X-xis. Drop B C Perpendiulrs from the tip of vetor C to X nd Y xes. O = P is the omponent of R long X-xis nd Q R is lled horizontl omponent of R. OB = Q is the θ omponent of R long Y-xis nd is lled O P X the vertil omponent of R. Fig (17) From fig (17), R = P + Q = i P + j Q Where i nd j re the unit vetors ting long X nd Y xes respetively. From fig(17), the mgnitude of O is O = osθ O = OC osθ P = Rosθ OC The mgnitude of OB is C = sinθ OC C = OC sinθ Q = R sinθ Thus, R = (R osθ) i + (Rsinθ) j C Q lso, = tnθ or tnθ =, where θ gives the resultnt diretion. O P From geometry of the figure, it n e shown tht, R 2 =P 2 + Q 2 ************* Bridge Course Phy I PUC 29

7 QUESTIONS: 1. Wht is slr? 2. Identify whether the following quntities re slrs or vetors? (i) Mss (ii) weight (iii)speed (iv)veloity (v)energy (vi)work (vii)fore (ix)temperture (x)pressure (xi)ngulr momentum (xii)wvelength. 3. Wht is vetor? 4. Find the mgnitude of 4î + 3 ĵ 5. Two vetors P nd Q t t n ngle 60 0 with eh other. If P = 20 units nd Q = 8 units find the mgnitude of resultnt R. 6. If = 6 î + 3 ĵ nd B = 3 î + 2 ĵ find + B nd B. 7. If = 3 î 2 ĵ + kˆ, find the unit vetor C 8. When will e P + Q = O 9. vetor of mgnitude 10 units mkes n ngle of 30 0 with the X-xis. Find its X nd Y omponents. 10. For wht ngle the mgnitude of X nd Y omponents of vetor eome equl? Bridge Course Phy I PUC 30

8 NSWERS : 1) physil quntity whih requires only mgnitude for their omplete desription re lled slrs. 2) SCLRS: mss, speed, energy, work, temperture nd pressure. VECTORS: Weight, veloity, fore, ngulr momentum nd wvelength. 3) Physil quntity whih requires oth the mgnitude nd diretion for their omplete desription re lled vetors. 4) Given vetor is 4î + 3 ĵ The mgnitude of the vetor, r = = 4 3 = 25 units. 5) Given: P = 20 units nd Q = 8 units nd θ =60 0 R =? Mgnitude of the resultnt vetor R is 2 2 R = P + Q + 2 P Q osθ 2 2 o = (20)(8) os 60 = (20)(8)( ) = units 6) Given: 6 î + 3 ĵ nd B = 3 î + 2 ĵ i) + B = (6 î + 3 ĵ ) + (3 î + 2 ĵ ) = 9 î + 5 ĵ ii ) B = (6 î + 3 ĵ ) - (3 î + 2 ĵ ) =3 î + ĵ 7) Given: = 3 î 2 ĵ + kˆ then Â =? We know tht, Â = 3ˆ i 2 ˆj + kˆ Â = ( 2) + 1 = 3i ˆ 2 ˆj + kˆ 14 8) P + Q = O if Q =- P i.e., the mgnitude of Q is equl to the mgnitude of P nd ts in opposite diretion. (Note: Two equl vetors ting in opposite diretion nel eh other) 9) Given: θ =30 0 with X-xis X-omponent=10os30 0 = 10(0.8660) =8.660 units Y-omponent =10sin30 0 = 10(0.5000) =5.000 units. 10) Let the vetor e R Its X-omponent is Ros nd Y- omponent is Rsin Given: Ros = Rsin os = sin this is possile only when θ =45 0 Bridge Course Phy I PUC 31

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