Scalars and Vectors Scalar
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2 Scalas and ectos Scala A phscal quantt that s completel chaacteed b a eal numbe (o b ts numecal value) s called a scala. In othe wods a scala possesses onl a magntude. Mass denst volume tempeatue tme eneg aea speed and length ae eamples to scala quanttes.
3 ecto Seveal quanttes that occu n mechancs eque a descpton n tems of the decton as well as the numecal value of the magntude. Such quanttes behave as vectos. Theefoe vectos possess both magntude and decton; and the obe the paallelogam law of addton. Foce moment dsplacement veloct acceleaton mpulse and momentum ae vecto quanttes.
4 Tpes of ectos: Phscal quanttes that ae vectos fall nto one of the thee classfcatons as fee sldng o fed. A fee vecto s one whose acton s not confned to o assocated wth a unque lne n space. Fo eample f a bod s n tanslatonal moton veloct of an pont n the bod ma be taen as a vecto and ths vecto wll descbe equall well the veloct of eve pont n the bod. Hence we ma epesent the veloct of such a bod b a fee vecto. In statcs couple moment s a fee vecto.
5 A sldng vecto s one fo whch a unque lne n space must be mantaned along whch the quantt acts. hen we deal wth the etenal acton of a foce on a gd bod the foce ma be appled at an pont along ts lne of acton wthout changng ts effect on the bod as a whole and hence consdeed as a sldng vecto.
6 A fed vecto s one fo whch a unque pont of applcaton s specfed and theefoe the vecto occupes a patcula poston n space. The acton of a foce on a defomable bod must be specfed b a fed vecto.
7 Pncple of Tansmssblt: The etenal effect of a foce on a gd bod wll eman unchanged f the foced s moved to act on ts lne of acton. In othe wods a foce ma be appled at an pont on ts gven lne of acton wthout alteng the esultant etenal effects on the gd bod on whch t acts.
8 Equalt and Equvalence of ectos Two vectos ae equal f the have the same dmensons magntudes and dectons. Two vectos ae equvalent n a cetan capact f each poduces the ve same effect n ths capact. To sum up the equalt of two vectos s detemned b the vectos themselves but the equvalence between two vectos s detemned b the stuaton at hand.
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10 PROPERTIES OF ECTORS Addton of ectos s done accodng to the paallelogam law of vecto addton.
11 Subtacton of ectos s done accodng to the paallelogam law. Multplcaton of a Scala and a ecto
12 nt ecto A unt vecto s a fee vecto havng a magntude of 1 (one) as n It descbes decton. The most convenent wa to descbe a vecto n a cetan decton s to multpl ts magntude wth ts unt vecto. n 1 n and have the same unt hence the unt vecto s dmensonless. Theefoe ma be epessed n tems of both ts magntude and decton sepaatel. (a scala) epesses the magntude and dectonal sense of. n (a dmensonless vecto) epesses the
13 ecto Components and Resultant ecto Let the sum of and be. Hee and ae named as the components and s named as the esultant.
14 Catesan Coodnates Catesan coodnate sstem s composed of 90 (othogonal) aes. It conssts of and aes n two dmensonal (plane) case and aes n thee dmensonal (spatal) case. - aes ae geneall taen wthn the plane of the pape the postve dectons can be selected abtal; the postve decton of the aes must be detemned n accodance wth the ght hand ule.
15
16 ecto Components n Two Dmensonal (Plane) Catesan Coodnates
17 ecto Components n Thee Dmensonal (Spatal) Catesan Coodnates 2 2 2
18 Poston ecto It s the vecto that descbes the locaton of one pont wth espect to anothe pont. A B A B A B A B 2 2 In two dmensonal case
19 In thee dmensonal case A B A B A B A B A B A B 2 2 2
20 Dot (Scala) Poduct A scala quantt s obtaned fom the dot poduct of two vectos. a a cos cos s elevant multplcaton of ode Coodnate s Catesan n of unt vectos In tems cos cos 0
21 Nomal and Paallel Components of a ecto wth espect to a Lne
22 Coss (ecto) Poduct The multplcaton of two vectos n coss poduct esults n a vecto. Ths multplcaton vecto s nomal to the plane contanng the othe two vectos. Its decton s detemned b the ght hand ule. Its magntude equal to the aea of the paallelogam that the vectos span. The ode of multplcaton s mpotant. Y Y a a a sn sn
23 1 sn sn 0 Coodnates Catesan n unt vectos of In tems
24 - - -
25 Med Tple Poduct It s used when tang the moment of a foce about a lne. o
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