Chapter 4 PDF - Vector Algebra

vè;k; 10 lfn'k chtxf.kr (Vector Algebra) v In most sciences one generation tears down what another has built and what one has established another undoes. In Mathematics alone each generation builds a new story to the old structure. – HERMAN HANKEL v...

vè;k; 10 lfn'k chtxf.kr (Vector Algebra) v In most sciences one generation tears down what another has built and what one has established another undoes. In Mathematics alone each generation builds a new story to the old structure. – HERMAN HANKEL v 10.1 Hkwfedk (Introduction) vius nSfud thou esa gesa vusd iz'u feyrs gSa tSls fd vkidh Å¡pkbZ D;k gS\ ,d iqQVcky osQ f[kykM+h dks viuh gh Vhe osQ nwljs f[kykM+h osQ ikl xsan igq¡pkus osQ fy, xsan ij fdl izdkj izgkj djuk pkfg,\ voyksdu dhft, fd izFke iz'u dk laHkkfor mÙkj 1-6 ehVj gks ldrk gSA ;g ,d ,slh jkf'k gS ftlesa osQoy ,d eku ifjek.k tks ,d okLrfod la[;k gS] lfEefyr gSA ,slh jkf'k;k¡ vfn'k dgykrh gSA rFkkfi nwljs iz'u dk mÙkj ,d ,slh jkf'k gS (ftls cy dgrs gSa) ftlesa ekalisf'k;ksa dh 'kfDr ifjek.k osQ lkFk&lkFk fn'kk (ftlesa nwljk f[kykM+h fLFkr gS) Hkh lfEefyr gSA ,slh jkf'k;ka¡ lfn'k dgykrh gSA xf.kr] HkkSfrdh ,oa vfHk;kaf=kdh esa ;s nksuksa izdkj dh jkf'k;k¡ uker% vfn'k jkf'k;k¡] tSls fd yackbZ] nzO;eku] W.R. Hamilton (1805-1865) le;] nwjh] xfr] {ks=kiQy] vk;ru] rkieku] dk;Z] /u] oksYVrk] ?kuRo] izfrjks/d bR;kfn ,oa lfn'k jkf'k;k¡ tSls fd foLFkkiu] osx] Roj.k] cy] Hkkj] laoxs ] fo|qr {ks=k dh rhozrk bR;kfn cgq/k feyrh gaSA bl vè;k; esa ge lfn'kksa dh oqQN vk/kjHkwr ladYiuk,¡] lfn'kksa dh fofHkUu lafØ;k,¡ vkSj buosQ chth; ,oa T;kferh; xq.k/eks± dk vè;;u djsaxsA bu nksuksa izdkj osQ xq.k/eks± dk lfEefyr :i lfn'kksa dh ladYiuk dk iw.kZ vuqHkwfr nsrk gS vkSj mi;qZDr p£pr {ks=kksa esa budh fo'kky mi;ksfxrk dh vksj izsfjr djrk gSA 10.2 oqQN vk/kjHkwr ladYiuk,¡ (Some Basic Concepts) eku yhft, fd fdlh ry vFkok f=k&foeh; varfj{k esa l dksbZ ljy js[kk gSA rhj osQ fu'kkuksa dh lgk;rk ls bl js[kk dks nks fn'kk,¡ iznku dh tk ldrh gSaA bu nksuksa esa ls fuf'pr fn'kk okyh dksbZ Rationalised 2023-24 350 xf.kr Hkh ,d js[kk fn"V js[kk dgykrh gS [vko`Qfr 10.1 (i), (ii)]A vko`Qfr 10-1 vc izsf{kr dhft, fd ;fn ge js[kk ‘l’ dks js[kk[kaM AB rd izfrcaf/r dj nsrs gSa rc nksuksa es ls fdlh ,d fn'kk okyh js[kk ‘l’ ij ifjek.k fu/kZfjr gks tkrk gSA bl izdkj gesa ,d fn"V js[kk[kaM izkIr gksrk gS (vko`Qfr 10.1(iii))A vr% ,d fn"V js[kk[kaM esa ifjek.k ,oa fn'kk nksuksa gksrs gSaA ifjHkk"kk 1 ,d ,slh jkf'k ftlesa ifjek.k ,oa fn'kk nksuksa gksrs gSa] lfn'k dgykrh gSA


 è;ku nhft, fd ,d fn"V js[kk[kaM lfn'k gksrk gS (vko`Qfr 10.1(iii)), ftls AB vFkok


 lk/kj.kr% a , osQ :i esa fufnZ"V djrs gSa vkSj bls lfn'k ^ AB * vFkok lfn'k ^ a * osQ :i esa iqdk gqvk gSA 15. ¯cnqvksa P ( iˆ + 2 ˆj − kˆ ) vkSj Q (– iˆ + ˆj + kˆ) dks feykus okyh js[kk dks 2%1 osQ vuqikr esas (i) var% (ii) cká] foHkkftr djus okys ¯cnq R dk fLFkfr lfn'k Kkr dhft,A 16. nks ¯cnqvksa P(2, 3, 4) vkSj Q(4, 1, –2) dks feykus okys lfn'k dk eè; ¯cnq Kkr dhft,A   17. n'kkZb, fd ¯cnq A, B vkSj C, ftuosQ fLFkfr lfn'k Øe'k% a = 3iˆ − 4 ˆj − 4kˆ, b = 2iˆ − ˆj + kˆ  vkSj c = iˆ − 3 ˆj − 5kˆ gS]a ,d ledks.k f=kHkqt osQ 'kh"kks± dk fuekZ.k djrs gSAa 18. f=kHkqt ABC (vko`Qfr 10.18), osQ fy, fuEufyf[kr esa ls dkSu lk dFku lR; ugha gSA









  (A) AB + BC + CA = 0








  (B) AB + BC − AC = 0








  (C) AB + BC − CA = 0








  (D) AB − CB + CA = 0 vko`Qfr 10-18   19. ;fn a vkjS b nks lajs[k lfn'k gSa rks fuEufyf[kr esa ls dkSu lk dFku lgh ugha gS%   (A) b = λa, fdlh vfn'k λ osQ fy,   (B) a = ± b   (C) a vkjS b osQ Øekxr ?kVd lekuqikrh ugha gSaA   (D) nksuksa lfn'kksa a rFkk b dh fn'kk leku gS ijarq ifjek.k fofHkUu gSaA 10.6 nks lfn'kksa dk xq.kuiQy (Product of Two Vectors) vHkh rd geus lfn'kksa osQ ;ksxiQy ,oa O;odyu osQ ckjs esa vè;;u fd;k gSA vc gekjk mn~ns'; lfn'kksa dk xq.kuiQy uked ,d nwljh chth; lafØ;k dh ppkZ djuk gSA ge Lej.k dj ldrs gSa fd nks la[;kvksa dk xq.kuiQy ,d la[;k gksrh gS] nks vkO;wgksa dk xq.kuiQy ,d vkO;wg gksrk gS ijarq iQyuksa dh fLFkfr esa ge mUgsa nks izdkj ls xq.kk dj ldrs gSa uker% nks iQyuksa dk ¯cnqokj xq.ku ,oa nks iQyuksa dk la;kstuA blh izdkj lfn'kksa dk xq.ku Hkh nks rjhosQ ls ifjHkkf"kr fd;k tkrk gSA uker% vfn'k xq.kuiQy tgk¡ ifj.kke ,d vfn'k gksrk gS vkSj lfn'k xq.kuiQy tgk¡ ifj.kke ,d lfn'k gksrk gSA lfn'kksa osQ bu nks izdkj osQ xq.kuiQyksa osQ vk/kj ij T;kferh] ;kaf=kdh ,oa vfHk;kaf=kdh esa buosQ fofHkUu vuqi;z ksx gSAa bl ifjPNsn esa ge bu nks izdkj osQ xq.kuiQyksa dh ppkZ djsxa As Rationalised 2023-24 366 xf.kr 10.6.1 nks lfn'kksa dk vfn'k xq.kuiQy [Scalar (or dot) product of two vectors]     ifjHkk"kk 2 nks 'kwU;srj lfn'kksa a vkjS b dk vfn'k xq.kuiQy a ⋅ b }kjk fufnZ"V fd;k tkrk gS    vkSj bls a ⋅ b = | a | | b | cos θ osQ :i es ifjHkkf"kr fd;k tkrk gSA  tgk¡ θ, a vkSj b , oQs chp dk dk.sk gS vkSj 0 ≤ θ ≤ π (vko`Qfr 10-19)A     ;fn a = 0 vFkok b = 0, rks θ ifjHkkf"kr ugha gS vkSj bl fLFkfr eas   ge a ⋅ b = 0 ifjHkkf"kr djrs gSaA vko`Qfr 10-19 izs{k.k   1. a ⋅ b ,d okLrfod la[;k gSA   2. eku yhft, fd a vkjS b nks 'kwU;srj lfn'k gSa rc a ⋅ b = 0 ;fn vkSj osQoy ;fn       a vkjS b ijLij yacor~~ gSa vFkkZr~ a ⋅ b = 0 ⇔ a ⊥ b   3. ;fn θ = 0, rc a ⋅ b = | a | | b | fof'k"Vr% a ⋅ a = | a |2 , D;ksafd bl fLFkfr esa θ = 0 gSA   4. ;fn θ = π, rc a ⋅ b = − | a | | b |    fof'k"Vr% a ⋅ ( − a ) = − | a |2 , tSlk fd bl fLFkfr esa θ, π osQ cjkcj gSA 5. isz{k.k 2 ,oa 3 osQ lanHkZ esa ijLij yacor~ ek=kd lfn'kksa iˆ, ˆj ,oa kˆ, osQ fy, ge ikrs gSa fd iˆ ⋅ iˆ = ˆj ⋅ ˆj = kˆ ⋅ kˆ = 1 rFkk iˆ ⋅ ˆj = ˆj ⋅ kˆ = kˆ ⋅ iˆ = 0   6. nks 'kwU;srj lfn'kksa a vkjS b osQ chp dk dks.k θ,   a.b –1  a.b  cos θ =   vFkok θ = cos     }kjk fn;k tkrk gSA | a || b |  | a || b |  7. vfn'k xq.kuiQy Øe fofues; gS vFkkZr~     a⋅b = b ⋅ a (D;ksa ?) vfn'k xq.kuiQy osQ nks egRoiw.kZ xq.k/eZ (Two important properties of scalar product)    xq.k/eZ 1 (vfn'k xq.kuiQy dh ;ksxiQy ij forj.k fu;e) eku yhft, a , b vkSj c rhu        lfn'k gSa rc a ⋅ (b + c ) = a ⋅ b + a ⋅ c  xq.k/eZ 2 eku yhft, a vkjS b nks lfn'k gSa vkSj λ ,d vfn'k gS] rks       (λa ) ⋅ b = λ ( a ⋅ b ) = a ⋅ (λ b ) Rationalised 2023-24  lfn'k chtxf.kr 367 ;fn nks lfn'k ?kVd :i esa a1iˆ + a2 ˆj + a3kˆ ,oa b1iˆ + b2 ˆj + b3 kˆ, fn, gq, gSa rc mudk vfn'k xq.kuiQy fuEufyf[kr :i esa izkIr gksrk gS   a ⋅ b = (a1iˆ + a2 ˆj + a3kˆ ) ⋅ (b1iˆ + b2 ˆj + b3kˆ ) = a1iˆ ⋅ (b1iˆ + b2 ˆj + b3kˆ ) + a2 ˆj ⋅ (b1iˆ + b2 ˆj + b3 kˆ ) + a3kˆ ⋅ (b1iˆ + b2 ˆj + b3 kˆ) = a b (iˆ ⋅ iˆ) + a b (iˆ ⋅ ˆj ) + a b (iˆ ⋅ kˆ ) + a b ( ˆj ⋅ iˆ) + a b ( ˆj ⋅ ˆj ) + a b ( ˆj ⋅ kˆ) 1 1 1 2 1 3 2 1 2 2 2 3 + a3b1 ( kˆ ⋅ iˆ) + a3b2 (kˆ ⋅ ˆj ) + a3b3 ( kˆ ⋅ kˆ ) (mi;qZDr xq.k/eZ 1 vkSj 2 dk mi;ksx djus ij) = a1b1 + a2b2 + a3b3 (iz{ks.k 5 dk mi;ksx djus ij)   bl izdkj a ⋅ b = a1b1 + a2b2 + a3b3 10.6.2 ,d lfn'k dk fdlh js[kk ij lkFk iz{ksi (Projection of a vector on a line)


 eku yhft, fd ,d lfn'k AB fdlh fn"V js[kk l (eku yhft,) osQ lkFk okekorZ fn'kk esa θ


  dks.k cukrk gSA (vko`Qfr 10-20 nsf[k,) rc AB dk l ij iz{ksi ,d lfn'k p (eku yhft,)


 gS ftldk ifjek.k | AB | | cos θ | gS vkSj ftldh fn'kk dk l dh fn'kk osQ leku vFkok foijhr gksuk  bl ckr ij fuHkZj gS fd cos θ /ukRed gS vFkok ½.kkRedA lfn'k p dks iz{ksi lfn'k dgrs gSa 


 vkSj bldk ifjek.k | p |, fu£n"V js[kk l ij lfn'k AB dk iz{ksi dgykrk gSA mnkgj.kr%





 fuEufyf[kr esa ls izR;sd vko`Qfr esa lfn'k AB dk js[kk l ij iz{ksi lfn'k AC gS A [vko` Q fr 10.20 (i) ls (iv) rd] vko`Qfr 10-20 Rationalised 2023-24 368 xf.kr izs{k.k   1. js[kk l osQ vuqfn'k ;fn p̂ ek=kd lfn'k gS rks js[kk l ij lfn'k a dk iz{ksi a. pˆ ls izkIr gksrk gSA    2. ,d lfn'k a dk nwljs lfn'k b, ij iz{ksi a ⋅ bˆ, vFkok ls izkIr gksrk gSA








 3- ;fn θ = 0, rks AB dk iz{ksi lfn'k Lo;a AB gksxk vkSj ;fn θ = π rks AB dk iz{ksi


 lfn'k BA gksxkA π 3π


 4- ;fn θ = vFkok θ = rks AB dk iz{ksi lfn'k 'kwU; lfn'k gksxkA 2 2  fVIi.kh ;fn α, β vkSj γ lfn'k a = a1iˆ + a2 ˆj + a3 kˆ osQ fno~Q&dks.k gSa rks bldh fno~Q&dkslkbu fuEufyf[kr :i esa izkIr dh tk ldrh gSA  a.iˆ a a a cos α =  = 1 , cos β = 2 , and cos γ = 3 ˆ | a || i | | a | |a| |a| ;g Hkh è;ku nhft, fd | a | cos α, | a | cosβ vkjS | a | cosγ Øe'k% OX, OY rFkk OZ osQ vuqfn'k a osQ iz{ksi gSa vFkkZr~ lfn'k a osQ vfn'k ?kVd a1, a2 vkSj a3 Øe'k% x, y, ,oa z v{k osQ vuqfn'k a osQ iz{ksi gSA blosQ vfrfjDr ;fn a ,d ek=kd lfn'k gS rc bldks fno~Q&dkslkbu dh lgk;rk ls  a = cos αiˆ + cos βˆj + cos γkˆ osQ :i esa vfHkO;Dr fd;k tk ldrk gSA     mnkgj.k 13 nks lfn'kksa a vkjS b osQ ifjek.k Øe'k% 1 vkSj 2 gS rFkk a ⋅ b =1 , bu lfn'kksa osQ chp dk dks.k Kkr dhft,A     gy fn;k gqvk gS a ⋅ b = 1, | a | = 1 vkSj | b | = 2. vr%  −1  a.b  1 π θ = cos     = cos−1   =  | a || b |  2 3   mnkgj.k 14 lfn'k a = iˆ + ˆj − kˆ rFkk b = iˆ − ˆj + kˆ osQ chp dk dks.k Kkr dhft,A   gy nks lfn'kksa a vkjS b osQ chp dk dks.k θ fuEu }kjk iznÙk gS   a ⋅b cosθ =   ls izkIr gksrk gSA | a || b | Rationalised 2023-24  lfn'k chtxf.kr 369   vc a ⋅ b = (iˆ + ˆj − kˆ) ⋅ ( iˆ − ˆj + kˆ ) = 1 − 1 − 1 = −1 −1 blfy,] ge ikrs gSa fd cosθ = 3 vr% vHkh"V dks.k θ= gSA     mnkgj.k 15 ;fn a = 5iˆ − ˆj − 3kˆ vkjS b = iˆ + 3 ˆj − 5kˆ , rks n'kkZb, fd lfn'k a + b vkSj   a − b yacor~ gSA gy ge tkurs gSa fd nks 'kwU;srj lfn'k yacor~~ gksrs gSa ;fn mudk vfn'k xq.kuiQy 'kwU; gSA   ;gk¡ a + b = (5iˆ − ˆj − 3kˆ ) + ( iˆ + 3 ˆj − 5kˆ) = 6iˆ + 2 ˆj − 8kˆ   vkSj a − b = (5iˆ − ˆj − 3kˆ) − ( iˆ + 3 ˆj − 5kˆ) = 4iˆ − 4 ˆj + 2kˆ     blfy, (a + b ) ⋅ ( a − b ) = (6iˆ + 2 ˆj − 8kˆ ) ⋅ (4iˆ − 4 ˆj + 2kˆ) = 24 − 8 − 16 = 0     vr% a + b vkjS a − b yacor~ lfn'k gSaA  mnkgj.k 16 lfn'k a = 2iˆ + 3 ˆj + 2kˆ dk] lfn'k b = iˆ + 2 ˆj + kˆ ij iz{ksi Kkr dhft,A  gy lfn'k a dk lfn'k b ij iz{ksi 1   1 10 5  (a ⋅ b ) = (2. 1 + 3. 2 + 2. 1) = = 6 gSA |b | 2 (1) + (2) + (1) 2 2 6 3     mnkgj.k 17 ;fn nks lfn'k a vkjS b bl izdkj gSa fd | a | = 2, | b | = 3 vkSj a ⋅ b = 4 rks | a − b | Kkr dhft,A gy ge ikrs gSa fd  2     a − b = (a − b ) ⋅ ( a − b )        = a.a − a ⋅ b − b ⋅ a + b ⋅ b   = | a |2 −2( a ⋅ b ) + | b |2 = (2)2 − 2(4) + (3) 2   blfy, |a−b | = 5      mnkgj.k 18 ;fn a ,d ek=kd lfn'k gS vkSj ( x − a ) ⋅ ( x + a ) = 8 , rks | x | Kkr dhft,A gy D;ksafd a ,d ek=kd lfn'k gS] blfy, | a | = 1. ;g Hkh fn;k gqvk gS fd     ( x − a ) ⋅ ( x + a) = 8         vFkok x ⋅ x + x⋅a − a⋅ x − a ⋅a = 8   vFkok | x |2 −1 = 8 vFkkZr ~ | x |2 = 9  blfy, | x | = 3 (D;ksafd lfn'k dk ifjek.k lnSo 'kwU;srj gksrk gS) Rationalised 2023-24 370 xf.kr      mnkgj.k 19 nks lfn'kksa a vkjS b , osQ fy, lnSo | a ⋅ b | ≤ | a | | b | (Cauchy-Schwartz vlfedk)A    gy nh gqbZ vlfedk lgt :i esa Li"V gS ;fn a = 0 vFkok b = 0. okLro esa bl fLFkfr esa     ge ikrs gSa fd | a ⋅ b | = 0 = | a | | b |. blfy, ge dYiuk djrs gSa fd | a | ≠ 0 ≠ | b | rc gesa   | a ⋅b |   = | cos θ | ≤ 1 feyrk gSA | a || b |    blfy, | a ⋅ b | ≤ | a | | b | vko`Qfr 10-21    mnkgj.k 20 nks lfn'kksa a rFkk b osQ fy, lnSo | a + b | ≤ | a | + | b | (f=kHkqt&vlfedk)  gy nh gqbZ vlfedk] nksuksa fLFkfr;ksa a = 0 ;k b = 0 esa lgt :i ls Li"V gS (D;ksa ?)A blfy,    eku yhft, fd | a | ≠ 0 ≠ | b | rc         | a + b |2 = ( a + b ) 2 = ( a + b ) ⋅ ( a + b )         = a ⋅a + a ⋅b + b ⋅a + b ⋅b     = | a |2 +2a ⋅ b + | b |2 (vfn'k xq.kuiQy Øe fofue; gS)     ≤ | a |2 +2 | a ⋅ b | + | b |2 (D;ksafd x ≤ | x | ∀x ∈ R ) 2   2 ≤ | a | +2 | a || b | + | b | (mnkgj.k 19 ls)   = (| a | + | b |) 2     vr% |a +b |≤|a |+|b | AfVIi.kh ;fn f=kHkqt&vlfedk esa lfedk /kj.k gksrh gS (mi;qZDr mnkgj.k 20 esa) vFkkZr~     | a + b | = | a | + | b | , rc








 | AC | = | AB | + | BC | ¯cnq A, B vkSj C lajs[k n'kkZrk gSA mnkgj.k 21 n'kkZb, fd ¯cnq A ( −2iˆ + 3 ˆj + 5kˆ ), B( iˆ + 2 ˆj + 3kˆ) vkSj C(7iˆ − kˆ) lajs[k gSA gy ge izkIr djrs gSa%


 AB = (1 + 2)iˆ + (2 − 3) ˆj + (3 − 5) kˆ = 3iˆ − ˆj − 2kˆ Rationalised 2023-24  lfn'k chtxf.kr 371


 BC = (7 − 1)iˆ + (0 − 2) ˆj + ( −1 − 3) kˆ = 6iˆ − 2 ˆj − 4kˆ


 ˆ ˆ AC = (7 + 2)iˆ + (0 − 3) ˆj + ( −1 − 5)k = 9iˆ − 3 ˆj − 6k








 | AB | = 14, BC = 2 14 vkSj | AC | = 3 14








 blfy, AC = | AB | + | BC | vr% ¯cnq A, B vkSj C lajs[k gSaA








  AfVIi.kh mnkgj.k 21 esa è;ku nhft, fd AB + BC + CA = 0 ijarq fiQj Hkh ¯cnq A, B vkSj C f=kHkqt osQ 'kh"kks± dk fuekZ.k ugha djrs gSaA iz'ukoyh 10-3       1. nks lfn'kksa a rFkk b osQ ifjek.k Øe'k% 3 ,oa 2 gSa vkSj a ⋅ b = 6 gS rks a rFkk b osQ chp dk dks.k Kkr dhft,A 2. lfn'kksa iˆ − 2 ˆj + 3kˆ vkjS 3iˆ − 2 ˆj + kˆ osQ chp dk dks.k Kkr dhft,A 3. lfn'k iˆ + ˆj ij lfn'k iˆ − ˆj dk iz{ksi Kkr dhft,A 4. lfn'k iˆ + 3 ˆj + 7kˆ dk] lfn'k 7iˆ − ˆj + 8kˆ ij iz{ksi Kkr dhft,A 5. n'kkZb, fd fn, gq, fuEufyf[kr rhu lfn'kksa esa ls izR;sd ek=kd lfn'k gS] 1 ˆ 1 1 (2i + 3 ˆj + 6kˆ ), (3iˆ − 6 ˆj + 2kˆ ), (6iˆ + 2 ˆj − 3kˆ) 7 7 7 ;g Hkh n'kkZb, fd ;s lfn'k ijLij ,d nwljs osQ yacor~ gSaA         6. ;fn (a + b ) ⋅ ( a − b ) = 8 vkSj | a | = 8 | b | gks rks | a | ,oa | b | Kkr dhft,A     7. (3a − 5b ) ⋅ (2a + 7b ) dk eku Kkr dhft,A   8. nks lfn'kksa a vkjS b osQ ifjek.k Kkr dhft,] ;fn buosQ ifjek.k leku gS vkSj bu osQ chp 1 dk dks.k 60° gS rFkk budk vfn'k xq.kuiQy gSA 2       9. ;fn ,d ek=kd lfn'k a , osQ fy, ( x − a ) ⋅ ( x + a ) = 12 gks rks | x | Kkr dhft,A       10. ;fn a = 2iˆ + 2 ˆj + 3kˆ, b = − iˆ + 2 ˆj + kˆ vkjS c = 3iˆ + ˆj bl izdkj gS fd a + λ b, c ij yac gS] rks λ dk eku Kkr dhft,A Rationalised 2023-24 372 xf.kr           11. n'kkZb, fd nks 'kwU;srj lfn'kksa a vkjS b osQ fy, | a | b + | b | a , | a | b − | b | a ij yac gSA      12. ;fn a ⋅ a = 0 vkjS a ⋅ b = 0 , rks lfn'k b osQ ckjs esa D;k fu"d"kZ fudkyk tk ldrk gS?              13. ;fn a , b , c ek=kd lfn'k bl izdkj gS fd a + b + c = 0 rks a ⋅ b + b ⋅ c + c ⋅ a dk eku Kkr dhft,A       14. ;fn a = 0 vFkok b = 0, rc a ⋅ b = 0 ijarq foykse dk lR; gksuk vko';d ugha gSA ,d mnkgj.k }kjk vius mÙkj dh iqf"V dhft,A 15. ;fn fdlh f=kHkqt ABC osQ 'kh"kZ A, B, C Øe'k% (1, 2, 3), (–1, 0, 0), (0, 1, 2) gSa rks





 ∠ABC Kkr dhft,A [∠ABC, lfn'kksa BA ,oa BC osQ chp dk dks.k gS ] 16. n'kkZb, fd ¯cnq A(1, 2, 7), B(2, 6, 3) vkSj C(3, 10, –1) lajs[k gSaA 17. n'kkZb, fd lfn'k 2iˆ − ˆj + kˆ, iˆ − 3 ˆj − 5kˆ vkSj 3iˆ − 4 ˆj − 4kˆ ,d ledks.k f=kHkqt osQ 'kh"kks± dh jpuk djrs gSaA   18. ;fn 'kwU;srj lfn'k a dk ifjek.k ‘a’ gS vkSj λ ,d 'kwU;rsj vfn'k gS rks λ a ,d ek=kd lfn'k gS ;fn (A) λ = 1 (B) λ = – 1 (C) a = | λ | (D) a = 1/| λ | 10.6.3 nks lfn'kksa dk lfn'k xq.kuiQy [Vector (or cross) product of two vectors] ifjPNsn 10-2 esa geus f=k&foeh; nf{k.kkorhZ ledksf.kd funsZ'kkad i¼fr dh ppkZ dh FkhA bl i¼fr esa /ukRed x-v{k dks okekorZ ?kqekdj /ukRed y-v{k ij yk;k tkrk gS rks /ukRed z-v{k dh fn'kk esa ,d nf{k.kkorhZ (izkekf.kd) isap vxzxr gks tkrh gS [vko`Qfr 10.22(i)]A ,d nf{k.kkorhZ funsZ'kkad i¼fr esa tc nk,¡ gkFk dh m¡xfy;ksa dks /ukRed x-v{k dh fn'kk ls nwj /ukRed y-v{k dh rji+Q oqQa ry fd;k tkrk gS rks v¡xBw k /ukRed z-v{k dh vksj laoQs r djrk [vko`Qfr 10-22 (ii)] gSA vko`Qfr 10-22 Rationalised 2023-24  lfn'k chtxf.kr 373    ifjHkk"kk 3 nks 'kwU;srj lfn'kksa a rFkk b , dk lfn'k xq.kuiQy a × b ls    fufnZ"V fd;k tkrk gS vkSj a × b = | a || b | sin θ nˆ osQ :i esa ifjHkkf"kr   fd;k tkrk gS tgk¡ θ, a vkjS b osQ chp dk dks.k gS vkSj 0 ≤ θ ≤ π gSA   ;gk¡ n̂ ,d ek=kd lfn'k gS tks fd lfn'k a vkjS b , nksuksa ij yac gSA  bl izdkj a , b rFkk nˆ ,d nf{k.kkorhZ i¼fr dks fufeZr djrs gSa vko`Qfr 10-23  (vko`Qfr 10-23) vFkkZr~ nf{k.kkorhZ i¼fr dks a ls b dh rji+Q ?kqekus ij ;g n̂ dh fn'kk esa pyrh gSA       ;fn a = 0 vFkok b = 0 , rc θ ifjHkkf"kr ugha gS vkSj bl fLFkfr eas ge a × b = 0 ifjHkkf"kr djrs gSaA izs{k.k%   1. a × b ,d lfn'k gSA      2. eku yhft, a vkjS b nks 'kwU;srj lfn'k gSa rc a × b = 0 ;fn vkSj osQoy ;fn   a vkjS b ,d nwljs osQ lekarj (vFkok lajs[k) gSa vFkkZr~      a×b = 0 ⇔ a b     fof'k"Vr% a × a = 0 vkSj a × (−a) = 0 , D;ksafd izFke fLFkfr esa θ = 0 rFkk f}rh; fLFkfr esa θ = π, ftlls nksuksa gh fLFkfr;ksa esa sin θ dk eku 'kwU; gks tkrk gSA π     3. ;fn θ = rks a × b = | a || b | 2 4. izs{k.k 2 vkSj 3 osQ lanHkZ esa ijLij yacor~ ek=kd lfn'kksa iˆ, ˆj vkSj kˆ osQ fy, (vko`Qfr 10-24), ge ikrs gSa fd  iˆ × iˆ = ˆj × ˆj = kˆ × kˆ = 0 iˆ × ˆj = kˆ, ˆj × kˆ = iˆ, kˆ × iˆ = ˆj   5. lfn'k xq.kuiQy dh lgk;rk ls nks lfn'kksa a rFkk b osQ chp dk dks.k vko`Qfr 10-24 θ fuEufyf[kr :i esa izkIr gksrk gS   | a ×b | sinθ =   | a || b |     6. ;g loZnk lR; gS fd lfn'k xq.kuiQy Øe fofue; ugha gksrk gS D;ksafd a × b = − b × a    okLro esa a × b = | a || b | sin θ nˆ , tgk¡ a , b vkSj nˆ ,d nf{k.kkorhZ i¼fr dks fu£er djrs Rationalised 2023-24 374 xf.kr  gSa vFkkZr~ θ, a ls b dh rjiQ pØh; Øe gksrk gSA vko`Qfr 10-25(i) tcfd       b × a =| a || b | sin θ nˆ1 , tgk¡ b , a vkjS nˆ1 ,d nf{k.kkorhZ i¼fr dks fufeZr djrs gSa  vFkkZr~ θ, b ls a dh vksj pØh; Øe gksrk gS vko`Qfr 10-25(ii)A vko`Qfr 10-25   vr% ;fn ge ;g eku ysrs gSa fd a vkjS b nksuksa ,d gh dkx”k osQ ry esa gSa rks nˆ vkjS nˆ1 nksuksa dkx”k osQ ry ij yac gksaxs ijarq n̂ dkx”k ls Åij dh rji+Q fn"V gksxk vkSj n̂1 dkx”k ls uhps dh rji+Q fn"V gksxk vFkkZr~ nˆ1 = − nˆ     bl izdkj a × b = | a || b | sin θnˆ     = − | a || b | sin θnˆ1 = − b × a 7. izs{k.k 4 vkSj 6 osQ lanHkZ esa ˆj × iˆ = − kˆ, kˆ × ˆj = − iˆ vkjS iˆ × kˆ = − ˆj gAS   8. ;fn a vkjS b f=kHkqt dh layXu Hkqtkvksa dks fu:fir djrs gSa rks f=kHkqt dk {ks=kiQy 1   | a × b | osQ :i esa izkIr gksrk gSA 2 f=kHkqt osQ {ks=kiQy dh ifjHkk"kk osQ vuqlkj ge vko`Qfr 10-26 1 ls ikrs gSa fd f=kHkqt ABC dk {ks=kiQy = AB ⋅ CD. 2   vko`Qfr 10-26 ijarq AB = | b | (fn;k gqvk gS) vkSj CD = | a | sinθ 1   1   vr% f=kHkqt ABC dk {ks=kiQy = | b || a | sin θ = | a × b | 2 2   9. ;fn a vkjS b lekarj prqHkZqt dh layXu Hkqtkvksa dks fu:fir djrs gSa rks lekarj prqHkqZt dk  {ks=kiQy | a × b | osQ :i esa izkIr gksrk gSA Rationalised 2023-24  lfn'k chtxf.kr 375 vko`Qfr 10-27 ls ge ikrs gSa fd lekarj prqHkaZt ABCD dk {ks=kiQy = AB. DE.  ija r q AB = | b | ( fn;k gq v k gS ) , vkS j  DE = | a | sin θ vr% leka r j prq H kq Z t ABCD dk {ks = kiQy =     vko`Qfr 10-27 | b || a | sin θ = | a × b | vc ge lfn'k xq.kuiQy osQ nks egRoiw.kZ xq.kksa dks vfHkO;Dr djsaxsA xq.k/eZ lfn'k xq.kuiQy dk ;ksxiQy ij forj.k fu;e (Distributivity of vector product    over addition) ;fn a , b vkSj c rhu lfn'k gSa vkSj λ ,d vfn'k gS rks        (i) a × ( b + c ) = a × b + a × c       (ii) λ (a × b ) = (λ a ) × b = a × (λ b )  eku yhft, nks lfn'k a vkjS b ?kVd :i esa Øe'k% a1iˆ + a2 ˆj + a3kˆ vkSj b1iˆ + b2 ˆj + b3 kˆ iˆ ˆj kˆ fn, gq, gSa rc mudk lfn'k xq.kuiQy   = a a2 a3 }kjk fn;k tk ldrk gSA a×b 1 b1 b2 b3 O;k[;k ge ikrs gSa   a × b = (a1iˆ + a2 ˆj + a3kˆ ) × (b1iˆ + b2 ˆj + b3kˆ ) = a1b1 (iˆ × iˆ) + a1b2 (iˆ × ˆj ) + a1b3 (iˆ × kˆ ) + a2b1 ( ˆj × iˆ) + a2 b2 ( ˆj × ˆj ) + a2b3 ( ˆj × kˆ) + a3b1 ( kˆ × iˆ) + a3b2 (kˆ × ˆj ) + a3b3 ( kˆ × kˆ) ( xq.k/eZ 1 ls) = a1b2 (iˆ × ˆj ) − a1b3 (kˆ × iˆ) − a2b1 (iˆ × ˆj ) + a2 b3 ( ˆj × kˆ ) + a3b1 (kˆ × iˆ) − a3b2 ( ˆj × kˆ) Rationalised 2023-24 376 xf.kr ( D;kfsad iˆ × iˆ = ˆj × ˆj = kˆ × kˆ = 0 vkjS iˆ × kˆ = − kˆ × iˆ, ˆj × iˆ = −iˆ × ˆj vkSj kˆ × ˆj = − ˆj × kˆ ) = a1b2 kˆ − a1b3 ˆj − a2b1kˆ + a2b3iˆ + a3b1 ˆj − a3b2iˆ ( D;kasfd iˆ × ˆj = kˆ, ˆj × kˆ = iˆ vkSj kˆ × iˆ = ˆj ) = (a2b3 − a3b2 )iˆ − (a1b3 − a3b1 ) ˆj + ( a1b2 − a2b1 ) kˆ iˆ ˆj kˆ = a1 a2 a3 b1 b2 b3   mnkgj.k 22 ;fn a = 2iˆ + ˆj + 3kˆ vkjS b = 3iˆ + 5 ˆj − 2kˆ, rks | a × b | Kkr dhft,A gy ;gk¡ iˆ ˆj kˆ   a×b = 2 1 3 3 5 −2 = iˆ( −2 − 15) − (− 4 − 9) ˆj + (10 – 3)kˆ = −17iˆ + 13 ˆj + 7kˆ   vr% a × b = ( −17) 2 + (13) 2 + (7)2 = 507     mnkgj.k 23 lfn'k (a + b ) vkSj ( a − b ) esa ls izR;sd osQ yacor~ ek=kd lfn'k Kkr  dhft, tgk¡ a = iˆ + ˆj + kˆ, b = iˆ + 2 ˆj + 3kˆ gSaA   gy ge ikrs gSa fd a + b = 2iˆ + 3 ˆj + 4kˆ vkSj a − b = − ˆj − 2kˆ     ,d lfn'k] tks a + b vkjS a − b nksuks ij yac gS] fuEufyf[kr }kjk iznÙk gS iˆ ˆj kˆ     ˆ  (a + b ) × ( a − b ) = 2 3 4 = −2iˆ + 4 ˆj − 2k (= c , eku yhft, ) 0 −1 −2  vc | c | = 4 + 16 + 4 = 24 = 2 6 vr% vHkh"V ek=kd lfn'k  c −1 ˆ 2 ˆ 1 ˆ  = i+ j− k gSA |c | 6 6 6 Rationalised 2023-24  lfn'k chtxf.kr 377   A fVIi.kh fdlh ry ij nks yacor~ fn'kk,¡ gksrh gSAa vr% a + b vkjS a − b ij nwljk yacor~ 1 ˆ 2 ˆ 1 ˆ   ek=kd lfn'k i− j+ k gksxkA ijarq ;g (a − b ) × ( a + b ) dk ,d ifj.kke gSA 6 6 6 mnkgj.k 24 ,d f=kHkqt dk {ks=kiQy Kkr dhft, ftlosQ 'kh"kZ ¯cnq A(1, 1, 1), B(1, 2, 3) vkSj C(2, 3, 1) gSaA





 gy ge ikrs gSa fd AB = ˆj + 2kˆ vkjS AC = iˆ + 2 ˆj. fn, gq, f=kHkqt dk {ks=kiQy 1





 | AB × AC | gSA 2 iˆ ˆj kˆ





 ˆ vc AB × AC = 0 1 2 = − 4iˆ + 2 ˆj − k 1 2 0





 blfy, | AB × AC | = 16 + 4 + 1 = 21 1 vr% vHkh"V {ks=kiQy 21 gSA 2 Z dk {ks=kiQy Kkr dhft, ftldh layXu Hkqtk,¡ a = 3iˆ + ˆj + 4kˆ mnkgj.k 25 ml lekarj prqHkqt  vkSj b = iˆ − ˆj + kˆ }kjk nh xbZ gSaA   gy fdlh lekarj prqHkqZt dh layXu Hkqtk,¡ a vkjS b gSa rks mldk {ks=kiQy | a × b | }kjk izkIr gksrk gSA iˆ ˆj kˆ   vc a × b = 3 1 4 = 5iˆ + ˆj − 4kˆ 1 −1 1   blfy, |a×b | = 25 + 1 + 16 = 42 bl izdkj vko';d {ks=kiQy 42 gSA Rationalised 2023-24 378 xf.kr iz'ukoyh 10-4     1. ;fn a = iˆ − 7 ˆj + 7 kˆ vkSj b = 3iˆ − 2 ˆj + 2kˆ rks | a × b | Kkr dhft,A      2. lfn'k a + b vkjS a − b dh yac fn'kk esa ek=kd lfn'k Kkr dhft, tgk¡ a = 3iˆ + 2 ˆj + 2kˆ  vkSj b = iˆ + 2 ˆj − 2kˆ gSA  π π 3. ;fn ,d ek=kd lfn'k a , iˆ oQs lkFk , ˆj osQ lkFk vkSj kˆ osQ lkFk ,d U;wu dks.k θ 3 4  cukrk gS rks θ dk eku Kkr dhft, vkSj bldh lgk;rk ls a osQ ?kVd Hkh Kkr dhft,A       4. n'kkZb, fd (a − b ) × ( a + b ) = 2( a × b )  5. λ vkSj µ Kkr dhft,] ;fn (2iˆ + 6 ˆj + 27kˆ) × (iˆ + λˆj + µkˆ) = 0        6. fn;k gqvk gS fd a ⋅ b = 0 vkSj a × b = 0. lfn'k a vkjS b osQ ckjs esa vki D;k fu"d"kZ fudky ldrs gS?a    7. eku yhft, lfn'k a, b , c Øe'k% a1iˆ + a2 ˆj + a3kˆ, b1iˆ + b2 ˆj + b3kˆ, c1iˆ + c2 ˆj + c3 kˆ osQ        :i esa fn, gq, gSa rc n'kkZb, fd a × ( b + c ) = a × b + a × c        8. ;fn a = 0 vFkok b = 0 rc a × b = 0 gksrk gSA D;k foykse lR; gS\ mnkgj.k lfgr vius mÙkj dh iqf"V dhft,A 9. ,d f=kHkqt dk {ks=kiQy Kkr dhft, ftlosQ 'kh"kZ A(1, 1, 2), B(2, 3, 5) vkSj C (1, 5, 5) gSAa  10. ,d lekarj prqHkqt Z dk {ks=kiQy Kkr dhft, ftldh layXu Hkqtk,¡ lfn'k a = iˆ − ˆj + 3kˆ vkSj  b = 2iˆ − 7 ˆj + kˆ }kjk fu/kZfjr gSaA     2   11. eku yhft, lfn'k a vkjS b bl izdkj gSa fd | a | = 3 vkjS | b | = , rc a × b ,d 3   ek=kd lfn'k gS ;fn a vkjS b osQ chp dk dks.k gS% (A) π/6 (B) π/4 (C) π/3 (D) π/2 12. ,d vk;r osQ 'kh"kks± A, B, C vkSj D ftuosQ fLFkfr lfn'k Øe'k% 1ˆ 1 1 1 – iˆ + j + 4kˆ, iˆ + ˆj + 4kˆ , iˆ − ˆj + 4kˆ vkSj – iˆ − ˆj + 4kˆ , gSa dk {ks=kiQy gS% 2 2 2 2 1 (A) (B) 1 2 (C) 2 (D) 4 Rationalised 2023-24  lfn'k chtxf.kr 379 fofo/ mnkgj.k mnkgj.k 26 XY-ry esa lHkh ek=kd lfn'k fyf[k,A  ∧ ∧ gy eku yhft, fd r = x i + y j , XY-ry esa ,d ek=kd lfn'k gS (vko`Qfr 10-28)A rc  vko`Qfr osQ vuqlkj ge ikrs gSa fd x = cos θ vkSj y = sin θ (D;ksafd | r | = 1). blfy, ge lfn'k  r dks ] 


 r ( = OP ) = cos θiˆ + sin θˆj... (1) osQ :i esa fy[k ldrs gSaA  Li"Vr% | r | = cos 2 θ + sin 2 θ = 1 vko`Qfr 10-28 tSls&tSls θ, 0 ls 2π, rd ifjofrZr gksrk gS ¯cnq P (vko`Qfr 10-28) okekorZ fn'kk esa o`r x2 + y2 = 1 dk vuqjs[k.k djrk gS vkSj bleas lHkh laHkkfor fn'kk,¡ lfEefyr gSaA vr% (1) ls XY- ry esa izR;sd ek=kd lfn'k izkIr gksrk gSA mnkgj.k 27 ;fn ¯cnqvksa A, B, C vkSj D, osQ fLFkfr lfn'k Øe'k% iˆ + ˆj + kˆ, 2iˆ + 5 ˆj , 3iˆ + 2 ˆj − 3kˆ vkSj iˆ − 6 ˆj − kˆ gS] rks ljy js[kkvksa AB rFkk CD osQ chp dk dks.k Kkr dhft,A fuxeu dhft, fd AB vkSj CD lajs[k gSaA





 gy uksV dhft, fd ;fn θ, AB vkSj CD, osQ chp dk dks.k gS rks θ, AB vkjS CD osQ chp dk Hkh dks.k gSA


 vc AB = B dk fLFkfr lfn'k – A dk fLFkfr lfn'k = (2iˆ + 5 ˆj ) − (iˆ + ˆj + kˆ) = iˆ + 4 ˆj − kˆ Rationalised 2023-24 380 xf.kr


 blfy, | AB | = (1) 2 + (4) 2 + ( −1) 2 = 3 2





 blh izdkj CD = − 2iˆ − 8 ˆj + 2kˆ vkSj |CD | = 6 2





 AB. CD vr% cosθ =





 |AB||CD| 1(−2) + 4(−8) + ( −1)(2) −36 = = = −1 (3 2)(6 2) 36 D;ksafd 0 ≤ θ ≤ π, blls izkIr gksrk gS fd θ = π. ;g n'kkZrk gS fd AB rFkk CD ,d nwljs osQ lajs[k gSaA


 1








 fodYir% AB = − CD ] blls dg ldrs fd AB vkSj CD lajs[k lfn'k gSaA 2     mnkgj.k 28 eku yhft, a , b vkSj c rhu lfn'k bl izdkj gSa fd | a |= 3, | b |= 4, | c |= 5  vkSj buesa ls izR;sd] vU; nks lfn'kksa osQ ;ksxiQy ij yacor~ gaS rks] | a + b + c | Kkr dhft,A    gy fn;k gqvk gS fd a ⋅ (b + c ) = 0, b ⋅ (c + a ) = 0, c ⋅ ( a + b ) = 0          vc | a + b + c |2 = (a + b + c ) ⋅ (a + b + c )           = a ⋅ a + a ⋅ (b + c ) + b ⋅ b + b ⋅ (a + c )     + c.(a + b ) + c.c    = | a |2 + | b |2 + | c |2 = 9 + 16 + 25 = 50    blfy, | a + b + c | = 50 = 5 2      mnkgj.k 29 rhu lfn'k a, b vkSj c izfrca/ a + b + c = 0 dks larq"V djrs gSaA ;fn          | a |= 3, | b |= 4 vkSj | c |= 2 rks jkf'k µ = a ⋅ b + b ⋅ c + c ⋅ a dk eku Kkr dhft,A   gy D;ksafd a + b + c = 0 , blfy, ge ikrs gSa fd     a ⋅ (a + b + c ) = 0       vFkok a ⋅ a + a ⋅b + a ⋅c = 0     2 blfy, a ⋅ b + a ⋅ c = − a = −9... (1)     iqu% b ⋅ (a + b + c ) = 0 Rationalised 2023-24  lfn'k chtxf.kr 381      2 vFkok a ⋅b + b ⋅c = − b = −16... (2)     blh izdkj a ⋅c + b⋅c = – 4... (3) (1)] (2) vkSj (3) dks tksM+us ij ge ikrs gSa fd       2 ( a ⋅ b + b ⋅ c + a ⋅ c ) = – 29 −29 ;k 2µ = – 29, i.e., µ = 2 mnkgj.k 30 ;fn ijLij yacor~ ek=kd lfn'kksa iˆ, ˆj vkSj kˆ, dh nf{k.kkorhZ i¼fr osQ lkis{k        α = 3iˆ − ˆj , β = 2iˆ + ˆj – 3kˆ , rks β dks β = β1 + β2 osQ :i esa vfHkO;Dr dhft, tgk¡ β1 ]    α osQ lekarj gS vkSj β 2 , α osQ yacor~ gSA    gy eku yhft, fd β1 = λα , λ ,d vfn'k gS vFkkZr~ β1 = 3λiˆ − λˆj    vc β2 = β − β1 = (2 − 3λ)iˆ + (1 + λ ) ˆj − 3kˆ     D;ksafd β 2 , α ij yac gS blfy, α ⋅β2 = 0 vFkkZr~ 3(2 − 3λ ) − (1 + λ ) = 0 1 vFkok λ= 2  3 1  1 3 blfy, β1 = iˆ − ˆj vkSj β2 = iˆ + ˆj – 3kˆ 2 2 2 2 vè;k; 10 ij fofo/ iz'ukoyh 1. XY-ry esa] x-v{k dh /ukRed fn'kk osQ lkFk okekorZ fn'kk esa 30° dk dks.k cukus okyk ek=kd lfn'k fyf[k,A 2. ¯cnq P (x1, y1, z1) vkSj Q (x2, y2, z2) dks feykus okys lfn'k osQ vfn'k ?kVd vkSj ifjek.k Kkr dhft,A 3. ,d yM+dh if'pe fn'kk esa 4 km pyrh gSA mlosQ i'pkr~ og mÙkj ls 30° if'pe dh fn'kk esa 3 km pyrh gS vkSj :d tkrh gSA izLFkku osQ izkjafHkd ¯cnq ls yM+dh dk foLFkkiu Kkr dhft,A     4. ;fn a = b + c, rc D;k ;g lR; gS fd | a | = | b | + | c | ? vius mÙkj dh iqf"V dhft,A 5. x dk og eku Kkr dhft, ftlosQ fy, x(iˆ + ˆj + kˆ) ,d ek=kd lfn'k gSA  6. lfn'kksa a = 2iˆ + 3 ˆj − kˆ vkSj b = iˆ − 2 ˆj + kˆ osQ ifj.kkeh osQ lekarj ,d ,slk lfn'k Kkr dhft, ftldk ifjek.k 5 bdkbZ gSA Rationalised 2023-24 382 xf.kr       7. ;fn a = iˆ + ˆj + kˆ , b = 2iˆ − ˆj + 3kˆ vkSj c = iˆ − 2 ˆj + kˆ , rks lfn'k 2a – b + 3c osQ lekarj ,d ek=kd lfn'k Kkr dhft,A 8. n'kkZb, fd ¯cnq A (1, – 2, – 8), B (5, 0, – 2) vkSj C (11, 3, 7) lajs[k gS vkSj B }kjk AC dks foHkkftr djus okyk vuqikr Kkr dhft,A   9. nks ¯cnqvksa P (2a + b ) vkjS Q (a – 3b ) dks feykus okyh js[kk dks 1% 2 osQ vuqikr es cká foHkkftr djus okys ¯cnq R dk fLFkfr lfn'k Kkr dhft,A ;g Hkh n'kkZb, fd ¯cnq P js[kk[kaM RQ dk eè; ¯cnq gSA 10. ,d lekarj prqHkqt Z dh layXu Hkqtk,¡ 2iˆ − 4 ˆj + 5kˆ vkjS iˆ − 2 ˆj − 3kˆ gaAS blosQ fod.kZ osQ lekarj ,d ek=kd lfn'k Kkr dhft,A bldk {ks=kiQy Hkh Kkr dhft,A 11. n'kkZb, fd OX, OY ,oa OZ v{kksa osQ lkFk cjkcj >qosQ gq, lfn'k dh fno~Q&dkslkbu 1 1 1 dksT;k,¡ ± , ,  gSA  3 3 3    12. eku yhft, a = iˆ + 4 ˆj + 2kˆ, b = 3iˆ − 2 ˆj + 7 kˆ vkjS c = 2iˆ − ˆj + 4kˆ. ,d ,slk lfn'k      d Kkr dhft, tks a vkSj b nksuksa ij yac gS vkSj c ⋅ d = 15 13. lfn'k iˆ + ˆj + kˆ dk] lfn'kksa 2iˆ + 4 ˆj − 5kˆ vkSj λiˆ + 2 ˆj + 3kˆ osQ ;ksxiQy dh fn'kk esa ek=kd lfn'k osQ lkFk vfn'k xq.kuiQy 1 osQ cjkcj gS rks λ dk eku Kkr dhft,A       14. ;fn a , b , c leku ifjek.kksa okys ijLij yacor~ lfn'k gSa rks n'kkZb, fd lfn'k a + b + c  lfn'kksa a , b rFkk c osQ lkFk cjkcj >qdk gqvk gSA         15. fl¼ dhft, fd (a + b ) ⋅ ( a + b ) =| a |2 + | b |2 , ;fn vkSj osQoy ;fn a, b yacor~ gSaA    ;g fn;k gqvk gS fd a ≠ 0, b ≠ 0 16 ls 19 rd osQ iz'uksa esa lgh mÙkj dk p;u dhft,A     16. ;fn nks lfn'kksa a vkSj b osQ chp dk dks.k θ gS rks a ⋅ b ≥ 0 gksxk ;fn% π π (A) 0 < θ < (B) 0 ≤ θ ≤ 2 2 (C) 0 < θ < π (D) 0 ≤ θ ≤ π     17. eku yhft, a vkjS b nks ek=kd lfn'k gSa vkSj muosQ chp dk dks.k θ gS rks a + b ,d ek=kd lfn'k gS ;fn% π π π 2π (A) θ = (B) θ = (C) θ = (D) θ = 4 3 2 3 Rationalised 2023-24  lfn'k chtxf.kr 383 18. iˆ.( ˆj × kˆ) + ˆj.(iˆ × kˆ) + kˆ.(iˆ × ˆj ) dk eku gS (A) 0 (B) –1 (C) 1 (D) 3       19. ;fn nks lfn'kksa a vkSj b osQ chp dk dks.k θ gS rks | a ⋅ b | = | a × b | tc θ cjkcj gS% π π (A) 0 (B) (C) (D) π 4 2 lkjak'k


  ® ,d ¯cnq P(x, y, z) dh fLFkfr lfn'k OP(= r ) = xiˆ + yjˆ + zkˆ gS vkSj ifjek.k x 2 + y 2 + z 2 gSA ® ,d lfn'k osQ vfn'k ?kVd blosQ fno~Q&vuqikr dgykrs gSa vkSj Øekxr v{kksa osQ lkFk blosQ iz{ksi dks fu:fir djrs gSaA ® ,d lfn'k dk ifjek.k (r), fno~Q&vuqikr a, b, c vkSj fno~Q&dkslkbu (l, m, n) fuEufyf[kr :i esa lacaf/r gSa% a b c l= , m= , n= r r r  ® f=kHkqt dh rhuksa Hkqtkvksa dks Øe esa ysus ij mudk lfn'k ;ksx 0 gSA ® nks lg&vkfne lfn'kksa dk ;ksx ,d ,sls lekarj prqHkqZt osQ fod.kZ ls izkIr gksrk gS ftldh layXu Hkqtk,¡ fn, gq, lfn'k gSaA ® ,d lfn'k dk vfn'k λ ls xq.ku blosQ ifjek.k dks | λ | osQ xq.kt esa ifjofrZr dj nsrk gS vkSj λ dk eku /ukRed vFkok ½.kkRed gksus osQ vuqlkj bldh fn'kk dks leku vFkok foijhr j[krk gSA  a ® fn, gq, lfn'k a osQ fy, lfn'k aˆ =  ] a dh fn'kk esa ek=kd lfn'k gSA |a |   ® fcnqvksa P vkSj Q ftuosQ fLFkfr lfn'k Øe'k% a vkjS b gSa] dks feykus okyh js[kk dks   na + mb m : n osQ vuqikr esa foHkkftr djus okys ¯cnq R dk fLFkfr lfn'k (i) var%  m + n  mb − na foHkktu ij (ii) cká foHkktu ij] osQ :i esa izkIr gksrk gSA m−n Rationalised 2023-24 384 xf.kr       ® nks lfn'kksa a vkjS b osQ chp dk dks.k θ gS rks mudk vfn'k xq.kuiQy a ⋅ b = | a || b | cos θ   osQ :i esa izkIr gksrk gSA ;fn a ⋅ b fn;k gqvk gS rks lfn'kksa a vkjS b osQ chp dk dks.k   a ⋅b ‘θ’, cos θ =   ls izkIr gksrk gSA | a || b |  ® ;fn nks lfn'kksa a vkjS b osQ chp dk dks.k θ gS rks mudk lfn'k xq.kuiQy     a × b = | a || b | sin θ nˆ osQ :i esa izkIr gksrk gSA tgk¡ n̂ ,d ,slk ek=kd lfn'k gS tks     a vkjS b dks lfEefyr djus okys ry osQ yacor~ gS rFkk a , b vkSj nˆ nf{k.kkorhZ ledksf.kd funsZ'kkad i¼fr dks fufeZr djrs gSaA   ® ;fn a = a1iˆ + a2 ˆj + a3 kˆ rFkk b = b1iˆ + b2 ˆj + b3kˆ vkSj λ ,d vfn'k gS rks   a + b = (a1 + b1 ) iˆ + (a2 + b2 ) ˆj + ( a3 + b3 ) kˆ  λa = (λa1 )iˆ + (λa2 ) ˆj + (λa3 ) kˆ  a.b = a1b1 + a2 b2 + a3 b3 iˆ ˆj kˆ   vkSj a × b = a1 b1 c1 a2 b2 c2 ,sfrgkfld i`"BHkwfe lfn'k 'kCn dk O;qRiUu ySfVu Hkk"kk osQ ,d 'kCn osDVl (vectus) ls gqvk gS ftldk vFkZ gS gLrxr djukA vk/qfud lfn'k fl¼kar osQ Hkzw.kh; fopkj dh frfFk lu~ 1800 osQ vklikl ekuh tkrh gS] tc Caspar Wessel (1745&1818 bZ-) vkSj Jean Robert Argand (1768-1822bZ-) us bl ckr dk o.kZu fd;k fd ,d funsZ'kkad ry esa fdlh fn"V js[kk[kaM dh lgk;rk ls ,d lfEeJ la[;k a + ib dk T;kferh; vFkZ fuoZpu dSls fd;k tk ldrk gSA ,d vk;fj'k xf.krK] William Rowen Hamilton (1805-1865 bZ-) us viuh iqLrd] "Lectures on Quaternions" (1853 bZ-) esa fn"V js[kk[kaM osQ fy, lfn'k 'kCn dk iz;ksx lcls igys fd;k FkkA prq"V;h;ksa (quaternians) [oqQN fuf'pr chth; fu;eksa dk ikyu djrs gq, a + b iˆ + c ˆj + d kˆ, iˆ, ˆj , kˆ osQ :i okys pkj okLrfod la[;kvksa dk Rationalised 2023-24  lfn'k chtxf.kr 385 leqPp;] dh gSfeYVu fof/ lfn'kksa dks f=k&foeh; varfj{k esa xq.kk djus dh leL;k dk ,d gy FkkA rFkkfi ge ;gk¡ bl ckr dk ftØ vo'; djsaxs fd lfn'k dh ladYiuk vkSj muosQ ;ksxiQy dk fopkj cgqr& fnuksa igys ls Plato (384-322 bZlk iwoZ) osQ ,d f'k"; ,oa ;wukuh nk'kZkfud vkSj oSKkfud Aristotle (427-348 bZlk iwoZ) osQ dky ls gh FkkA ml le; bl tkudkjh dh dYiuk Fkh fd nks vFkok vf/d cyksa dh la;qDr fØ;k mudks lekarj prqHkqZt osQ fu;ekuqlkj ;ksx djus ij izkIr dh tk ldrh gSA cyksa osQ la;kstu dk lgh fu;e] fd cyksa dk ;ksx lfn'k :i esa fd;k tk ldrk gS] dh [kkst Sterin Simon(1548-1620bZ-) }kjk yacor~ cyksa dh fLFkfr esa dh xbZA lu~ 1586 esa mUgksaus viuh 'kks/iqLrd] "DeBeghinselen der Weeghconst" (otu djus dh dyk osQ fl¼kar) esa cyksa osQ ;ksxiQy osQ T;kferh; fl¼kar dk fo'ys"k.k fd;k Fkk ftlosQ dkj.k ;kaf=kdh osQ fodkl esa ,d eq[; ifjorZu gqvkA ijarq blosQ ckn Hkh lfn'kksa dh O;kid ladYiuk osQ fuekZ.k esa 200 o"kZ yx x,A lu~ 1880 esa ,d vesfjdh HkkSfrd 'kkL=kh ,oa xf.krK Josaih Willard Gibbs (1839-1903 bZ-) vkSj ,d vaxzst vfHk;ark Oliver Heaviside (1850-1925 bZ-) us ,d prq"V;h osQ okLrfod (vfn'k) Hkkx dks dkYifud (lfn'k) Hkkx ls i`Fko~Q djrs gq, lfn'k fo'ys"k.k dk l`tu fd;k FkkA lu~ 1881 vkSj 1884 esa Gibbs us "Entitled Element of Vector Analysis" uked ,d 'kks/ iqfLrdk NiokbZA bl iqLrd esa lfn'kksa dk ,d Øec¼ ,oa laf{kIr fooj.k fn;k gqvk FkkA rFkkfi lfn'kksa osQ vuqiz;ksx dk fu:i.k djus dh dhfrZ D. Heaviside vkSj P.G. Tait (1831-1901 bZ-) dks izkIr gS ftUgksaus bl fo"k; osQ fy, lkFkZd ;ksxnku fn;k gSA —v— Rationalised 2023-24