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 

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] 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] laosx] 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 Hkh ,d js[kk fn"V js[kk dgykrh gS [vko`Qfr 10.1 (i), (ii)]A

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  vFkok 

lk/kj.kr% , osQ :i esa fufnZ"V djrs gSa vkSj bls lfn'k ^* vFkok lfn'k ^* osQ :i esa i<+rs gSaA

og ¯cnq A tgk¡ ls lfn'k  izkjaHk gksrk gS] izkjafHkd ¯cnq dgykrk gS vkSj og ¯cnq B tgk¡ ij lfn'k , lekIr gksrk gS vafre ¯cnq dgykrk gSA fdlh lfn'k osQ izkjafHkd ,oa vafre ¯cnqvksa osQ chp dh nwjh lfn'k dk ifjek.k (vFkok yackbZ) dgykrk gS vkSj bls || vFkok || osQ :i esa fufnZ"V fd;k tkrk gSA rhj dk fu'kku lfn'k dh fn'kk dks fufnZ"V djrk gSA

fLFkfr lfn'k (Position Vector)

d{kk XI ls] f=k&foeh; nf{k.kkorhZ ledksf.kd funsZ'kkad i¼fr dks Lej.k dhft,
(vko`Qfr 10-2(i))A varfj{k esa ewy ¯cnq O(0, 0, 0) osQ lkis{k ,d ,slk ¯cnq P yhft, ftlosQ funsZ'kkad (x, y, z) gSA rc lfn'k  ftlesa O vkSj P Øe'k% izkjafHkd ,oa vafre ¯cnq gSa] O osQ

lkis{k ¯cnq P dk fLFkfr lfn'k dgykrk gSA nwjh lw=k (d{kk XI ls) dk mi;ksx djrs gq,  (vFkok) dk ifjek.k fuEufyf[kr :i esa izkIr gksrk gS%

 =

O;ogkj esa ewy ¯cnq O osQ lkis{k] ¯cnqvksa A, B, C bR;kfn osQ fLFkfr lfn'k Øe'k%  ls fufnZ"V fd, tkrs gaS [vko`Qfr 10.2(ii)]A

fno~Q&dkslkbu (Direction Cosines)

,d ¯cnq P(x, y, z) dk fLFkfr lfn'k  yhft, tSlk fd vko`Qfr 10-3 esa n'kkZ;k x;k gSA lfn'k  }kjk x, y ,oa z-v{k dh /ukRed fn'kkvksa osQ lkFk cuk, x, Øe'k% dks.k α, β, ,oa γ fn'kk dks.k dgykrs gSaA bu dks.kksa osQ dkslkbu eku vFkkZr~ cosα, cosβ ,oa cosγ lfn'k  osQ fno~Q&dkslkbu dgykrs gSa vkSj lkekU;r% budks Øe'k% l, m ,oa n ls fufnZ"V fd;k tkrk gSA

vko`Qfr 10.3, ls ge ns[krs gSa fd f=kHkqt OAP ,d ledks.k f=kHkqt gS vkSj bl f=kHkqt ls ge izkIr djrs gSaA blh izdkj ledks.k f=kHkqtksa OBP ,oa OCP ls ge fy[k ldrs gSaA bl izdkj ¯cnq P osQ funsZ'kkadksa dks (lr, mr, nr) osQ :i esa vfHkO;Dr fd;k tk ldrk gSA fno~Q&dkslkbu osQ lekuqikrh la[;k,¡ lr, mr ,oa nr lfn'k  osQ fno~Q&vuqikr dgykrs gSa vkSj budks Øe'k% a, b rFkk c ls fufnZ"V fd;k tkrk gSA

10.3 lfn'kksa osQ izdkj (Types of Vectors)

'kwU; lfn'k [Zero (null) Vector] ,d lfn'k ftlosQ izkjafHkd ,oa vafre ¯cnq laikrh gksrs gSa] 'kwU; lfn'k dgykrk gS vkSj bls  osQ :i esa fufnZ"V fd;k tkrk gSA 'kwU; lfn'k dks dksbZ fuf'pr fn'kk iznku ugha dh tk ldrh D;ksafd bldk ifjek.k 'kwU; gksrk gS vFkok fodYir% bldks dksbZ Hkh fn'kk /kj.k fd, gq, ekuk tk ldrk gSA lfn'k  'kwU; lfn'k dks fu:fir djrs gSaA

ek=kd lfn'k (Unit Vector) ,d lfn'k ftldk ifjek.k ,d (vFkok 1 bdkbZ) gS ek=kd lfn'k dgykrk gSA fdlh fn, gq, lfn'k  dh fn'kk esa ek=kd lfn'k dks ls fufnZ"V fd;k tkrk gSA

lg&vkfne lfn'k (Co-initial Vectors) nks vFkok vf/d lfn'k ftudk ,d gh izkjafHkd ¯cnq gS] lg vkfne lfn'k dgykrs gSaA

lajs[k lfn'k (Collinear Vectors) nks vFkok vf/d lfn'k ;fn ,d gh js[kk osQ lekarj gS rks os lajs[k lfn'k dgykrs gSaA

leku lfn'k (Equal Vectors) nks lfn'k  leku lfn'k dgykrs gSa ;fn muosQ ifjek.k ,oa fn'kk leku gSaA budks  osQ :i esa fy[kk tkrk gSA

½.kkRed lfn'k (Negative of a Vector) ,d lfn'k ftldk ifjek.k fn, gq, lfn'k (eku yhft, ) osQ leku gS ijarq ftldh fn'kk fn, gq, lfn'k dh fn'kk osQ foijhr gS] fn, gq, lfn'k dk ½.kkRed dgykrk gSA mnkgj.kr% lfn'k , lfn'k  dk ½.kkRed gS vkSj bls osQ :i esa fy[kk tkrk gSA

fVIi.kh mi;qZDr ifjHkkf"kr lfn'k bl izdkj gS fd muesa ls fdlh dks Hkh mlosQ ifjek.k ,oa fn'kk dks ifjofrZr fd, fcuk Lo;a osQ lekarj foLFkkfir fd;k tk ldrk gSA bl izdkj osQ lfn'k Lora=k lfn'k dgykrs gSaA bl iwjs vè;k; esa ge Lora=k lfn'kksa dh gh ppkZ djsaxsA

mnkgj.k 1 nf{k.k ls 30° if'pe esa] 40 km osQ foLFkkiu dk vkys[kh; fu:i.k dhft,A

gy lfn'k  vHkh"V foLFkkiu dks fu:fir djrk gS (vko`Qfr 10-4 nsf[k,)A

mnkgj.k 2 fuEufyf[kr ekiksa dks vfn'k ,oa lfn'k osQ :i esa Js.khc¼ dhft,A

(i) 5 s (ii) 1000 cm3 (iii) 10 N

(iv) 30 km/h (v) 10 g/cm3

(vi) 20 m/s mÙkj dh vksj

gy

(i) le;&vfn'k (ii) vk;ru&vfn'k (iii) cy&lfn'k

(iv) xfr&vfn'k (v) ?kuRo&vfn'k (vi) osx&lfn'k

mnkgj.k 3 vko`Qfr 10-5 esa dkSu ls lfn'k

(i) lajs[k gSa

(ii) leku gSa

(iii) lg&vkfne gSa

gy

iz'ukoyh 10-1

1. mÙkj ls 30° iwoZ esa 40 km osQ foLFkkiu dk vkys[kh; fu:i.k dhft,A

2. fuEufyf[kr ekikas dks vfn'k ,oa lfn'k osQ :i esa Js.khc¼ dhft,A

(i) 10 kg (ii) 2 ehVj mÙkj&if'pe (iii) 40°

(iv) 40 okV (v) 10–19 owQyac (vi) 20 m/s2

3. fuEufyf[kr dks vfn'k ,oa lfn'k jkf'k;ksa osQ :i esa Js.khc¼ dhft,A

(i) le; dkyka'k (ii) nwjh (iii) cy

(iv) osx (v) dk;Z

4. vko`Qfr 10-6 (,d oxZ) esa fuEufyf[kr lfn'kksa dks igpkfu,A

(i) lg&vkfne (ii) leku

(iii) lajs[k ijarq vleku

5. fuEufyf[kr dk mÙkj lR; vFkok vlR; osQ :i esa nhft,A

(i)  rFkk lajs[k gSaA

(ii) nks lajs[k lfn'kksa dk ifjek.k lnSo leku gksrk gSA

(iii) leku ifjek.k okys nks lfn'k lajs[k gksrs gSaA

(iv) leku ifjek.k okys nks lajs[k lfn'k leku gksrs gSaA

10.4 lfn'kksa dk ;ksxiQy (Addition of Vectors)

lfn'k  ls lk/kj.kr% gekjk rkRi;Z gS ¯cnq A ls ¯cnq B rd foLFkkiuA vc ,d ,slh fLFkfr dh ppkZ dhft, ftlesa ,d yM+dh ¯cnq A ls ¯cnq B rd pyrh gS vkSj mlosQ ckn ¯cnq B ls ¯cnq C rd pyrh gS (vko`Qfr 10.7)A ¯cnq A ls ¯cnq C rd yM+dh }kjk fd;k x;k oqQy foLFkkiu lfn'k,  ls izkIr gksrk gS vkSj bls  =  osQ :i esa vfHkO;Dr fd;k tkrk gSA

;g lfn'k ;ksx dk f=kHkqt fu;e dgykrk gSA

lkekU;r%] ;fn gekjs ikl nks lfn'k  rFkk  gSa [vko`Qfr 10.8 (i)], rks mudk ;ksx Kkr djus osQ fy, mUgsa bl fLFkfr esa yk;k tkrk gS] rkfd ,d dk izkjafHkd ¯cnq nwljs osQ vafre ¯cnq osQ laikrh gks tk, [vko`Qfr 10.8(ii)]A

mnkgj.kr% vko`Qfr 10.8 (ii) esa] geus lfn'k  osQ ifjek.k ,oa fn'kk dks ifjofrZr fd, fcuk bl izdkj LFkkukarfjr fd;k gS rkfd bldk izkjafHkd ¯cnq]  osQ vafre ¯cnq osQ laikrh gS rc f=kHkqt ABC dh rhljh Hkqtk AC }kjk fu:fir lfn'k  gesa lfn'kksa  rFkk  dk ;ksx (vFkok ifj.kkeh) iznku djrk gS] vFkkZr~~ f=kHkqt ABC esa ge ikrs gSa fd =  [vko`Qfr 10.8 (ii)]A

vc iqu% D;ksafd, blfy, mi;qZDr lehdj.k ls ge ikrs gSa fd

bldk rkRi;Z ;g gS fd fdlh f=kHkqt dh Hkqtkvksa dks ;fn ,d Øe esa fy;k tk, rks ;g 'kwU; ifj.kkeh dh vksj izsfjr djrk gS D;ksafd izkjafHkd ,oa vafre ¯cnq laikrh gks tkrs gSa [vko`Qfr 10.8(iii)]A

vc ,d lfn'k  dh jpuk bl izdkj dhft, rkfd bldk ifjek.k lfn'k , osQ ifjek.k osQ leku gks] ijarq bldh fn'kk dh fn'kk osQ foijhr gks vko`Qfr 10-8(iii) vFkkZr~~

 =  rc f=kHkqt fu;e dk vuqiz;ksx djrs gq, [vko`Qfr 10-8(iii)] ls ge ikrs gSa fd  

lfn'k,  osQ varj dks fu:fir djrk gSA

vc fdlh unh osQ ,d fdukjs ls nwljs fdukjs rd ikuh osQ cgko dh fn'kk osQ yacor~~ tkus okyh ,d uko dh ppkZ djrs gSaA rc bl uko ij nks osx lfn'k dk;Z dj jgs gSa] ,d batu }kjk uko dks fn;k x;k osx vkSj nwljk unh osQ ikuh osQ cgko dk osxA bu nks osxksa osQ ;qxir izHkko ls uko okLro esa ,d fHkUu osx ls pyuk 'kq: djrh gSA bl uko dh izHkkoh xfr ,oa fn'kk (vFkkZr~~ ifj.kkeh osx) osQ ckjs esa ;FkkFkZ fopkj ykus osQ fy, gekjs ikl lfn'k ;ksxiQy dk fuEufyf[kr fu;e gSA

;fn gekjs ikl ,d lekarj prqHkqZt dh
nks layXu Hkqtkvksa ls fu:fir fd, tkus okys (ifjek.k ,oa fn'kk lfgr) nks lfn'k  gS (vko`Qfr 10-9) rc lekarj prqHkqZt dh bu nksuksa Hkqtkvksa osQ mHk;fu"B ¯cnq ls xqtjus okyk fod.kZ bu nksuksa lfn'kksa osQ ;ksx  dks ifjek.k ,oa fn'kk lfgr fu:fir djrk gSA ;g lfn'k ;ksx dk lekarj prqHkqZt fu;e dgykrk gSA

lfn'k ;ksxiQy osQ xq.k/eZ (Properties of vector addition)

xq.k/eZ 1 nks lfn'kksa  osQ fy,

 =  (Øefofue;rk)

miifÙk lekarj prqHkqZt ABCD dks yhft, (vko`Qfr 10-10) eku yhft,  rc f=kHkqt ABC esa f=kHkqt fu;e dk mi;ksx djrs gq, ge ikrs gSa fd=

vc] D;ksafd lekarj prqHkqZt dh lEeq[k Hkqtk,¡ leku ,oa lekarj gS] blfy, vko`Qfr 10-10 esa gSA iqu% f=kHkqt ADC esa f=kHkqt fu;e osQ iz;ksx ls   = 

vr%  = 

xq.k/eZ 2 rhu lfn'kksa osQ fy,  (lkgp;Z xq.k)

miifÙk eku yhft,] lfn'kksa  dks Øe'k%  ls fu:fir fd;k x;k gS tSlk fd vko`Qfr 10.11(i) vkSj (ii) esa n'kkZ;k x;k gSA

rc 

vkSj

blfy,

vkSj

vr%

uksV dhft, fd fdlh lfn'k  osQ fy, ge ikrs gSa%

;gk¡ 'kwU; lfn'k  lfn'k ;ksxiQy osQ fy, ;ksT; loZlfedk dgykrk gSA

10.5 ,d vfn'k ls lfn'k dk xq.ku (Multiplication of a Vector by a Scalar)

eku yhft, fd  ,d fn;k gqvk lfn'k gS vkSj λ ,d vfn'k gSA rc lfn'k  dk vfn'k λ, ls xq.kuiQy ftls λ osQ :i esa fufnZ"V fd;k tkrk gS] lfn'k  dk vfn'k λ ls xq.ku dgykrk gSA uksV dhft, fd λ Hkh lfn'k  osQ lajs[k ,d lfn'k gSA λ osQ eku /ukRed vFkok ½.kkRed gksus osQ vuqlkj λ dh fn'kk]  osQ leku vFkok foijhr gksrh gSA λ dk ifjek.k osQ ifjek.k dk |λ| xq.kk gksrk gS] vFkkZr~~

,d vfn'k ls lfn'k osQ xq.ku dk T;kferh; pk{kq"khdj.k [:i dh dYiuk (visualisation)] vko`Qfr 10-12 esa nh xbZ gSA

tc λ = – 1, rc tks ,d ,slk lfn'k gS ftldk ifjek.k  osQ leku gS vkSj fn'kk dh fn'kk osQ foijhr gSA lfn'k – lfn'k dk ½.kkRed (vFkok ;ksT; izfrykse)dgykrk gS vkSj ge ges'kk 

vkSj ;fn , fn;k gqvk gS fd  ,d 'kwU; lfn'k ugha gS rc

bl izdkj λ, dh fn'kk esa ek=kd lfn'k dks fu:fir djrk gSA ge bls

= osQ :i esa fy[krs gSaA

10.5.1 ,d lfn'k osQ ?kVd (Components of a vector)

vkbZ, ¯cnqvksa A(1, 0, 0), B(0, 1, 0) vkSj C(0, 0, 1) dks Øe'k% x-v{k, y-v{k ,oa z-v{k ij ysrs gSaA rc Li"Vr%

lfn'k  ftuesa ls izR;sd dk ifjek.k 1 gSa Øe'k% OX, OY vkSj OZ v{kksa osQ vuqfn'k ek=kd lfn'k dgykrs gSa vkSj budks Øe'k% }kjk fufnZ"V fd;k tkrk gS (vko`Qfr 10-13)A

vc ,d ¯cnq P(x, y, z) dk fLFkfr lfn'k  yhft, tSlk fd vko`Qfr 10-14 esa n'kkZ;k x;k gSA eku yhft, fd ¯cnq P1 ls ry XOY ij [khaps x, yac dk ikn ¯cnq P1 gSA bl izdkj ge ns[krs gSa fd P1 P, z-v{k osQ lekarj gSA D;ksafd Øe'k% x, y ,oa z-v{kosQ vuqfn'k ek=kd lfn'k gS vkSj P osQ funsZ'kkadksa dh ifjHkk"kk osQ vuqlkj ge ikrs gSa fd. blh izdkj. bl izdkj ge ikrs gSa fd

vkSj

bl izdkj O osQ lkis{k P dk fLFkfr lfn'k osQ :i esa izkIr gksrk gSA

fdlh Hkh lfn'k dk ;g :i ?kVd :i dgykrk gSA ;gk¡ x, y ,oa z,   osQ vfn'k ?kVd dgykrs gSa vkSj Øekxr v{kksa osQ vuqfn'k  osQ lfn'k ?kVd dgykrs gSaA dHkh&dHkh x, y ,oa z dks ledksf.kd ?kVd Hkh dgk tkrk gSA

fdlh lfn'k , dh yackbZ ikbFkkxksjl izes; dk nks ckj iz;ksx djosQ rqjar Kkr dh tk ldrh gSA ge uksV djrs gSa fd ledks.k f=kHkqt OQP1 esa (vko`Qfr 10.14)

vkSj ledks.k f=kHkqt OP1P, esa ge ikrs gSa fd

vr% fdlh lfn'k   osQ :i esa izkIr gksrh gSA

;fn nks lfn'k  ?kVd :i esa Øe'k% vkSj }kjk fn, x, gSa rks

(i) lfn'kksa dks ;ksx

osQ :i esa izkIr gksrk gSA

(ii) lfn'k   dk varj

 osQ :i esa izkIr gksrk gSA

(iii) lfn'k   leku gksrs gSa ;fn vkSj osQoy ;fn

a1 = b1, a2 = b2 vkSj a3 = b3

(iv) fdlh vfn'k λ ls lfn'k  dk xq.ku

= }kjk iznÙk gSA

lfn'kksa dk ;ksxiQy vkSj fdlh vfn'k ls lfn'k dk xq.ku lfEefyr :i esa fuEufyf[kr forj.k&fu;e ls feyrk gS

eku yhft, fd   dksbZ nks lfn'k gSa vkSj k ,oa m nks vfn'k gSa rc

fVIi.kh

1. vki izsf{kr dj ldrs gSa fd λ osQ fdlh Hkh eku osQ fy, lfn'k ges'kk lfn'k  osQ lajs[k gSA okLro esa nks lfn'k  lajs[k rHkh gksrs gSa ;fn vkSj osQoy ;fn ,d ,sls 'kwU;srj vfn'k λ dk vfLrRo gSa rkfd  gksA ;fn lfn'k  ?kVd :i esa fn, gq, gSa] vFkkZr~~  rc nks lfn'k lajs[k gksrs gSa ;fn vkSj osQoy ;fn

=

⇔ =

⇔ ,

⇔ =

2. ;fn  = rc a1, a2, a3 lfn'k  osQ fno~Q&vuqikr dgykrs gSaA

3. ;fn l, m, n fdlh lfn'k osQ fno~Q&dkslkbu gSa rc

=

fn, gq, lfn'k dh fn'kk esa ek=kd lfn'k gS tgk¡ α, β ,oa γ fn, gq, lfn'k }kjk Øe'k% x, y ,oa z v{k osQ lkFk cuk, x, dks.k gSaA

mnkgj.k 4 x, y vkSj z osQ eku Kkr dhft, rkfd lfn'k  leku gSaA

gy è;ku nhft, fd nks lfn'k leku gksrs gSa ;fn vkSj osQoy ;fn muosQ laxr ?kVd leku gSA vr% fn, gq, lfn'k  leku gksaxs ;fn vkSj osQoy ;fn x = 2, y = 2, z = 1

mnkgj.k 5 eku yhft, gS? D;k lfn'k   leku gSa\

gy ;gk¡ 

blfy,   ijarq fn, gq, lfn'k leku ugha gSa D;ksafd buosQ laxr ?kVd fHkUu gSaA

mnkgj.k 6 lfn'k osQ vuqfn'k ek=kd lfn'k Kkr dhft,A

gy lfn'k   osQ vuqfn'k ek=kd lfn'k }kjk izkIr gksrk gSA

vc =

blfy, =

mnkgj.k 7 lfn'k osQ vuqfn'k ,d ,slk lfn'k Kkr dhft, ftldk ifjek.k 7 bdkbZ gSA

gy fn, gq, lfn'k  osQ vuqfn'k ek=kd lfn'k =gSA

blfy,  osQ vuqfn'k vkSj 7 ifjek.k okyk lfn'k = = gSA

mnkgj.k 8 lfn'kksa osQ ;ksxiQy osQ vuqfn'k ek=kd lfn'k Kkr dhft,A

gy fn, gq, lfn'kksa dk ;ksxiQy

vkSj =

vr% vHkh"V ek=kd lfn'k

gSA

mnkgj.k 9 lfn'k osQ fno~Q&vuqikr fyf[k, vkSj bldh lgk;rk ls fno~Q&dkslkbu Kkr dhft,A

gy è;ku nhft, fd lfn'k osQ fno~Q&vuqikr a, b, c lfn'k osQ] Øekxr ?kVd x, y, z gksrs gSaA blfy, fn, gq, lfn'k osQ fy, ge ikrs gSa fd a = 1, b = 1 vkSj c = –2 gSA iqu% ;fn l, m vkSj n fn, gq, lfn'k osQ fno~Q&dkslkbu gSa rks%

vr% fno~Q&dkslkbu gSaA

10.5.2 nks ¯cnqvksa dks feykus okyk lfn'k (Vector joining two points)

;fn P1(x1, y1, z1) vkSj P2(x2, y2, z2) nks ¯cnq gSa rc P1 dks P2 ls feykus okyk lfn'k  gS (vko`Qfr 10-15)A P1 vkSj P2 dks ewy ¯cnq O ls feykus ij vkSj f=kHkqt fu;e dk iz;ksx djus ij ge f=kHkqt OP1P2 ls ikrs gSa fd 

lfn'k ;ksxiQy osQ xq.k/eks± dk mi;ksx djrs gq, mi;qZDr lehdj.k fuEufyf[kr :i ls fy[kk tkrk gSA

vFkkZr~~

lfn'k dk ifjek.k osQ :i esa izkIr gksrk gSA

mnkgj.k10 ¯cnqvksa P(2, 3, 0) ,oa Q(– 1, – 2, – 4) dks feykus okyk ,oa P ls Q dh rjiQ fn"V lfn'k Kkr dhft,A

gy D;ksafd lfn'k P ls Q dh rjiQ fn"V gS] Li"Vr% P izkjafHkd ¯cnq gS vkSj Q vafre ¯cnq gS] blfy, P vkSj Q dks feykus okyk vHkh"V lfn'k ,fuEufyf[kr :i esa izkIr gksrk gSA

vFkkZr~~ 

10.5.3 [kaM lw=k (Section Formula)

eku yhft, ewy ¯cnq O osQ lkis{k P vkSj Q nks ¯cnq gSa ftudks fLFkfr lfn'k  ls fu:fir fd;k x;k gSA ¯cnqvksa P ,oa Q dks feykus okyk js[kk [kaM fdlh rhljs ¯cnq R }kjk nks izdkj ls foHkkftr fd;k tk ldrk gSA var% (vko`Qfr 10-16) ,oa cká (vko`Qfr 10-17)A ;gk¡ gekjk mn~ns'; ewy ¯cnq O osQ lkis{k ¯cnq R dk fLFkfr lfn'k  Kkr djuk gSA ge nksuksa fLFkfr;ksa dks ,d&,d djosQ ysrs gSaA

fLFkfr 1 tc R, PQ dks var% foHkkftr djrk gS (vko`Qfr 10-16)A ;fn R, dks bl izdkj foHkkftr djrk gS fd ,tgk¡ m vkSj n /ukRed vfn'k gSa rks ge dgrs gSa fd ¯cnq R,  dks m : n osQ vuqikr esa var% foHkkftr djrk gSA vc f=kHkqtksa ORQ ,oa OPR ls

vkSj

blfy, = (D;ksa\)

vFkok  = (ljy djus ij)

vr% ¯cnq R tks fd P vkSj Q dks m : n osQ vuqikr esa var% foHkkftr djrk gS dk fLFkfr lfn'k

fLFkfr II tc R, PQ dks cká foHkkftr djrk gS (vko`Qfr 10-17)A ;g lR;kiu djuk ge ikBd osQ fy, ,d iz'u osQ :i esa NksM+rs gSa fd js[kk[kaM PQ dks m : n osQ vuqikr esa cká foHkkftr djus okys ¯cnq R dk fLFkfr lfn'k  = osQ :i esa izkIr gksrk gSA

fVIi.kh ;fn R, PQ dk eè; ¯cnq gS rks m = n vkSj blfy, fLFkfr i ls  osQ eè; ¯cnq R dk fLFkfr lfn'k  = osQ :i esa gksxkA

mnkgj.k 11 nks ¯cnq P vkSj Q yhft, ftuosQ fLFkfr lfn'k  gSaA ,d ,sls ¯cnq R dk fLFkfr lfn'k Kkr dhft, tks P ,oa Q dks feykus okyh js[kk dks 2:1 osQ vuqikr esa (i) var% (ii) cká foHkkftr djrk gSA

gy

(i) P vkSj Q dks feykus okyh js[kk dks 2:1 osQ vuqikr esa var% foHkkftr djus okys ¯cnq R dk fLFkfr lfn'k gS%

(ii) P vkSj Q dks feykus okyh js[kk dks 2:1 osQ vuqikr esa cká foHkkftr djus okys ¯cnq R dk fLFkfr lfn'k gS%

mnkgj.k 12 n'kkZb, fd ¯cnq ,d ledks.k f=kHkqt osQ 'kh"kZ gSaA

gy ge ikrs gSa fd

 =

 =

vkSj =

blosQ vfrfjDr è;ku nhft, fd

vr% fn;k gqvk f=kHkqt ,d ledks.k f=kHkqt gSA

iz'ukoyh 10-2

1. fuEufyf[kr lfn'kksa osQ ifjek.k dk ifjdyu dhft,%

2. leku ifjek.k okys nks fofHkUu lfn'k fyf[k,A

3. leku fn'kk okys nks fofHkUu lfn'k fyf[k,A

4. x vkSj y osQ eku Kkr dhft, rkfd lfn'k leku gksaA

5. ,d lfn'k dk izkjafHkd ¯cnq (2] 1) gS vkSj vafre ¯cnq (&5] 7) gSA bl lfn'k osQ vfn'k ,oa lfn'k ?kVd Kkr dhft,A

6. lfn'k  dk ;ksxiQy Kkr dhft,A

7. lfn'k osQ vuqfn'k ,d ek=kd lfn'k Kkr dhft,A

8. lfn'k  osQ vuqfn'k ek=kd lfn'k Kkr dhft, tgk¡ ¯cnq P vkSj Q Øe'k% (1] 2] 3) vkSj (4] 5] 6) gSaA

9. fn, gq, lfn'kksa   osQ vuqfn'k ek=kd lfn'k Kkr dhft,A

10. lfn'k osQ vuqfn'k ,d ,slk lfn'k Kkr dhft, ftldk ifjek.k 8 bdkbZ gSA

11. n'kkZb, fd lfn'k lajs[k gSaA

12. lfn'k dh fno~Q cosine Kkr dhft,A

13. ¯cnqvksa A(1, 2, –3) ,oa B(–1, –2, 1) dks feykus okys ,oa A ls B dh rji+Q fn"V lfn'k dh fno~Q cosine Kkr dhft,A

14. n'kkZb, fd lfn'k v{kksa OX, OY ,oa OZ osQ lkFk cjkcj >qdk gqvk gSA

15. ¯cnqvksa P (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%  vkSj gSa] ,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

19. ;fn  nks lajs[k lfn'k gSa rks fuEufyf[kr esa ls dkSu lk dFku lgh ugha gS%

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 vuqiz;ksx gSaA bl ifjPNsn esa ge bu nks izdkj osQ xq.kuiQyksa dh ppkZ djsaxsA

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  dk vfn'k xq.kuiQy  }kjk fufnZ"V fd;k tkrk gS vkSj bls   =osQ :i es ifjHkkf"kr fd;k tkrk gSA

tgk¡ θ,  (vko`Qfr 10-19)A

;fn   rks θ ifjHkkf"kr ugha gS vkSj bl fLFkfr eas ge ifjHkkf"kr djrs gSaA

izs{k.k

1.  ,d okLrfod la[;k gSA

2. eku yhft, fd   nks 'kwU;srj lfn'k gSa rc  ;fn vkSj osQoy ;fn  ijLij yacor~~ gSa vFkkZr~~ 

3.

4. ;fn θ = π, rc 

fof'k"Vr%  , 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 osQ fy, ge ikrs gSa fd

=

rFkk =

6. nks 'kwU;srj lfn'kksa osQ chp dk dks.k θ,

7. vfn'k xq.kuiQy Øe fofues; gS vFkkZr~~

 (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,  rhu lfn'k gSa rc   = 

xq.k/eZ 2 eku yhft,   nks lfn'k gSa vkSj λ ,d vfn'k gS] rks

;fn nks lfn'k ?kVd :i esa ,oa , fn, gq, gSa rc mudk vfn'k xq.kuiQy fuEufyf[kr :i esa izkIr gksrk gS

=

= +

=

+

(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 =

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  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  dk l ij iz{ksi ,d lfn'k (eku yhft,) gS ftldk ifjek.k  gS vkSj ftldh fn'kk dk l dh fn'kk osQ leku vFkok foijhr gksuk bl ckr ij fuHkZj gS fd  /ukRed gS vFkok ½.kkRedA lfn'k dks iz{ksi lfn'k dgrs gSa vkSj bldk ifjek.k ||, fu£n"V js[kk l ij lfn'k  dk iz{ksi dgykrk gSA mnkgj.kr% fuEufyf[kr esa ls izR;sd vko`Qfr esa lfn'k  dk js[kk l ij iz{ksi lfn'k  gSA [vko`Qfr 10.20 (i) ls (iv) rd]

izs{k.k

1. js[kk l osQ vuqfn'k ;fn ek=kd lfn'k gS rks js[kk l ij lfn'k   dk iz{ksi ls izkIr gksrk gSA

2. ,d lfn'k   dk nwljs lfn'k , ij iz{ksi vFkok ls izkIr gksrk gSA

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

4- ;fn vFkok rks  dk iz{ksi lfn'k 'kwU; lfn'k gksxkA

fVIi.kh ;fn α, β vkSj γ lfn'k  osQ fno~Q&dks.k gSa rks bldh fno~Q&dkslkbu fuEufyf[kr :i esa izkIr dh tk ldrh gSA

;g Hkh è;ku nhft, fd  Øe'k% OX, OY rFkk OZ osQ vuqfn'k  osQ iz{ksi gSa vFkkZr~~ lfn'k  osQ vfn'k ?kVd a1, a2 vkSj a3 Øe'k% x, y, ,oa z v{k osQ vuqfn'k  osQ iz{ksi gSA blosQ vfrfjDr ;fn  ,d ek=kd lfn'k gS rc bldks fno~Q&dkslkbu dh lgk;rk ls

osQ :i esa vfHkO;Dr fd;k tk ldrk gSA

mnkgj.k 13 nks lfn'kksa  osQ ifjek.k Øe'k% 1 vkSj 2 gS rFkk  , bu lfn'kksa osQ chp dk dks.k Kkr dhft,A

gy fn;k gqvk gS . vr%

mnkgj.k 14 lfn'k  osQ chp dk dks.k Kkr dhft,A

gy nks lfn'kksa  osQ chp dk dks.k θ fuEu }kjk iznÙk gS

ls izkIr gksrk gSA

vc  =

blfy,] ge ikrs gSa fd cosθ =

vr% vHkh"V dks.k θ = gSA

mnkgj.k 15 ;fn  vkSj yacor~ gSA

gy ge tkurs gSa fd nks 'kwU;srj lfn'k yacor~~ gksrs gSa ;fn mudk vfn'k xq.kuiQy 'kwU; gSA

mnkgj.k 16 lfn'k dk] lfn'k ij iz{ksi Kkr dhft,A

gy lfn'k  dk lfn'k  ij iz{ksi

 = gSA

mnkgj.k 17 ;fn nks lfn'kbl izdkj gSa fdKkr dhft,A

gy ge ikrs gSa fd

mnkgj.k 18 ;fn   ,d ek=kd lfn'k gS vkSj Kkr dhft,A

gy D;ksafd   ,d ek=kd lfn'k gS] blfy,  ;g Hkh fn;k gqvk gS fd

mnkgj.k 19 nks lfn'kksa , osQ fy, lnSo  (Cauchy-Schwartz vlfedk)A

gy nh gqbZ vlfedk lgt :i esa Li"V gS ;fn   okLro esa bl fLFkfr esa ge ikrs gSa fd . blfy, ge dYiuk djrs gSa fd rc gesa

feyrk gSA

blfy,

mnkgj.k 20 nks lfn'kksa  (f=kHkqt&vlfedk)

gy nh gqbZ vlfedk] nksuksa fLFkfr;ksa  esa lgt :i ls Li"V gS (D;ksa ?)A blfy, eku yhft, fd  rc

mnkgj.k 21 n'kkZb, fd ¯cnq vkSj lajs[k gSA

gy ge izkIr djrs gSa%

=

=

=

blfy,=

vr% ¯cnq A, B vkSj C lajs[k gSaA

iz'ukoyh 10-3

1. nks lfn'kksa  osQ chp dk dks.k Kkr dhft,A

2. lfn'kksa osQ chp dk dks.k Kkr dhft,A

3. lfn'k ij lfn'k dk iz{ksi Kkr dhft,A

4. lfn'k dk] lfn'k 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]

;g Hkh n'kkZb, fd ;s lfn'k ijLij ,d nwljs osQ yacor~ gSaA

6. ;fn Kkr dhft,A

7. dk eku Kkr dhft,A

8. nks lfn'kksa   osQ ifjek.k Kkr dhft,] ;fn buosQ ifjek.k leku gS vkSj bu osQ chp dk dks.k 60° gS rFkk budk vfn'k xq.kuiQy gSA

9. ;fn ,d ek=kd lfn'k  Kkr dhft,A

10. ;fn   ij yac gS] rks λ dk eku Kkr dhft,A

11. n'kkZb, fd nks 'kwU;srj lfn'kksa  osQ fy,  ij yac gSA

12. ;fn , rks lfn'k  osQ ckjs esa D;k fu"d"kZ fudkyk tk ldrk gS?

13. ;fn dk eku Kkr dhft,A

14. ;fn  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 ,oa  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 ,d ledks.k f=kHkqt osQ 'kh"kks± dh jpuk djrs gSaA

18. ;fn 'kwU;srj lfn'k   dk ifjek.k ‘a’ gS vkSj λ ,d 'kwU;rsj vfn'k gS rks λ ,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 oqaQry fd;k tkrk gS rks v¡xwBk /ukRed z-v{k dh vksj laosQr djrk [vko`Qfr 10-22 (ii)] gSA

ifjHkk"kk 3 nks 'kwU;srj lfn'kksa , dk lfn'k xq.kuiQy  ls fufnZ"V fd;k tkrk gS vkSj   =  osQ :i esa ifjHkkf"kr fd;k tkrk gS tgk¡ θ,   osQ chp dk dks.k gS vkSj gSA ;gk¡ ,d ek=kd lfn'k gS tks fd lfn'k , nksuksa ij yac gSA bl izdkj  ,d nf{k.kkorhZ i¼fr dks fufeZr djrs gSa (vko`Qfr 10-23) vFkkZr~~ nf{k.kkorhZ i¼fr dks  dh rji+Q ?kqekus ij ;g dh fn'kk esa pyrh gSA

;fn  rc θ ifjHkkf"kr ugha gS vkSj bl fLFkfr eas ge  ifjHkkf"kr djrs gSaA

izs{k.k%

1.  ,d lfn'k gSA

2. eku yhft,  nks 'kwU;srj lfn'k gSa rc  ;fn vkSj osQoy ;fn ,d nwljs osQ lekarj (vFkok lajs[k) gSa vFkkZr~~

fof'k"Vr%   vkSj , D;ksafd izFke fLFkfr esa θ = 0 rFkk f}rh; fLFkfr esa θ = π, ftlls nksuksa gh fLFkfr;ksa esa sinθ dk eku 'kwU; gks tkrk gSA

3. ;fn rks 

4. izs{k.k 2 vkSj 3 osQ lanHkZ esa ijLij yacor~ ek=kd lfn'kksa osQ fy, (vko`Qfr 10-24), ge ikrs gSa fd

=

=

5. lfn'k xq.kuiQy dh lgk;rk ls nks lfn'kksa osQ chp dk dks.k θ fuEufyf[kr :i esa izkIr gksrk gS

sinθ =

6. ;g loZnk lR; gS fd lfn'k xq.kuiQy Øe fofue; ugha gksrk gS D;ksafd   =  okLro esa , tgk¡  ,d nf{k.kkorhZ i¼fr dks fu£er djrs gSa vFkkZr~~ θ,  dh rjiQ pØh; Øe gksrk gSA vko`Qfr 10-25(i) tcfd , tgk¡   ,d nf{k.kkorhZ i¼fr dks fufeZr djrs gSa vFkkZr~~ θ, dh vksj pØh; Øe gksrk gS vko`Qfr 10-25(ii)A

vr% ;fn ge ;g eku ysrs gSa fd   nksuksa ,d gh dkx”k osQ ry esa gSa rks  nksuksa dkx”k osQ ry ij yac gksaxs ijarq dkx”k ls Åij dh rji+Q fn"V gksxk vkSj dkx”k ls uhps dh rji+Q fn"V gksxk vFkkZr~~

bl izdkj

7. izs{k.k 4 vkSj 6 osQ lanHkZ esa

8. ;fn  f=kHkqt dh layXu Hkqtkvksa dks fu:fir djrs gSa rks f=kHkqt dk {ks=kiQy osQ :i esa izkIr gksrk gSA

f=kHkqt osQ {ks=kiQy dh ifjHkk"kk osQ vuqlkj ge vko`Qfr 10-26 ls ikrs gSa fd f=kHkqt ABC dk {ks=kiQy = . ijarq (fn;k gqvk gS) vkSj CD = sinθ

vr% f=kHkqt ABC dk {ks=kiQy

9. ;fn   lekarj prqHkZqt dh layXu Hkqtkvksa dks fu:fir djrs gSa rks lekarj prqHkqZt dk {ks=kiQy  osQ :i esa izkIr gksrk gSA

vko`Qfr 10-27 ls ge ikrs gSa fd lekarj prqHkaZt ABCD dk {ks=kiQy = AB. DE.

ijarq   (fn;k gqvk gS), vkSj   vr%

lekarj prqHkqZt ABCD dk {ks=kiQy = 

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  rhu lfn'k gSa vkSj λ ,d vfn'k gS rks

eku yhft, nks lfn'k  ?kVd :i esa Øe'k% vkSj fn, gq, gSa rc mudk lfn'k xq.kuiQy 

  = }kjk fn;k tk ldrk gSA

O;k[;k ge ikrs gSa

=

=

+

+ ( xq.k/eZ 1 ls)

=

+

=

=

=

mnkgj.k 22 ;fn Kkr dhft,A

gy ;gk¡

=

=

vr% =

mnkgj.k 23 lfn'k vkSj esa ls izR;sd osQ yacor~ ek=kd lfn'k Kkr dhft, tgk¡ 

gy ge ikrs gSa fd 

,d lfn'k] tks   nksuks ij yac gS] fuEufyf[kr }kjk iznÙk gS

vc =

vr% vHkh"V ek=kd lfn'k

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. fn, gq, f=kHkqt dk {ks=kiQy  gSA

vc

blfy,

vr% vHkh"V {ks=kiQy gSA

mnkgj.k 25 ml lekarj prqHkqZt dk {ks=kiQy Kkr dhft, ftldh layXu Hkqtk,¡  vkSj   }kjk nh xbZ gSaA

gy fdlh lekarj prqHkqZt dh layXu Hkqtk,¡  gSa rks mldk {ks=kiQy  }kjk izkIr gksrk gSA

vc

blfy, =

bl izdkj vko';d {ks=kiQy gSA

iz'ukoyh 10-4

1. Kkr dhft,A

2. lfn'k  dh yac fn'kk esa ek=kd lfn'k Kkr dhft, tgk¡  vkSj  gSA

3. ;fn ,d ek=kd lfn'k , osQ lkFk ,d U;wu dks.k θ cukrk gS rks θ dk eku Kkr dhft, vkSj bldh lgk;rk ls osQ ?kVd Hkh Kkr dhft,A

4. n'kkZb, fd 

5. λ vkSj µ Kkr dhft,] ;fn

6. fn;k gqvk gS fd  � osQ ckjs esa vki D;k fu"d"kZ fudky ldrs gSa?

7. eku yhft, lfn'k   Øe'k% osQ :i esa fn, gq, gSa rc n'kkZb, fd

8. ;fn   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 prqHkqZt dk {ks=kiQy Kkr dhft, ftldh layXu Hkqtk,¡ lfn'k vkSj }kjk fu/kZfjr gSaA

11. eku yhft, lfn'k   bl izdkj gSa fd ,d ek=kd lfn'k gS ;fn  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%

, vkSj , gSa dk {ks=kiQy gS%

(A) (B) 1

(C) 2 (D) 4

fofo/ mnkgj.k

mnkgj.k 26 XY-ry esa lHkh ek=kd lfn'k fyf[k,A

gy eku yhft, fd   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 || 1). blfy, ge lfn'k dks]

osQ :i esa fy[k ldrs gSaA

Li"Vr% =

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% , 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,osQ chp dk Hkh dks.k gSA

vc

blfy,

blh izdkj 

=

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

fodYir% ] blls dg ldrs fd  vkSj  lajs[k lfn'k gSaA

mnkgj.k 28 eku yhft,  rhu lfn'k bl izdkj gSa fd  vkSj buesa ls izR;sd] vU; nks lfn'kksa osQ ;ksxiQy ij yacor~ gaS rks]  Kkr dhft,A

gy fn;k gqvk gS fd 

mnkgj.k 29 rhu lfn'k  izfrca/  dks larq"V djrs gSaA ;fn 

gy D;ksafd  , blfy, ge ikrs gSa fd

;k 2µ = – 29, i.e., µ=

mnkgj.k 30 ;fn ijLij yacor~ ek=kd lfn'kksa dh nf{k.kkorhZ i¼fr osQ lkis{k

osQ :i esa vfHkO;Dr dhft, tgk¡ osQ yacor~ gSA

��gy eku yhft, fd 

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  ? vius mÙkj dh iqf"V dhft,A

5. x dk og eku Kkr dhft, ftlosQ fy, ,d ek=kd lfn'k gSA

6. lfn'kksa   osQ ifj.kkeh osQ lekarj ,d ,slk lfn'k Kkr dhft, ftldk ifjek.k 5 bdkbZ gSA

7. ;fn 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 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 prqHkqZt dh layXu Hkqtk,¡ gaSA 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 dksT;k,¡ gSA

12. eku yhft,   ,d ,slk lfn'k  Kkr dhft, tks �� nksuksa ij yac gS vkSj 

13. lfn'k dk] lfn'kksa vkSj osQ ;ksxiQy dh fn'kk esa ek=kd lfn'k osQ lkFk vfn'k xq.kuiQy 1 osQ cjkcj gS rks dk eku Kkr dhft,A

14. ;fn   leku ifjek.kksa okys ijLij yacor~ lfn'k gSa rks n'kkZb, fd lfn'k  lfn'kksa osQ lkFk cjkcj >qdk gqvk gSA

15. fl¼ dhft, fd  , ;fn vkSj osQoy ;fn  yacor~ gSaA ;g fn;k gqvk gS fd 

16 ls 19 rd osQ iz'uksa esa lgh mÙkj dk p;u dhft,A

16. ;fn nks lfn'kksa  osQ chp dk dks.k θ gS rks gksxk ;fn%

(A) (B)

(C) 0 < θ < π (D) 0 ≤ θ ≤ π

17. eku yhft,   nks ek=kd lfn'k gSa vkSj muosQ chp dk dks.k θ gS rks   ,d ek=kd lfn'k gS ;fn%

(A) (B) (C) (D)

18. dk eku gS

(A) 0 (B) –1 (C) 1 (D) 3

19. ;fn nks lfn'kksa osQ chp dk dks.k θ gS rks  tc θ cjkcj gS%

(A) 0 (B) (C) (D) π