The moving power of mathematical invention is not reasoning but imagination. – A.DEMORGAN 

11.1 Hkwfedk (Introduction)

d{kk XI esa] oS'ysf"kd T;kfefr dk vè;;u djrs le; f}&foeh; vkSj f=k&foeh; fo"k;ksa osQ ifjp; esa geus Lo;a dks osQoy dkrhZ; fof/ rd lhfer j[kk gSA bl iqLrd osQ fiNys vè;k; esa geus lfn'kksa dh ewy ladYiukvksa dk vè;;u fd;k gSA vc ge lfn'kksa osQ chtxf.kr dk f=k&foeh; T;kfefr esa mi;ksx djsaxsA f=k&foeh; T;kfefr esa bl mikxe dk mn~ns'; gS fd ;g blosQ vè;;u dks vR;ar ljy ,oa lq#fpiw.kZ (lqxzkg;) cuk nsrk gSA*

bl vè;k; esa ge nks ¯cnqvksa dks feykus okyh js[kk osQ fnd~&dksT;k o fno~Q&vuqikr dk vè;;u djsaxs vkSj fofHkUu fLFkfr;ksa esa varfj{k esa js[kkvksa vkSj ryksa osQ lehdj.kksa] nks js[kkvksa] nks ryksa o ,d js[kk vkSj ,d ry osQ chp dk dks.k] nks fo"keryh; js[kkvksa osQ chp U;wure nwjh o ,d ry dh ,d ¯cnq ls nwjh osQ fo"k; esa Hkh fopkj foe'kZ djsaxsA mijksDr ifj.kkeksa esa ls vf/dka'k ifj.kkeksa dks lfn'kksa osQ :i esa izkIr djrs gSaA rFkkfi ge budk dkrhZ; :i esa Hkh vuqokn djsaxs tks dkykarj esa fLFkfr dk Li"V T;kferh; vkSj fo'ys"k.kkRed fp=k.k izLrqr dj losQxkA

11.2 js[kk osQ fno~Q&dkslkbu vkSj fno~Q&vuqikr (Direction Cosines and Direction Ratios of a Line)

vè;k; 10 esa] Lej.k dhft,] fd ewy ¯cnq ls xqtjus okyh lfn'k js[kk L }kjk x, y vkSj z-v{kksa osQ lkFk Øe'k α] β vkSj γ cuk, x, dks.k fno~Q&dks.k dgykrs gSa rc bu dks.kksa dh dkslkbu uker% cosα, cosβ vkSj cosγ js[kk L osQ fno~Q&dkslkbu (direction cosines or dc's)dgykrh gSaA

;fn ge L dh fn'kk foijhr dj nsrs gSa rks fno~Q&dks.k] vius laiwjdksa esa vFkkZr~ π-α, π-β vkSj π-γ ls cny tkrs gSaA bl izdkj] fno~Q&dkslkbu osQ fpÉ cny tkrs gSaA

è;ku nhft,] varfj{k esa nh xbZ js[kk dks nks foijhr fn'kkvksa esa c<+k ldrs gSa vkSj blfy, blosQ fno~Q&dkslkbu osQ nks lewg gSaA blfy, varfj{k esa Kkr js[kk osQ fy, fno~Q&dkslkbu osQ vf}rh; lewg osQ fy,] gesa Kkr js[kk dks ,d lfn'k js[kk ysuk pkfg,A bu vf}rh; fno~Q&dkslkbu dks
l, m vkSj n osQ }kjk fufnZ"V fd, tkrs gSaA

fVIi.kh varfj{k esa nh xbZ js[kk ;fn ewy ¯cnq ls ugha xqtjrh gS rks bldh fno~Q&dkslkbu dks Kkr djus osQ fy,] ge ewy ¯cnq ls nh xbZ js[kk osQ lekarj ,d js[kk [khaprs gSaA vc ewy ¯cnq ls buesa ls ,d lfn'k js[kk osQ fno~Q&vuqikr Kkr djrs gSa D;ksafd nks lekarj js[kkvksa osQ fno~Q&vuqikrksa osQ lewg leku (ogh) gksrs gSaA

,d js[kk osQ fno~Q&dkslkbu osQ lekuqikrh la[;kvksa dks js[kk osQ fno~Q&vuqikr (direction ratios or dr's) dgrs gSaA ;fn ,d js[kk osQ fno~Q&dkslkbu l, m, n o fno~Q&vuqikr a, b, c gksa rc fdlh 'kwU;srj λ ∈ R osQ fy, a = λl, b=λm vkSj c = λn

eku yhft, ,d js[kk osQ fno~Q&vuqikr a, b, c vkSj js[kk dh fno~Q&dkslkbu l, m, n gSA rc

= = (eku yhft,), k ,d vpj gSA

blfy, l = ak, m = bk, n = ck ... (1)

ijarq l2 + m2 + n2 = 1

blfy, k2 (a2 + b2 + c2) = 1

;k k =

vr% (1) ls] js[kk dh fno~Q&dkslkbu (d.c.’s )

fdlh js[kk osQ fy, ;fn js[kk osQ fno~Q&vuqikr Øe'k% a, b, c gS] rks ka, kb, kc; k ≠ 0 Hkh fno~Q&vuqikrksa dk ,d lewg gSA blfy, ,d js[kk osQ fno~Q&vuqikrksa osQ nks lewg Hkh lekuqikrh gksaxsA vr% fdlh ,d js[kk osQ fno~Q&vuqikrksa osQ vla[; lewg gksrs gSaA

11.2.1 js[kk dh fno~Q&dkslkbu esa laca/ (Relation between the direction cosines of a line)

eku yhft, fd ,d js[kk RS dh fno~Q&dkslkbu l, m, n gSA ewy ¯cnq ls nh xbZ js[kk osQ lekarj ,d js[kk [khafp, vkSj bl ij ,d ¯cnq P(x, y, z) yhft,A P ls x-v{k ij yac PA [khafp,
(vko`Qfr 11-2)A

;fn OP = r. rks . ftlls x = lr izkIr gksrk gSA blh izdkj y = mr vkSj z = nr.

blfy, x2 + y2 + z2 = r2 (l2 + m2 + n2)

ijarq x2 + y2 + z2 = r2

vr% l2 + m2 + n2 = 1

11.2.2 nks ¯cnqvksa dks feykus okyh js[kk dh fno~Q&dkslkbu (Direction cosines of a line passing through two points)

D;ksafd nks fn, ¯cnqvksa ls gksdj tkus okyh js[kk vf}rh; gksrh gSA blfy, nks fn, x, ¯cnqvksa
P(x1, y1, z1) vkSj Q(x2, y2, z2) ls xqtjus okyh js[kk dh fno~Q&dkslkbu dks fuEu izdkj ls Kkr dj ldrs gSa (vko`Qfr 11-3(a)A

eku yhft, fd js[kk PQ dh fno~Q&dkslkbu l, m, n gSa vkSj ;g x, y vkSj z-v{k osQ lkFk dks.k Øe'k% α, β, γ cukrh gSaA

eku yhft, P vkSj Q ls yac [khafp, tks XY-ry dks R rFkk S ij feyrs gSaA P ls ,d vU; yac [khafp, tks QS dks N ij feyrk gSA vc ledks.k f=kHkqt PNQ esa, ∠PQN = γ (vko`Qfr 11.3 (b)) blfy,

cosγ =

blh izdkj cosα =

vr% ¯cnqvksa P(x1, y1, z1) rFkk Q(x2, y2, z2) dks tksM+us okys js[kk[kaM PQ fd fno~Q&dkslkbu

, , gSaA

tgk¡ PQ =

fVIi.kh ¯cnqvksa P(x1, y1, z1) rFkk Q(x2, y2, z2) dks tksM+us okys js[kk[kaM osQ fno~Q&vuqikr fuEu izdkj ls fy, tk ldrs gSaA

x2 – x1, y2 – y1, z2 – z1, ;k x1 – x2, y1 – y2, z1 – z2

mnkgj.k 1 ;fn ,d js[kk x, y rFkk z-v{kksa dh /ukRed fn'kk osQ lkFk Øe'k% 90°, 60° rFkk 30° dk dks.k cukrhs gS rks fno~Q&dkslkbu Kkr dhft,A

gy eku yhft, js[kk dh fno~Q&dkslkbu l, m, n gSA rc l = cos 90° = 0, m = cos 60° = ,
n = cos 30° =

mnkgj.k 2 ;fn ,d js[kk osQ fno~Q&vuqikr 2] &1] &2 gSa rks bldh fno~Q&dkslkbu Kkr dhft,A

gy fno~Q&dkslkbu fuEuor~ gSa

, ,

vFkkZr~

mnkgj.k 3 nks ¯cnqvksa (– 2, 4, – 5) vkSj (1, 2, 3) dks feykus okyh js[kk dh fno~Q&dkslkbu Kkr dhft,A

gy ge tkurs gSa fd nks ¯cnqvksa P (x1, y1, z1) vkSj Q(x2, y2, z2) dks feykus okyh js[kk dh fno~Q&dkslkbu

gSa] tgk¡ PQ =

;gk¡ P vkSj Q Øe'k% (– 2, 4, – 5) vkSj (1, 2, 3) gSaA

blfy, PQ = =

blfy, nks ¯cnqvksa dks feykus okyh js[kk dh fno~Q&dkslkbu gSa%

mnkgj.k 4 x, y vkSj z-v{kksa dh fno~Q&dkslkbu Kkr dhft,A

gy x-v{k Øe'k% x, y vkSj z-v{k osQ lkFk 0°, 90° vkSj 90° osQ dks.k cukrk gSA blfy, x-v{k dh fno~Q&dkslkbu cos 0°, cos 90°, cos 90° vFkkZr~ 1,0,0 gSaA

blh izdkj y-v{k vkSj z-v{k dh fno~Q&dkslkbu Øe'k% 0, 1, 0 vkSj 0, 0, 1 gSaA

mnkgj.k 5 n'kkZb, fd ¯cnq A (2, 3, – 4), B (1, – 2, 3) vkSj C (3, 8, – 11) lajs[k gSaA

gy A vkSj B (dks feykus okyh js[kk osQ fno~Q&vuqikr

1 –2, –2 –3, 3 + 4 vFkkZr~ – 1, – 5, 7 gSaA

B (vkSj C dks feykus okyh js[kk osQ fno~Q&vuqikr 3 –1, 8 + 2, – 11 – 3, vFkkZr~, 2, 10, – 14 gSaA

Li"V gS fdABvkSj BC osQ fno~Q&vuqikr lekuqikrh gSaA vr% AB vkSj BC lekarj gSaAq ijarq ABvkSj BC nksuksa esa B (mHk;fu"B gSA vr% A, B, vkSj C lajs[k ¯cnq gSaA

iz'ukoyh 11-1

1. ;fn ,d js[kk x, y vkSj z-v{k osQ lkFk Øe'k% 90°, 135°, 45° osQ dks.k cukrh gS rks bldh fno~Q&dkslkbu Kkr dhft,A

2. ,d js[kk dh fno~Q&dkslkbu Kkr dhft, tks funsZ'kka{kksa osQ lkFk leku dks.k cukrh gSA

3. ;fn ,d js[kk osQ fno~Q&vuqikr –18, 12, – 4, gSa rks bldh fno~Q&dkslkbu D;k gSa\

4. n'kkZb, fd ¯cnq (2, 3, 4), (– 1, – 2, 1), (5, 8, 7) lajs[k gSaA

5. ,d f=kHkqt dh Hkqtkvksa dh fno~Q&dkslkbu Kkr dhft, ;fn f=kHkqt osQ 'kh"kZ ¯cnq
(3, 5, – 4), (– 1, 1, 2) vkSj (– 5, – 5, – 2) gSaA

11.3 varfj{k esa js[kk dk lehdj.k (Equation of a Line in Space)

d{kk XI esa f}&foeh; ry esa js[kkvksa dk vè;;u djus osQ i'pkr~ vc ge varfj{k esa ,d js[kk osQ lfn'k rFkk dkrhZ; lehdj.kksa dks Kkr djsaxsA

,d js[kk vf}rh;r% fu/kZfjr gksrh gS] ;fn

(i);g fn, ¯cnq ls nh xbZ fn'kk ls gksdj tkrh gS] ;k

(ii);g nks fn, x, ¯cnqvksa ls gksdj tkrh gSA

11.3.1 fn, x, ¯cnq A ls tkus okyh rFkk fn, x, lfn'k osQ lekarj js[kk dk lehdj.k (Equation of a line through a given point A and parallel to a given vector )

ledksf.kd funsZ'kka{k fudk; osQ ewy ¯cnq O osQ lkis{k eku yhft, fd ¯cnq A dk lfn'k   gSA eku yhft, fd ¯cnq A ls tkus okyh rFkk fn, x, lfn'k  osQ lekarj js[kk l gSA eku yhft, fd l ij fLFkr fdlh LosPN ¯cnq P dk fLFkfr lfn'k  gS (vko`Qfr 11-4)A

rc  lfn'k   osQ lekarj gS vFkkZr~ =λ, tgk¡ λ ,d okLrfod la[;k gSA

ijarq

vFkkZr~ λ = 

foykser% izkpy λ osQ izR;sd eku osQ fy, ;g lehdj.k js[kk osQ fdlh ¯cnq P dh fLFkfr iznku djrk gSA vr% js[kk dk lfn'k lehdj.k gS%

= ... (1)

fVIi.kh ;fn  gS rks js[kk osQ fno~Q&vuqikr a, b, c gS vkSj foykser% ;fn ,d js[kk osQ fno~Q&vuqikr a, b, c gksa rksjs[kk osQ lekarj gksxkA ;gk¡ b dks u le>k tk,A

lfn'k :i ls dkrhZ; :i O;qRiUu djuk (Derivation of Cartesian Form from Vector Form)

eku yhft, fd fn, ¯cnq A osQ funsZ'kkad (x1, y1, z1) gSa vkSj js[kk dh fno~Q&dkslkbu
a, b, c gSa eku yhft, fdlh ¯cnq P osQ funsZ'kkad (x, y, z) gSaA rc

vkSj 

bu ekuksa dks (1) esa izfrLFkkfir djosQ vkSj , osQ xq.kkadksa dh rqyuk djus ij ge ikrs gSa fd

x = x1 + λa;y = y1 + λ b;z = z1+ λc ... (2)

;s js[kk osQ izkpy lehdj.k gSaA (2) ls izkpy λ dk foyksiu djus ij] ge ikrs gSa%

= ... (3)

;g js[kk dk dkrhZ; lehdj.k gSA

fVIi.kh ;fn js[kk dh fno~Q&dkslkbu l, m, n gSa] rks js[kk dk lehdj.k

= gSaA

mnkgj.k 6 ¯cnq (5, 2, – 4) ls tkus okyh rFkk lfn'k osQ lekarj js[kk dk lfn'k rFkk dkrhZ; lehdj.kksa dks Kkr dhft,A

gy gesa Kkr gS] fd

 =

blfy,] js[kk dk lfn'k lehdj.k gS%

= [(1) ls]

pw¡fd js[kk ij fLFkr fdlh ¯cnq P x, y, z) dh fLFkfr lfn'k  gS] blfy,

=

λ dk foyksiu djus ij ge ikrs gSa fd

=

tks js[kk osQ lehdj.k dk dkrhZ; :i gSA

11.3.2 nks fn, x, ¯cnqvksa ls tkus okyh js[kk dk lehdj.k (Equation of a line passing through two given points)

eku yhft, ,d js[kk ij fLFkr nks ¯cnqvksa A(x1, y1, z1) vkSj B(x2, y2, z2), osQ fLFkfr lfn'k Øe'k%  vkSj   gSa (vko`Qfr 11.5)A

eku yhft,  ,d LosPN ¯cnq P dk fLFkfr lfn'k gSA rc P js[kk ij gS ;fn vkSj osQoy ;fn  rFkk  lajs[k lfn'k gSaA blfy, P js[kk ij fLFkr gS ;fn vkSj osQoy ;fn

;k , λ ∈ R ... (1)

tks js[kk dk lfn'k lehdj.k gSA

lfn'k :i ls dkrhZ; :i O;qRiUu djuk

ge ikrs gSa fd

bu ekuksa dks (1) esa izfrLFkkfir djus ij ge ikrs gSa fd

 

  osQ xq.kkadksa dh rqyuk djus ij ge ikrs gSa fd

x = x1 + λ (x2 – x1); y = y1 + λ (y2 – y1); z = z1 + λ (z2 – z1)

λ dk foyksiu djus ij ge ikrs gSa fd

tks js[kk osQ lehdj.k dk dkrhZ; :i gSA

mnkgj.k 7 ¯cnqvksa (–1, 0, 2) vkSj (3, 4, 6) ls gksdj tkus okyh js[kk dk lfn'k lehdj.k Kkr dhft,A

gy eku yhft, vkSj¯cnqvksa A(– 1, 0, 2) vkSj B(3, 4, 6) osQ fLFkfr lfn'k gSaA

eku yhft, fd js[kk ij fLFkr fdlh LosPN ¯cnq P dk fLFkfr lfn'k gSA vr% js[kk dk lfn'k lehdj.k

mnkgj.k 8 ,d js[kk dk dkrhZ; lehdj.k gSA bl js[kk dk lfn'k lehdj.k Kkr dhft,A

gy fn, x, lehdj.k dk ekud :i

ls rqyuk djus ij ge ikrs gSa fd x1 = – 3, y1 = 5, z1 = – 6; a = 2, b = 4, c = 2

bl izdkj vHkh"V js[kk ¯cnq (– 3, 5, – 6) ls gksdj tkrh gS rFkk lfn'k osQ lekarj gSA eku yhft, fd js[kk ij fLFkr fdlh ¯cnq dh fLFkfr lfn'k gS rks js[kk dk lfn'k lehdj.k

}kjk iznÙk gSA

  

11.4 nks js[kkvksa osQ eè; dks.k (Angle between two lines)

eku yhft, fdL1 vkSjL2 ewy ¯cnq ls xqtjus okyh nks js[kk,¡ gSa ftuosQ fno~Q&vuqikr Øe'k% a1, b1, c1 vkSj a2, b2, c2, gSA iqu% eku yhft,fd L1 ij ,d ¯cnq P rFkk L2 ij ,d ¯cnq QgSA vko`Qfr 11-6 esa fn, x, lfn'k OP vkSj OQ ij fopkj dhft,A eku yhft, fd OP vkSj OQ osQ chp U;wu dks.k θ gSA vc Lej.k dhft, fd lfn'kksa OP vkSj OQ osQ ?kVd Øe'k% a1, b1, c1 vkSj a2, b2, c2 gSaA blfy, muosQ chp dk dks.k θ

 cosθ = }kjk iznÙk gSA

iqu% sin θosQ :i esa] js[kkvksa osQ chp dk dks.k

sin θ=

.... (2)

cosθ = |l1 l2 + m1m2 + n1n2| (D;ksafd)... (3)

vkSj sin θ =... (4)

fno~Q&vuqikr a1, b1, c1 vkSj a2, b2, c2 okyh js[kk,¡

(i)yacor~~ gS] ;fn θ = 90°, vFkkZr~ (1) ls a1a2 + b1b2 + c1c2 = 0

(ii)lekarj gS] ;fn θ = 0, vFkkZr~ (2) ls 

vc ge nks js[kkvksa osQ chp dk dks.k Kkr djsaxs ftuosQ lehdj.k fn, x, gSaA ;fn mu js[kkvksa  =vkSj = osQ chp U;wu dks.k θ g

rc 

dkrhZ; :i esa ;fn js[kkvksa%=... (1)

vkSj=... (2)

cos θ ==

osQ chp dk dks.k θ gS tgk¡ js[kk,¡ (1) o (2) osQ fno~Q&vuqikr Øe'k% a1, b1, c1 rFkk a2, b2, c2 gS rc

mnkgj.k 9 fn, x, js[kk&;qXe

osQ eè; dks.k Kkr dhft,

gy eku yhft, 

nksuksa js[kkvksa osQ eè; dks.k θ gS] blfy,

cosθ =

=

vr% θ = cos–1 

mnkgj.k 10 js[kk&;qXe%

=

vkSj ==

osQ eè; dks.k Kkr dhft,A

gy igyh js[kk osQ fno~Q&vuqikr 3] 5] 4 vkSj nwljh js[kk osQ fno~Q&vuqikr 1] 1] 2 gSaA ;fn muosQ chp dk dks.k θ gks rc

cosθ =  =

vr% vHkh"V dks.k cos–1 gSA

11.5 nks js[kkvksa osQ eè; U;wure nwjh (Shortest Distance between two lines)

varfj{k esa ;fn nks js[kk,¡ ijLij izfrPNsn djrh gS rks muosQ chp dh U;wure nwjh 'kwU; gSA vkSj varfj{k esa ;fn nks js[kk,¡ lekarj gS rks muosQ chp dh U;wure nwjh] muosQ chp yacor~~ nwjh gksxh vFkkZr~ ,d js[kk osQ ,d ¯cnq ls nwljh js[kk ij [khapk x;k yacA

blosQ vfrfjDr varfj{k esa] ,slh Hkh js[kk,¡ gksrh gS tks u rks izfrPNsnh vkSj u gh lekarj gksrh gSA okLro esa ,slh js[kkvksa osQ ;qXe vleryh; gksrs gSa vkSj bUgsa fo"keryh; js[kk,¡ (skew lines) dgrs gSaA mnkgj.kr;k ge vko`Qfr 11-7 esa x, y vkSj z-v{k osQ vuqfn'k Øe'k% 1] 3] 2 bdkbZ osQ vkdkj okys dejs ij fopkj djrs gSaA

js[kk GE Nr osQ fod.kZ osQ vuqfn'k gS vkSj js[kk DB, A osQ Bhd Åij Nr osQ dksus ls xqtjrh gqbZ nhokj osQ fod.kZ osQ vuqfn'k gSA ;s js[kk,¡ fo"keryh; gSa D;ksafd os lekarj ugha gS vkSj dHkh feyrh Hkh ugha gSaA

nks js[kkvksa osQ chp U;wure nwjh ls gekjk vfHkizk; ,d ,sls js[kk[kaM ls gS tks ,d js[kk ij fLFkr ,d ¯cnq dks nwljh js[kk ij fLFkr vU; ¯cnq dks feykus ls izkIr gksa rkfd bldh yackbZ U;wure gksA

U;wure nwjh js[kk[kaM nksuksa fo"keryh; js[kkvksa ij yac gksxkA

11.5.1 nks fo"keryh; js[kkvksa osQ chp dh nwjh (Distance between two skew lines)

vc ge js[kkvksa osQ chp dh U;wure nwjh fuEufyf[kr fof/ ls Kkr djrs gSaA eku yhft, l1 vkSj l2 nks fo"keryh; js[kk,¡ gS ftuosQ lehdj.k (vko`Qfr 11.8) fuEufyf[kr gSa%

js[kk l1 ij dksbZ ¯cnq S ftldh fLFkfr lfn'k vkSj l2 ij dksbZ ¯cnq T ftldh fLFkfr lfn'k . gS] yhft,A rc U;wure nwjh lfn'k dk ifjek.k] ST dk U;wure nwjh dh fn'kk esa iz{ksi dh eki osQ leku gksxk (vuqPNsn 10.6.2)A

;fn l1 vkSj l2 osQ chp dh U;wure nwjh lfn'k gS rks ;g nksuksa b1 vkSj b2 ij yac gksxhA  dh fn'kk esa bdkbZ lfn'k  bl izdkj gksxh fd

 =

blfy, vHkh"V U;wure nwjh 

dkrhZ; :i (Cartesian Form)

js[kkvksa%

l2 :=

osQ chp dh U;wure nwjh gS%

11.5.2 lekarj js[kkvksa osQ chp dh nwjh (Distance between parallel lines)

;fn nks js[kk,¡ l1 ;fn l2 lekarj gSa rks os leryh; gksrh gSaA ekuk nh xbZ js[kk,¡ Øe'k%

... (1)

vkSj

gSa] tgk¡ l1 ij ¯cnq S dk fLFkfr lfn'kvkSj l2 ij ¯cnq T dk fLFkfr lfn'k gS (vko`Qfr 11.9)

D;ksafd l1, vkSj l2 leryh; gSA ;fn ¯cnq T ls l1 ij Mkys x, yac dk ikn P gS rc js[kkvksa l1 vkSj l2 osQ chp dh nwjh = |TP|

eku yhft, fd lfn'kksa vkSj osQ chp dk dks.k θ gSA rc]

stgk¡ js[kkvksa l1 vkSj l2 osQ ry ij yac bdkbZ lfn'k gSa

ijarq 

blfy, (3) ls ge ikrs gSa fd

blfy, Kkr js[kkvksa osQ chp U;wure nwjh

mnkgj.k 11 js[kkvksa l1 vkSj l2 osQ chp dh U;wure nwjh Kkr dhft, ftuosQ lfn'k lehdj.k gS%

vkSj 

gy lehdj.k (1) o (2) dh  vkSj , ls rqyuk djus ij ge ikrs gSa fd

blfy,

vkSj

=

bl izdkj  =

blfy, nh xbZ js[kkvksa osQ chp dh U;wure nwjh

mnkgj.k 12 fuEufyf[kr nh xbZ js[kkvksa l1 vkSj l2 :

vkSj ==osQ chp U;wure nwjh Kkr dhft,A

gy nksuksa js[kk,¡ lekraj gSaA (D;ksa\) gesa izkIr gS fd

blfy, js[kkvksa osQ chp dh nwjh

d=  =  

= gSA

iz'ukoyh 11-2

1. n'kkZb, fd fno~Q&dkslkbu  okyh rhu js[kk,¡ ijLij yacor~ gSaA

2. n'kkZb, fd ¯cnqvksa (1, – 1, 2), (3, 4, – 2) ls gksdj tkus okyh js[kk ¯cnqvksa (0, 3, 2) vkSj (3, 5, 6) ls tkus okyh js[kk ij yac gSA

3. n'kkZb, fd ¯cnqvksa (4, 7, 8), (2, 3, 4) ls gksdj tkus okyh js[kk] ¯cnqvksa (– 1, – 2, 1), (1, 2, 5) ls tkus okyh js[kk osQ lekarj gSA

4.

5. ¯cnq ftldh fLFkfr lfn'kls xq”kjus o lfn'k  dh fn'kk esa tkus okyh js[kk dk lfn'k vkSj dkrhZ; :iksa esa lehdj.k Kkr dhft,A

6. ml js[kk dk dkrhZ; lehdj.k Kkr dhft, tks ¯cnq (– 2, 4, – 5) ls tkrh gS vkSjosQ lekarj gSA

7. ,d js[kk dk dkrhZ; lehdj.k  gSA bldk lfn'k lehdj.k Kkr dhft,A

8. ewy ¯cnq vkSj (5, – 2, 3) ls tkus okyh js[kk dk lfn'k rFkk dkrhZ; :iksa esa lehdj.k Kkr dhft,A

9. ¯cnqvksa (3, – 2, – 5), vkSj (3, – 2, 6) ls xq”kjus okyh js[kk dk lfn'k rFkk dkrhZ; :iksa esa lehdj.k dks Kkr dhft,A

10.fuEufyf[kr js[kk&;qXeksa osQ chp dk dks.k Kkr dhft,%

11. fuEufyf[kr js[kk&;qXeksa osQ chp dk dks.k Kkr dhft,%

(i)

12. p dk eku Kkr dhft, rkfd js[kk,¡ 

vkSj  ijLij yac gksaA

13. fn[kkb, fd js[kk,¡   ijLij yac gSaA

14. js[kkvksa  osQ chp dh U;wure nwjh Kkr dhft,%

15. js[kkvksa  osQ chp dh U;wure nwjh Kkr dhft,A

16. js[kk,¡] ftuosQ lfn'k lehdj.k fuEufyf[kr gS] osQ chp dh U;wure nwjh Kkr dhft,%

17. js[kk,¡] ftudh lfn'k lehdj.k fuEufyf[kr gSa] osQ chp dh U;wure Kkr dhft,%

11.6 lery (Plane)

,d lery dks vf}rh; :i ls Kkr fd;k tk ldrk gS ;fn fuEufyf[kr esa ls dksbZ ,d 'krZ Kkr gks%

(i)lery dk vfHkyac vkSj ewy ¯cnq ls lery dh nwjh Kkr gS] vFkkZr~ vfHkyac :i esa lery dk lehdj.k

(ii);g ,d ¯cnq ls xq”kjrk gS vkSj nh xbZ fn'kk osQ yacor~ gSA

(iii);g fn, x, rhu vlajs[k ¯cnqvksa ls xq”kjrk gSA

vc ge leryksa osQ lfn'k vkSj dkrhZ; lehdj.kksa dks izkIr djsaxsA

�11.6.1 vfHkyac :i esa lery dk lehdj.k (Equation of a Plane in normal form)

,d lery ij fopkj dhft, ftldh ewy ¯cnq ls yacor~ nwjh d (d ≠ 0) gS (vko`Qfr 11-10)A

;fn  ewy ¯cnq ls ry ij yac gS rFkk  osQ vuqfn'k  ek=kd vfHkyac lfn'k gS rc = d   gSA eku yhft, fd lery ij dksbZ ¯cnq P gSA blfy,] ,  ij yac gSA

vr%

eku yhft, P dh fLFkfr lfn'k  gS rks 

bl izdkj (1) dk :i fuEufyf[kr gS%

;g lery dk lfn'k lehdj.k gSA

dkrhZ; :i (Cartesian Form)

lery dk lfn'k lehdj.k gS tgkW lery osQ vfHkyac bdkbZ lfn'k gSA eku yhft, lery ij dksbZ ¯cnq P (x, y, z) gSA rc

eku yhft,  dh fno~Q&dkslkbu l, m, n gSaA rc

 osQ ekuksa dks (2) esa izfrLFkkfir djus ij ge ikrs gSa]

vFkkZr~ lx + my + nz = d ... (3)

;g lery dk dkrhZ; lehdj.k gSA

mnkgj.k 13 ml lery dk lfn'k lehdj.k Kkr dhft, tks ewy ¯cnq ls dh nwjh ij gS 

vkSj ewy ¯cnq ls bldk vfHkyac lfn'k  gSA

gy eku yhft,  gSA rc

blfy, lery dk vHkh"V lehdj.k

mnkgj.k 14 lery = 0 ij ewy ¯cnq ls Mkys x, yac bdkbZ lfn'k dh fno~Q&dkslkbu Kkr dhft,A

gy lery osQ Kkr lehdj.k dks bl izdkj O;Dr fd;k tk ldrk gS%

vc  ... (1)

blfy, (1) osQ nksuksa i{kksa dks 7 ls Hkkx djus ij ge ikrs gSa fd

=

tks fd lery dk lehdj.k osQ :i dk gSA

blls Li"V gS fdlery osQ yac bdkbZ lfn'k gS tks ewy ¯cnq ls xqtjrk gSA bl izdkj dh fno~Q&dkslkbu gS

mnkgj.k 15 lery 2x – 3y + 4z – 6 = 0 dh ewy ¯cnq ls nwjh Kkr dhft,A

gy D;ksafd ry osQ vfHkyac osQ fno~Q&vuqikr 2, –3, 4 gSa blfy, bldh fno~Q&dkslkbu gSa%

vFkkZr~ 

blfy, lehdj.k 2x – 3y + 4z – 6 = 0 vFkkZr~ 2x – 3y + 4z = 6 dks ls Hkkx djus ij ge izkIr djrs gSa%aA

vkSj ;g lx + my + nz = d, osQ :i esa gS tgk¡ ewy ¯cnq ls lery dh nwjh d gSA blfy, lery dh ewy ¯cnq ls nwjh gSaAs

mnkgj.k 16 ewy ¯cnq ls lery 2x – 3y + 4z – 6 = 0 ij Mkys x, yac osQ ikn osQ funsZ'kkad Kkr dhft,A

gy eku yhft, ewy ¯cnq ls lery in Mkys x, yac osQ ikn P osQ funsZ'kkad (x1, y1, z1) gS (vko`Qfr 11.11)A

rc js[kk OP osQ fno~Q&vuqikr x1, y1, z1 gSaA

lery dh lehdj.k dks vfHkyac osQ :i esa fy[kus ij ge ikrs gSa fd

tgk¡ OP osQ fno~Q&vuqikr  gSaA

D;ksafd ,d js[kk osQ fno~Q&dkslkbu vkSj fno~Q&vuqikr lekuqikrh gksrs gSaA vr%

vFkkZr~ 

bu ekuksa dks lery osQ lehdj.k esa izfrLFkkfir djus ij ge ikrs gSa fd k = 

vr% yac osQ ikn osQ funsZ'kkad  gSA

11.6.2 ,d fn, lfn'k osQ vuqyac rFkk fn, ¯cnq ls gksdj tkus okys lery dk lehdj.k (Equation of a plane perpendicular to a given vector and passing through a given point)

varfj{k esa] ,d fn, x, lfn'k osQ vuqyac vusoQ lery gks ldrs gSa ijarq ,d fn, x, ¯cnq P x1, y1, z1) ls bl izdkj dk osQoy ,d lery dk vfLrRo gksrk gS (nsf[k, vko`Qfr 11-12)A

eku yhft, fd lery ,d ¯cnq A, ftldh fLFkfr lfn'k  gS] ls tkrk gS vkSj lfn'k  osQ vuqyac gSA eku yhft, fd lery ij fdlh ¯cnq P dk fLFkfr lfn'k gS (vko`Qfr 11-13)A

rc ¯cnq P lery esa fLFkr gksrk gS] ;fn vkSj osQoy ;fn , ij yac gS] vFkkZr~ = 0. ijarq  blfy,

 ......(1)

;g lery dk lfn'k lehdj.k gSA

dkrhZ; :i (Cartesian Form)

eku yhft, fd fn;k ¯cnq A (x1, y1, z1) vkSj lery ij dksbZ ¯cnq P (x, y, z) osQ

fno~Q&vuqikr  A, B rFkk  C gSa] rc

osQ fno~Q&vuqikr A, B rFkk C gSa] rc

vc 

blfy,

vFkkZr~ A(x – x1) + B(y – y1) + C(z – z1) = 0

mnkgj.k 17 ml lery dk lfn'k vkSj dkrhZ; lehdj.k Kkr dhft,] tks ¯cnq (5, 2, – 4) ls tkrk gS vkSj 2, 3, – 1 fno~Q&vuqikr okyh js[kk ij yac gSA

gy ge tkurs gSa fd ¯cnq (5, 2, – 4) dk fLFkfr lfn'k  gS vkSj lery osQ yac dk vfHkyac lfn'k  gSA

blfy, lery dk lfn'k lehdj.k   ls iznÙk gSA

;k 

(1) dks dkrhZ; :i esa :ikarj.k djus ij ge ikrs gSa] fd

;k

vFkkZr~ 2x + 3y – z = 20

tks lery dk dkrhZ; lehdj.k gSA

11.6.3 rhu vlajs[kh; ¯cnqvksa ls gksdj tkus okys lery dk lehdj.k (Equation of a plane passing through three non-collinear points)

eku yhft, lery ij fLFkr rhu vlajs[k ¯cnqvksa R, S vkSj T osQ fLFkfr lfn'k Øe'k% gSa (vko`Qfr 11.14)A

lfn'k  vkSj fn, lery esa gSaA blfy, lfn'k  ¯cnqvksa R, S vkSj T dks vUrfoZ"V djus okys lery ij yac gksxkA eku yhft, lery esa dksbZ ¯cnq P dk fLFkfr lfn'k gSA blfy, R ls tkus okys rFkk lfn'k  ij yac] lery dk lehdj.k gSA

;k 

;g rhu vlajs[k ¯cnqvksa ls xqtjus okys lery osQ lehdj.k dk lfn'k izk:i gSA

;s lery ,d iqLrd osQ i`"Bksa dh Hkk¡fr gksaxs tgk¡ ¯cnqvksa R, S vkSj T dks varfoZ"V djus okyh js[kk iqLrd osQ i`"Bksa osQ ca/u okys LFkku dk lnL; gSA

dkrhZ; :i (Cartesian Form)

eku yhft, ¯cnqvksa R, S vkSj T osQ funsZ'kkad Øe'k% (x1, y1, z1), (x2, y2, z2) vkSj (x3, y3, z3) gSaA eku yhft, fd lery ij fdlh ¯cnq P osQ funsZ'kkad (x, y, z) o bldk fLFkfr lfn'k gSA rc

bu ekuksa dks lfn'k izk:i osQ lehdj.k (1) esa izfrLFkkiu djus ij ge ikrs gSa fd

tks rhu ¯cnqvksa (x1, y1, z1), (x2, y2, z2) vkSj (x3, y3, z3) ls xq”kjus okys lery osQ lehdj.k dk dkrhZ; izk:i gSA

mngkj.k 18 ¯cnqvksa R(2, 5, – 3), S(– 2, – 3, 5) vkSj T(5, 3,– 3) ls tkus okys lery dk lfn'k lehdj.k Kkr dhft,A

gy eku yhft, 

rcls tkus okys lery dk lfn'k lehdj.k fuEufyf[kr gSa%

;k 

vFkkZr~ 

11.6.4 lery osQ lehdj.k dk var% [kaM&:i (Intercept form of the equation of a plane)

bl vuqPNsn esa] ge lery osQ lehdj.k dks] mlosQ }kjk funsZ'kka{kksa ij dVs var% [kaM osQ :i esa Kkr djsaxsA eku yhft, lery dk lehdj.k

eku yhft, lery }kjk x, y, vkSj z-v{kksa ij dVs var% [kaM Øe'k% a, b vkSj c (vko`Qfr 11.16) gSaA

Li"Vr% lery x, y vkSj z-v{kksa ls Øe'k% ¯cnqvksa (a, 0, 0), (0, b, 0), vkSj (0, 0, c) ij feyrk gSA

blfy, 

bu ekuksa dks lery osQ lehdj.k (1) esa izfrLFkkfir djus vkSj ljy djus ij ge ikrs gSa fd

tks var% [kaM :i esa lery dk vHkh"V lehdj.k gSA

mnkgj.k 19 ml lery dk lehdj.k Kkr dhft, tks x, y vkSj z-v{kksa ij Øe'k% 2] 3 vkSj 4 var% [kaM dkVrk gSA

;gk¡ a = 2, b = 3, c = 4 Kkr gSaA

a, b vkSj c osQ bu ekuksa dks (1) esa izfrLFkkfir djus ij ge lery dk vHkh"V lehdj.k  ;k 6x + 4y + 3z = 12 izkIr djrs gSaA

11.6.5 nks fn, leryksa osQ izfrPNsnu ls gksdj tkus okyk lery (Plane passing through the intersection of two given planes)

eku yhft, π1 vkSj π2 nks lery] ftuosQ lehdj.k Øe'k% gSa buosQ izfrPNsnu js[kk ij fLFkr fdlh ¯cnq dk fLFkfr lfn'k bu nksauksa lehdj.kksa dks larq"V djsxk (vko`Qfr 11.17)A

;fn bl js[kk ij fLFkr fdlh ¯cnq dh fLFkfr lfn'k gS] rks

blhfy, λ osQ lHkh okLrfod ekuksa osQ fy, ge ikrs gSa fd

D;ksafd  LosPN gS blfy, ;g js[kk osQ fdlh ¯cnq dks larq"V djrk gSA

bl izdkj lehdj.k  lery π3 dks fu:fir djrk gS tks ,slk gS fd ;fn dksbZ lfn'k, π1 vkSj π2, osQ lehdj.kksa dks larq"V djrk gS rks og π3 dks vo'; larq"V djsxkA vr% leryksa  osQ izfrPNsnu js[kk ls tkus okys fdlh lery dk lehdj.k = d1 + λd2 gSA ... (1)

dkrhZ; :i (Cartesian Form)

dkrhZ; :i osQ fy, ekuk

vkSj 

rks (1) dk ifjofrZr :i gS%

  

;k (A1x + B1y + C1z – d1) + λ(A2x + B2y + C2z – d2) = 0 ... (2)

tks izR;sd λ osQ fy, fn, leryksa osQ izfrPNsnu js[kk ls gksdj tkus okys fdlh lery dk dkrhZ; lehdj.k gSA

mnkgj.k 20 leryksa osQ izfrPNsnu rFkk ¯cnq (1,1,1) ls tkus okys lery dk lfn'k lehdj.k Kkr dhft,A

gy ;gk¡  vkSj vkSj d1 = 6 vkSj d2 = –5 gSaA

blfy, lw=k  dk iz;ksx djus ij]

;k 

tgk¡ λ ,d okLrfod la[;k gSA

;k 

;k 

vc iz'ukuqlkj vHkh"V lery ¯cnq (1, 1, 1) ls tkrk gS] vr% ;g ¯cnq] (2) dks larq"V djsxk vFkkZr~

;k λ = 

λ osQ bl eku dks (1) esa izfrLFkkfir djus ij ge ikrs gSa] fd

;k 

;k 

tks lery dk vHkh"V lfn'k lehdj.k gSA

11.7 nks js[kkvksa dk lg&ryh; gksuk (Coplanarity of two lines)

eku yhft, fd nks Kkr js[kk,¡%

rFkk 

js[kk (1) ¯cnq A, ftldh fLFkfr lfn'k gS] ls gksdj tkrh gS rFkk  osQ lekarj gSA js[kk (2) ¯cnq B (ftldh fLFkfr lfn'k  gS] ls gksdj tkrh gS rFkk  osQ lekarj gSA rc

Kkr js[kk,¡ lg&ryh; gSa] ;fn osQoy ;fn lg&ryh; gSaA vFkkZr~

dkrhZ; :i (Cartesian Form)

eku yhft, fd ¯cnqvksa A vkSj B (osQ funsZ'kkad Øe'k% (x1, y1, z1) vkSj (x2, y2, z2) gSaA eku yhft, fd  vkSj  osQ fno~Q&vuqikr Øe'k% a1, b1, c1 rFkk a2, b2, c2 gSA rc

Kkr js[kk,¡ lg&ryh; gSa] ;fn vkSj osQoy ;fn  ftls fuEufyf[kr dkrhZ; :i esa O;Dr dj ldrs gSaA

... (4)

mnkgj.k 21 n'kkZb, fd js[kk,¡

gy ;gk¡ gesa Kkr gS fd x1 = – 3, y1 = 1, z1 = 5, a1 = – 3, b1 = 1, c1 = 5

x2 = – 1, y2 = 2, z2 = 5, a2 = –1, b2 = 2, c2 = 5

vc fuEufyf[kr lkjf.kd ysus ij ge ikrs gSa fd

blfy, js[kk,¡ le&ryh; gSaA

11.8 nks leryksa osQ chp dk dks.k (Angle between two planes)

ifjHkk"kk 2 nks leryksa osQ chp dk dks.k muosQ vfHkyacksa osQ eè;LFk dks.k }kjk ifjHkkf"kr gS (vko`Qfr 11.18 (a))A è;ku nhft, fd ;fn nks leryksa osQ chp dk dks.k θ gS rks 180 – θ (vko`Qfr 11.18 (b)) Hkh muosQ chp dk dks.k gSA ge U;wu dks.k dks gh leryksa osQ chp dk dks.k ysaxsA

eku yhft, fd leryksa]  osQ chp dk dks.k θ gSA rc fdlh lkoZ ¯cnq ls leryksa ij [khaps x, vfHkyacksa osQ chp dk dks.k θ gSA

rc 

dkrhZ; :i (Cartesian Form)

eku yhft, leryksa%

A1 x + B1 y + C1z + D1 = 0 vkSj A2x + B2 y + C2 z + D2 = 0

osQ chp dk dks.k θ gSA

cosθ = 

mnkgj.k 22 nks leryksa  osQ chp dk dks.k lfn'k fof/ }kjk Kkr dhft,A

gy nks leryksa osQ chp dk dks.k ogh gS tks muosQ vfHkyacksa osQ chp dk dks.k gSA leryksa osQ fn, x, lehdj.kksa ls leryksa osQ lfn'k vfHkyac

mnkgj.k 23 nks leryksa  osQ chp dk dks.k Kkr dhft,A

gy leryksa dh Kkr lehdj.kksa dh rqyuk lehdj.kksa

ls djus ij ge ikrs gSa fd%  

iqu% cos θ 

blfy,  

11.9 lery ls fn, x, ¯cnq dh nwjh (Distance of a point from a plane)

lfn'k :i (Vector Form)

,d ¯cnq P ftldk fLFkfr lfn'k  vkSj ,d lery π1 ftldk lehdj.k  (vko`Qfr 11.19) ij fopkj dhft,A

iqu% ¯cnq P ls lery π1 osQ lekarj lery π2 ij fopkj dhft,A lery π2 osQ vfHkyac bdkbZ lfn'k  gSA vr% bldk lehdj.k  gSA

vFkkZr~  

vr%] ewy ¯cnq ls bl lery dh nwjh ON′  gSA blfy, P ls lery π1 ls nwjh (vko`Qfr 11.21 (a))

gS] tks ,d ¯cnq ls Kkr lery ij yac dh yackbZ gSA vko`Qfr 11.19 (b) osQ fy, ge blh izdkj dk ifj.kke LFkkfir dj ldrs gSaA

dkrhZ; :i (Cartesian Form)

eku yhft, fd P (x1, y1, z1) ,d fn;k ¯cnq gS ftldk fLFkfr lfn'k  gS rFkk fn, lery dk dkrhZ; lehdj.k

rc 

vr% (1) osQ }kjk P ls lery ij yac dh yackbZ

 

=

mnkgj.k 24 ¯cnq (2, 5, – 3) dh lery ls nwjh Kkr dhft,A

gy ;gk¡ 

blfy, ¯cnq (2, 5, – 3) dh fn, lery ls nwjh gS%

11.10 ,d js[kk vkSj ,d lery osQ chp dk dks.k (Angle between a line and a plane)

ifjHkk"kk 2 ,d js[kk vkSj ,d lery osQ chp dk dks.k] js[kk vkSj lery osQ vfHkyac osQ chp osQ dks.k dk dks.k (complementary angle) iwjd gksrk gS (vko`Qfr 11.20)A

lfn'k :i (Vector Form)

eku yhft, fd js[kk dk lehdj.k  gS rFkk lery dk lehdj.k gSA rc js[kk vkSj lery osQ vfHkyac osQ chp dk dks.k θ, fuEufyf[kr lw=k }kjk O;Dr fd;k tk ldrk gSA

vkSj bl izdkj js[kk vkSj lery osQ chp dk dks.k φ, 90° – θ, }kjk iznÙk gS vFkkZr~

sin (90° – θ) = cos θ

vFkkZr~] s 

mnkgj.k 25 js[kk vkSj lery 10 x+2y – 11 z = 3 osQ chp dk dks.k Kkr dhft,A

gy eku yhft, fd js[kk vksj lery osQ vfHkyac osQ chp dk dks.k θ gSA fn, x, js[kk rFkk lery osQ lehdj.kksa dks lfn'k :i esa O;Dr djus ij ge

vkSj 

;gk¡ 

vr% 

iz'ukoyh 11-3

1. fuEufyf[kr iz'uksa esa ls izR;sd esa lery osQ vfHkyac dh fno~Q&dkslkbu vkSj ewy ¯cnq ls nwjh Kkr dhft,%

(a) z = 2 (b) x + y + z = 1

(c) 2x + 3y-z = 5 (d) 5y + 8 = 0

2. ml lery dk lfn'k lehdj.k Kkr dhft,] tks ewy ¯cnq ls 7 ek=kd nwjh ij gS] vkSj lfn'k ij vfHkyac gSA

3. fuEufyf[kr leryksa dk dkrhZ; lehdj.k Kkr dhft,%

4. fuEufyf[kr fLFkfr;ksa esa] ewy ¯cnq ls [khaps x, yac osQ ikn osQ funsZ'kkad Kkr dhft,A

5. fuEufyf[kr izfrca/ksa osQ varxZr leryksa dk lfn'k ,oa dkrhZ; lehdj.k Kkr dhft, tks%

6. mu leryksa dk lehdj.k Kkr dhft, tks fuEufyf[kr rhu ¯cnqvksa ls xqtjrk gSA

(a) (1, 1, – 1), (6, 4, – 5), (– 4, – 2, 3)

(b) (1, 1, 0), (1, 2, 1), (– 2, 2, – 1)

7. lery 2x + y – z = 5 }kjk dkVs x, var% [kaMksa dks Kkr dhft,A

8. ml lery dk lehdj.k Kkr dhft, ftldk y&v{k ij var%[kaM 3 vkSj tks ry Z OX osQ lekarj gSA

9. ml lery dk lehdj.k Kkr dhft, tks leryksa 3x – y + 2z – 4 = 0 vkSj
x + y + z – 2 = 0 osQ izfrPNsnu rFkk ¯cnq (2, 2, 1) ls gksdj tkrk gSA

10. ml lery dk lfn'k lehdj.k Kkr dhft, tks leryksa 

  osQ izfrPNsnu js[kk vkSj (2, 1, 3) ls gksdj tkrk gSA

11. ryksa x + y + z = 1 vkSj 2x + 3y + 4z = 5 osQ izfrPNsnu js[kk ls gksdj tkus okys rFkk ry x – y + z = 0 ij yacor~ ry dk lehdj.k Kkr dhft,A

12. leryksa] ftuosQ lfn'k lehdj.k  vkSj 

 gSa] osQ chp dk dks.k Kkr dhft,A

13. fuEufyf[kr iz'uksa esa Kkr dhft, fd D;k fn, x, leryksa osQ ;qXe lekarj gS vFkok yacor~ gSa] vkSj ml fLFkfr esa] tc ;s u rks lekarj gS vkSj u gh yacor~~ rks muosQ chp dk dks.k Kkr dhft,A

(a) 7x + 5y + 6z + 30 = 0 vkSj 3x – y – 10z + 4 = 0

(b) 2x + y + 3z – 2 = 0 vkSj x – 2y + 5 = 0

(c) 2x – 2y + 4z + 5 = 0 vkSj 3x – 3y + 6z – 1 = 0

(d) 2x – y + 3z – 1 = 0 vkSj 2x – y + 3z + 3 = 0

(e) 4x + 8y + z – 8 = 0 vkSj y + z – 4 = 0

14. fuEufyf[kr iz'uksa esa izR;sd fn, x, ¯cnq ls fn, x, laxr leryksa dh nwjh Kkr dhft,A

¯cnq lery

(a) (0, 0, 0) 3x – 4y + 12 z = 3

(b) (3, – 2, 1) 2x – y + 2z + 3 = 0

(c) (2, 3, – 5) x + 2y – 2z = 9

(d) (– 6, 0, 0) 2x – 3y + 6z – 2 = 0

fofo/ mnkgj.k

mnkgj.k 26 ,d js[kk] ,d ?ku osQ fod.kks± osQ lkFk α, β, γ, δ, dks.k cukrh gS rks fl¼ dhft, fd

cos2 α + cos2 β + cos2 γ + cos2 δ = 

gy ,d ?ku] ,d ledksf.kd "kV~iQydh; gksrk gS ftldh yackbZ] pkSM+kbZ vkSj Å¡pkbZ leku gksrs gSaA

eku yhft, fd OADBEFCG ,d ?ku ftldh izR;sd Hkqtk a yackbZ dh gS (vko`Qfr 11.21)A

OE, AF, BG vkSj CD pkj fod.kZ gSaA

nks ¯cnqvksa O rFkk E dks feykus okyh js[kk OE vFkkZr~ fod.kZ OE osQ fno~Q&dkslkbu

vFkkZr~ , ,

gSaA blh izdkj AF, BG vkSj CD dh fno~Q&dkslkbu Øe'k%

– , , ; , – ,  vkSj  , , – , gSaA

eku yhft, nh xbZ js[kk tks OE, AF, BG, vkSj CD, osQ lkFk Øe'k% α, β, γ, vkSj δ dks.k cukrh gS] dh fno~Q&dkslkbu l, m, n gSaA

rc cosα = (l + m+ n); cos β = (– l + m + n)

cosγ = (l – m + n); cos δ = (l + m – n)

oxZ djosQ tksM+us ij ge ikrs gSa fd

cos2 α + cos2 β + cos2 γ + cos2 δ

mnkgj.k 27 ml ry dk lehdj.k Kkr dhft, ftlesa ¯cnq (1, – 1, 2) varfoZ"V gS vkSj tks leryksa 2x + 3y-2z=5vkSj x+2y – 3z = 8 esa ls izR;sd ij yac gSA

gy fn, x, ¯cnq dks varfoZ"V djus okys lery dk lehdj.k

leryksa 2x + 3y-2z=5 vkSj x+2y – 3z = 8, osQ lkFk (1) }kjk iznÙk lery ij yac gksus osQ izfrca/ dk iz;ksx djus ij ge ikrs gSa fd

bu lehdj.kksa dks gy djus ij ge ikrs gSa fd A = – 5C vkSj B = 4C

vr% vHkh"V lehdj.k gS%

– 5C (x – 1) + 4 C (y + 1) + C(z – 2) = 0

vFkkZr~  

mnkgj.k 28 ¯cnq P(6, 5, 9) ls ¯cnqvksa A (3, – 1, 2), B (5, 2, 4) vkSj C(– 1, – 1, 6) }kjk
fu/kZfjr lery dh nwjh Kkr dhft,A

gy eku yhft, fd lery esa rhu ¯cnq A, B, rFkk C gSaA ¯cnq P ls lery ij yac dk ikn D gSA gesa vHkh"V nwjh PD Kkr djuh gS tgk¡ PD , ij iz{ksi gSA

vr%

iqu% 

vkSj 

vr% 

fodYir% ¯cnq A, B vkSj C ls xq”kjus okys lery dk lehdj.k Kkr dhft, vkSj rc ¯cnq P dh lery ls nwjh Kkr dhft,A

mnkgj.k 29 n'kkZb, fd js[kk,¡

vkSj  lg&ryh; gSaA

gy ;gk¡ Kkr gS fd

vc lkjf.kd

ij fopkj dhft,A

rhljs LraHk dks igys LraHk esa tksM+us ij ge ikrs gSaA

D;ksafd izFke vkSj f}rh; LraHk leku gSaA vr% nksuksa js[kk,¡ lg&ryh; gSaA

mnkgj.k 30 ml ¯cnq osQ funsZ'kkad Kkr dhft, tgk¡ ¯cnqvksa A(3, 4, 1) vkSj B(5, 1, 6) dks feykus okyh js[kk XY-ry dks dkVrh gSaA

gy ¯cnqvksa A vkSj B (ls tkus okyh js[kk dk lfn'k lehdj.k%

vFkkZr~ eq 

eku yhft, P og ¯cnq gS tgk¡ js[kk AB, XY-ry dks izfrPNsn djrh gSA rc ¯cnq P dk fLFkfr lfn'k  osQ :i esa gSA

;g ¯cnq vo'; gh lehdj.k (1) dks larq"V djrk gSA (D;ksa?)

vFkkZr~ 

 osQ xq.kkadksa dh rqyuk djus ij ge ikrs gSa

x = 3 + 2 λ

y = 4 – 3 λ

0 = 1 + 5 λ

mijksDr lehdj.kksa dks gy djus ij ge ikrs gSa fd

vr% vHkh"V ¯cnq osQ funsZ'kkad  gSaA

vè;k; 11 ij fofo/ iz'ukoyh

1. fn[kkb, fd ewy ¯cnq ls (2] 1] 1) feykus okyh js[kk] ¯cnqvksa (3] 5 &1) vkSj
(4] 3] &1) ls fu/kZfjr js[kk ij yac gSA

2. ;fn nks ijLij yac js[kkvksa dh fno~Q&dkslkbu l1, m1, n1 vkSj l2, m2, n2 gkas rks fn[kkb, fd bu nksuksa ij yac js[kk dh fno~Q&dkslkbu

m1 n2 – m2 n1, n1 l2 – n2 l1, l1 m2 – l2 – m1 gSaA

3. mu js[kkvksa osQ eè; dks.k Kkr dhft,] ftuosQ fno~Q&vuqikr a, b, c vkSj b – c, c – a, a-b gSaA

4. x-v{k osQ lekarj rFkk ewy&¯cnq ls tkus okyh js[kk dk lehdj.k Kkr dhft,A

5. ;fn ¯cnqvksa A, B, C, vkSj D osQ funsZ'kkad Øe'k% (1, 2, 3), (4, 5, 7), (– 4, 3, – 6) vkSj (2, 9, 2) gSa rks ABvkSj CD js[kkvksa osQ chp dk dks.k Kkr dhft,A

6. ;fn js[kk,¡  ijLij yac gksa rks k dk eku Kkr dhft,A

7. ¯cnq (1, 2, 3) ls tkus okyh rFkk ry  ij yacor~ js[kk dk

lfn'k lehdj.k Kkr dhft,A

8. ¯cnq (a, b, c) ls tkus okys rFkk ry   osQ lekarj ry dk lehdj.k Kkr dhft,A

9.

osQ chp dh U;wure nwjh Kkr dhft,A

10. ml ¯cnq osQ funsZ'kkad Kkr dhft, tgk¡ ¯cnqvksa (5, 1, 6) vkSj (3, 4, 1) dks feykus okyh js[kk YZ-ry dks dkVrh gSA

11. ml ¯cnq osQ funsZ'kkad Kkr dhft, tgk¡ ¯cnqvksa (5, 1, 6) vkSj (3, 4, 1) dks feykus okyh js[kk ZX-ry dks dkVrh gSA

12. ml ¯cnq osQ funsZ'kkad Kkr dhft, tgk¡ ¯cnqvksa (3, – 4, – 5) vkSj (2, – 3, 1) ls xq”kjus okyh js[kk] lery 2x + y + z=7osQ ikj tkrh gSA

13. ¯cnq (– 1, 3, 2) ls tkus okys rFkk leryksa x+2y + 3z=5vkSj 3x + 3y + z = 0 esa ls izR;sd ij yac lery dk lehdj.k Kkr dhft,A

14. ;fn ¯cnq (1, 1, p) vkSj (– 3, 0, 1) lery  ls leku nwjh ij fLFkr gksa] rks p dk eku Kkr dhft,A

15. leryksa  osQ izfrPNsnu js[kk ls tkus okys rFkk x-v{k osQ lekarj ry dk lehdj.k Kkr dhft,A

16. ;fn O ewy ¯cnq rFkk ¯cnq P osQ funsZ'kkad (1, 2, – 3), gSa rks ¯cnq P ls tkus okys rFkk OP osQ yacor~ ry dk lehdj.k Kkr dhft,A 

17. leryksa  osQ izfrPNsnu js[kk dks varfoZ"V djus okys rFkk ry  osQ yacor~ ry dk lehdj.k Kkr dhft,A

18. ¯cnq (– 1, – 5, – 10) ls js[kk  vkSj lery  osQ izfrPNsnu ¯cnq osQ eè; dh nwjh Kkr dhft,A

19. ¯cnq (1, 2, 3) ls tkus okyh rFkk leryksa  vkSj  osQ lekarj js[kk dk lfn'k lehdj.k Kkr dhft,A

20. ¯cnq (1, 2, – 4) ls tkus okyh vkSj nksuksa js[kkvksa  vkSj 

 ij yac js[kk dk lfn'k lehdj.k Kkr dhft,A

21. ;fn ,d lery osQ var%[kaM a, b, c gSa vkSj bldh ewy ¯cnq ls nwjh p bdkbZ gSa rks fl¼ dhft, fd 

iz'uksa 22 vkSj 23 esa lgh mÙkj dk pquko dhft,A

22. 

23. 

(A) ijLij yac (B) lekarj