1300mathformulas

CHAPTER 4. TRIGONOMETRY = =± N − Åçë Oα = O í~å α = N + Åçë Oα O N + í~åO α O = 399. Åçí α = Åçë α = ± ÅëÅO α − N = N + Åçë Oα = ëáå Oα = ëáå α ëáå Oα N − Åçë Oα = =± N + Åçë Oα = N − í~åO α = N − Åçë Oα O O í~å α O = ëÉÅ α = N = ± N + í~åO α Åçë α O 400. N+ í~å O α = α = N− í~å O O = N N + í~åO α ëáå α O 401. ÅëÅ α = = ± N + ÅçíO α = O í~å α = O = = = 4.9 Addition and Subtraction Formulas = 402. ëáå(α + β) = ëáåαÅçëβ + ëáåβÅçëα = = 403. ëáå(α − ó) = ëáåαÅçëβ − ëáåβÅçëα = = 404. Åçë(α + β) = Åçëα Åçëβ − ëáåα ëáåβ = = 405. Åçë(α − β) = Åçëα Åçëβ + ëáåα ëáåβ = 91

CHAPTER 4. TRIGONOMETRY 406. í~å(α + β) = í~åα + í~åβ = N− í~åα í~åβ = 407. í~å(α − β) = í~åα − í~åβ = N+ í~åα í~åβ = 408. Åçí(α + β) = N− í~åα í~åβ = í~åα + í~åβ = 409. Åçí(α − β) = N+ í~åα í~åβ = í~åα − í~åβ = = = 4.10 Double Angle Formulas = 410. ëáå Oα = Oëáå α ⋅ Åçë α = = 411. Åçë Oα = ÅçëO α − ëáåO α = N− OëáåO α = OÅçëO α −N = = 412. í~å Oα = Oí~å α = Åçí α O α = N− í~åO α − í~å = 413. Åçí Oα = ÅçíO α −N = Åçí α − í~å α = OÅçí α O = = = = = = 92

CHAPTER 4. TRIGONOMETRY 4.11 Multiple Angle Formulas = 414. ëáå Pα = Pëáå α − QëáåP α = PÅçëO α ⋅ ëáå α − ëáåP α = = 415. ëáå Qα = Qëáå α ⋅ Åçë α − UëáåP α ⋅ Åçë α = = 416. ëáå Rα = Rëáå α − OMëáåP α + NSëáåR α = = 417. Åçë Pα = QÅçëP α − PÅçë α = ÅçëP α − PÅçë α ⋅ ëáåO α = = 418. Åçë Qα = UÅçëQ α − UÅçëO α + N= = 419. Åçë Rα = NSÅçëR α − OMÅçëP α + RÅçë α = = 420. í~å Pα = Pí~å α − í~åP α = N− Pí~åO α = 421. í~å Qα = N Q í~å α − Q í~åP α = − Sí~åO α + í~åQ α = 422. í~å Rα = í~åR α −NMí~åP α + Rí~å α = N−NMí~åO α + Rí~åQ α = 423. Åçí Pα = ÅçíP α − PÅçí α = PÅçíO α −N = 424. Åçí Qα = N− Sí~åO α + í~åQ α == Q í~å α − Q í~åP α = 93

CHAPTER 4. TRIGONOMETRY 425. Åçí Rα = N−NMí~åO α + Rí~åQ α α = í~åR α −NMí~åP α + Rí~å = = = 4.12 Half Angle Formulas = 426. ëáå α = ± N− Åçë α = OO = 427. Åçë α = ± N+ Åçë α = O O = 428. í~å α = ± N− Åçë α = ëáå α = N− Åçë α = ÅëÅ α − Åçí α = O N+ Åçë α N+ Åçë α ëáå α = 429. Åçí α = ± N+ Åçë α = ëáå α = N+ Åçë α = ÅëÅ α + Åçí α = O N− Åçë α N− Åçë α ëáå α = = = 4.13 Half Angle Tangent Identities = Oí~å α 430. O ëáå α = í~åO α = N + O = 94

CHAPTER 4. TRIGONOMETRY N− í~åO α 431. O Åçë α = α = N+ í~åO O = Oí~å α 432. O í~å α = í~åO α = N− O = N− í~åO α 433. O Åçí α = Oí~å α = O = = = 4.14 Transforming of Trigonometric Expressions to Product = 434. ëáå α + ëáå β = Oëáå α + β Åçë α − β = OO = 435. ëáå α − ëáå β = OÅçë α + β ëáå α − β = OO = 436. Åçë α + Åçë β = OÅçë α + β Åçë α − β = OO = 437. Åçë α − Åçë β = −Oëáå α + β ëáå α − β = OO = 95

CHAPTER 4. TRIGONOMETRY 438. í~å α + í~å β = ëáå(α + β) = Åçë α ⋅ Åçë β = í~å α − í~å β = ëáå(α − β) = 439. Åçë α ⋅ Åçë β = Åçí α + Åçí β = ëáå(β + α) = 440. ëáå α ⋅ ëáå β = Åçí α − Åçí β = ëáå(β − α) = 441. ëáå α ⋅ ëáå β = 442. Åçë α + ëáå α = O Åçë π − α  = O ëáå π + α  =  Q   Q  = 443. Åçë α − ëáå α = O ëáå π − α  = O Åçë π + α  =  Q   Q  = í~å α + Åçí β = Åçë(α − β) = 444. Åçë α ⋅ ëáå β = í~å α − Åçí β = − Åçë(α + β) = 445. Åçë α ⋅ ëáå β = 446. N+ Åçë α = OÅçëO α = O = 447. N− Åçë α = OëáåO α = O = 96

CHAPTER 4. TRIGONOMETRY 448. N + ëáå α = O Åçë O  π − α  =  Q O  = 449. N − ëáå α = O ëáå O  π − α  =  Q O  = = = 4.15 Transforming of Trigonometric Expressions to Sum = 450. ëáå α ⋅ ëáå β = Åçë(α − β)− Åçë(α + β) = O = 451. Åçë α ⋅ Åçë β = Åçë(α − β)+ Åçë(α + β) = O = 452. ëáå α ⋅ Åçë β = ëáå(α − β)+ ëáå(α + β) = O = 453. í~å α ⋅ í~å β = í~å α + í~å β = Åçí α + Åçí β = 454. Åçí α ⋅ Åçí β = Åçí α + Åçí β = í~å α + í~å β = 455. í~å α ⋅ Åçí β = í~å α + Åçí β = Åçí α + í~å β = = = 97

CHAPTER 4. TRIGONOMETRY 4.16 Powers of Trigonometric Functions = 456. ëáåO α = N− ÅçëOα = O = 457. ëáåP α = Pëáå α − ëáåPα = Q = 458. ëáåQ α = Åçë Qα − Q Åçë Oα + P = U = 459. ëáåR α = NM ëáå α − Rëáå Pα + ëáå Rα = NS = 460. ëáåS α = NM −NRÅçë Oα + SÅçë Qα − Åçë Sα = PO = 461. ÅçëO α = N+ Åçë Oα = O = 462. ÅçëP α = PÅçë α + Åçë Pα = Q = 463. ÅçëQ α = Åçë Qα + QÅçë Oα + P = U = 464. ÅçëR α = NMÅçë α + Rëáå Pα + Åçë Rα = NS = 465. ÅçëS α = NM + NRÅçë Oα + SÅçë Qα + Åçë Sα = PO = 98

CHAPTER 4. TRIGONOMETRY 4.17 Graphs of Inverse Trigonometric Functions = 466. fåîÉêëÉ=páåÉ=cìåÅíáçå== ó = ~êÅëáå ñ I= −N≤ ñ ≤ N I= − π ≤ ~êÅëáå ñ ≤ π K= O O = = = Figure 66. = 467. fåîÉêëÉ=`çëáåÉ=cìåÅíáçå== ó = ~êÅÅçë ñ I= −N≤ ñ ≤ NI= M ≤ ~êÅÅçë ñ ≤ π K= = 99

CHAPTER 4. TRIGONOMETRY = = Figure 67. = 468. fåîÉêëÉ=q~åÖÉåí=cìåÅíáçå== ó = ~êÅí~å ñ I= − ∞ ≤ ñ ≤ ∞ I= − π < ~êÅí~å ñ < π K= OO = ===== = = Figure 68. 100

CHAPTER 4. TRIGONOMETRY 469. fåîÉêëÉ=`çí~åÖÉåí=cìåÅíáçå== ó = ~êÅ Åçí ñ I= − ∞ ≤ ñ ≤ ∞ I= M < ~êÅÅçí ñ < π K= ===== = Figure 69. = 470. fåîÉêëÉ=pÉÅ~åí=cìåÅíáçå== ó = ~êÅëÉÅ=ñI ñ ∈(− ∞I −N]∪ [NI∞)I ~êÅ ëÉÅ ñ ∈ MI π  ∪  π I πK O   O = Figure 70. 101

CHAPTER 4. TRIGONOMETRY 471. fåîÉêëÉ=`çëÉÅ~åí=cìåÅíáçå== ó = ~êÅÅëÅ ñI ñ ∈(− ∞I −N]∪ [NI∞)I ~êÅ ÅëÅ ñ ∈ − π I M ∪  MI π K O   O = = Figure 71. = = 4.18 Principal Values of Inverse Trigonometric Functions 472. ñ= M= N = O = P N= O OO ~êÅëáå ñ = M° = PM° = QR° = SM° VM° ~êÅÅçë ñ = VM° SM° = QR° = PM° M° = ñ= −N − O − P −N= = O O O ~êÅëáå ñ = − PM° − QR° − SM° − VM° = = = ~êÅÅçë ñ = NOM° NPR° = NRM° = NUM° = = = 102

CHAPTER 4. TRIGONOMETRY 473. ñ= M= P N= P = − P −N= − P = P P ~êÅí~å ñ = M° = PM° QR° SM° − PM° − QR° − SM° = = ~êÅÅçí ñ = VM° SM° QR° PM° NOM° = NPR° NRM° = = = = = 4.19 Relations between Inverse Trigonometric Functions = 474. ~êÅëáå(− ñ) = −~êÅëáå ñ = = 475. ~êÅëáå ñ = π − ~êÅÅçë ñ = O = 476. ~êÅëáå ñ = ~êÅÅçë N− ñO I= M ≤ ñ ≤ NK= = 477. ~êÅëáå ñ = −~êÅÅçë N− ñO I= −N≤ ñ ≤ M K= = 478. ~êÅëáå ñ = ~êÅí~å ñ I= ñO < NK= N− ñO = 479. ~êÅëáå ñ = ~êÅ Åçí N− ñO I= M < ñ ≤ NK= ñ = 480. ~êÅëáå ñ = ~êÅ Åçí N− ñO − π I= −N≤ ñ < M K= ñ = 481. ~êÅÅçë(− ñ) = π − ~êÅÅçë ñ = 103

CHAPTER 4. TRIGONOMETRY 482. ~êÅÅçë ñ = π − ~êÅëáå ñ = O = 483. ~êÅÅçë ñ = ~êÅëáå N− ñO I= M ≤ ñ ≤ NK= = 484. ~êÅÅçë ñ = π − ~êÅëáå N− ñO I= −N≤ ñ ≤ M K= = 485. ~êÅÅçë ñ = ~êÅí~å N− ñO I= M < ñ ≤ NK= ñ = 486. ~êÅÅçë ñ = π + ~êÅí~å N− ñO I= −N≤ ñ < M K= ñ = 487. ~êÅÅçë ñ = ~êÅÅçí ñ I= −N≤ ñ ≤ NK= N− ñO = 488. ~êÅí~å(− ñ) = −~êÅí~å ñ = = 489. ~êÅí~å ñ = π − ~êÅÅçí ñ = O = 490. ~êÅí~å ñ = ~êÅëáå ñ = N+ ñO = 491. ~êÅí~å ñ = ~êÅÅçë N I= ñ ≥ M K= N+ ñO = 492. ~êÅí~å ñ = −~êÅÅçë N I= ñ ≤ M K= N+ ñO = 104

CHAPTER 4. TRIGONOMETRY 493. ~êÅí~å ñ = π − ~êÅí~å N I= ñ > M K= Oñ = 494. ~êÅí~å ñ = − π − ~êÅí~å N I= ñ < M K= Oñ = 495. ~êÅí~å ñ = ~êÅÅçí N I= ñ > M K= ñ = 496. ~êÅí~å ñ = ~êÅÅçí N − π I= ñ < M K= ñ = 497. ~êÅÅçí(− ñ) = π − ~êÅÅçí ñ = = 498. ~êÅÅçí ñ = π − ~êÅí~å ñ = O = 499. ~êÅÅçí ñ = ~êÅëáå N I= ñ > M K= N+ ñO = 500. ~êÅÅçí ñ = π − ~êÅëáå N I= ñ < M K= N+ ñO = 501. ~êÅÅçí ñ = ~êÅÅçë ñ = N+ ñO = 502. ~êÅÅçí ñ = ~êÅí~å N I= ñ > M K= ñ = 503. ~êÅÅçí ñ = π + ~êÅí~å N I= ñ < M K= ñ = = 105

CHAPTER 4. TRIGONOMETRY 4.20 Trigonometric Equations = tÜçäÉ=åìãÄÉêW=å= = = 504. ëáå ñ = ~ I= ñ = (−N)å ~êÅëáå~ + πå = = 505. Åçë ñ = ~ I= ñ = ± ~êÅÅçë ~ + Oπå = = 506. í~å ñ = ~ I= ñ = ~êÅí~å~ + πå = = 507. Åçí ñ = ~ I= ñ = ~êÅ Åçí ~ + πå = = = = 4.21 Relations to Hyperbolic Functions = fã~Öáå~êó=ìåáíW=á= = = 508. ëáå(áñ) = á ëáåÜ ñ = = 509. í~å(áñ) = á í~åÜ ñ = = 510. Åçí(áñ) = −á ÅçíÜ ñ = = 511. ëÉÅ(áñ) = ëÉÅÜ ñ = = 512. ÅëÅ(áñ) = −á ÅëÅÜ ñ = = = = 106

Chapter 5 Matrices and Determinants = = = = j~íêáÅÉëW=^I=_I=`= bäÉãÉåíë=çÑ=~=ã~íêáñW= ~á I= Äá I= ~áà I= Äáà I= Åáà = aÉíÉêãáå~åí=çÑ=~=ã~íêáñW= ÇÉí ^ = jáåçê=çÑ=~å=ÉäÉãÉåí= ~áà W= jáà = `çÑ~Åíçê=çÑ=~å=ÉäÉãÉåí= ~áà W= `áà = qê~åëéçëÉ=çÑ=~=ã~íêáñW= ^q I= ^ú = ^Çàçáåí=çÑ=~=ã~íêáñW= ~Çà ^ = qê~ÅÉ=çÑ=~=ã~íêáñW= íê ^ = fåîÉêëÉ=çÑ=~=ã~íêáñW= ^−N = oÉ~ä=åìãÄÉêW=â= oÉ~ä=î~êá~ÄäÉëW= ñá = k~íìê~ä=åìãÄÉêëW=ãI=å=== = = 5.1 Determinants = 513. pÉÅçåÇ=lêÇÉê=aÉíÉêãáå~åí= ÇÉí ^ = ~N ÄN = ~NÄO − ~OÄN = ~O ÄO = = = = = 107

CHAPTER 5. MATRICES AND DETERMINANTS 514. qÜáêÇ=lêÇÉê=aÉíÉêãáå~åí= ~NN ~NO ~NP ÇÉí ^ = ~ON ~OO ~OP = ~NN~ ~OO PP + ~NO~ ~OP PN + ~NP~ ~ON PO − = ~PN ~PO ~PP − ~NN~ ~OP PO − ~NO~ON~PP − ~NP~OO~PN = = 515. p~êêìë=oìäÉ=E^êêçï=oìäÉF= = = Figure 72. = 516. k-íÜ=lêÇÉê=aÉíÉêãáå~åí= ~NN ~NO K ~Nà K ~Nå ~ON ~OO K ~Oà K ~Oå ÇÉí ^ = K K KKK K = ~ áN ~áO K ~áà K ~ áå K KKKKK ~åN ~åO K ~åà K ~åå = 517. jáåçê= qÜÉ=ãáåçê= jáà =~ëëçÅá~íÉÇ=ïáíÜ=íÜÉ=ÉäÉãÉåí= ~áà =çÑ=å-íÜ=çêÇÉê= ã~íêáñ= ^= áë= íÜÉ= (å −N) -íÜ= çêÇÉê= ÇÉíÉêãáå~åí= ÇÉêáîÉÇ= Ñêçã= íÜÉ=ã~íêáñ=^=Äó=ÇÉäÉíáçå=çÑ=áíë=á-íÜ=êçï=~åÇ=à-íÜ=ÅçäìãåK=== = 108

CHAPTER 5. MATRICES AND DETERMINANTS 518. `çÑ~Åíçê= ( )`áà = − N á +à jáà = = 519. i~éä~ÅÉ=bñé~åëáçå=çÑ=å-íÜ=lêÇÉê=aÉíÉêãáå~åí= i~éä~ÅÉ=Éñé~åëáçå=Äó=ÉäÉãÉåíë=çÑ=íÜÉ=á-íÜ=êçï= å ∑ÇÉí ^ = ~ `áà áà I= á = NI OIKIå K= à=N i~éä~ÅÉ=Éñé~åëáçå=Äó=ÉäÉãÉåíë=çÑ=íÜÉ=à-íÜ=Åçäìãå= å ∑ÇÉí ^ = ~ `áà áà I= à = NI OIKIå K== á=N = = = 5.2 Properties of Determinants = 520. qÜÉ==î~äìÉ==çÑ=~=ÇÉíÉêãáå~åí=êÉã~áåë==ìåÅÜ~åÖÉÇ=áÑ=êçïë=~êÉ= ÅÜ~åÖÉÇ=íç=Åçäìãåë=~åÇ=Åçäìãåë=íç=êçïëK= = ~N ~O = ~N ÄN == ÄN ÄO ~O ÄO = 521. fÑ=íïç==êçïë==Eçê=íïç=ÅçäìãåëF=~êÉ==áåíÉêÅÜ~åÖÉÇI=íÜÉ=ëáÖå=çÑ= íÜÉ=ÇÉíÉêãáå~åí=áë=ÅÜ~åÖÉÇK= ~N ÄN = − ~O ÄO = ~O ÄO ~N ÄN = 522. fÑ=íïç=êçïë==Eçê=íïç=ÅçäìãåëF=~êÉ==áÇÉåíáÅ~äI=íÜÉ=î~äìÉ=çÑ=íÜÉ= ÇÉíÉêãáå~åí=áë=òÉêçK= ~N ~N = M = ~O ~O = 109

CHAPTER 5. MATRICES AND DETERMINANTS 523. fÑ==íÜÉ===ÉäÉãÉåíë==çÑ==~åó=êçï==Eçê=ÅçäìãåF=~êÉ=ãìäíáéäáÉÇ=Äó===== ~==Åçããçå==Ñ~ÅíçêI==íÜÉ==ÇÉíÉêãáå~åí==áë==ãìäíáéäáÉÇ==Äó==íÜ~í= Ñ~ÅíçêK= â~N âÄN = â ~N ÄN = ~O ÄO ~O ÄO = 524. fÑ==íÜÉ==ÉäÉãÉåíë==çÑ==~åó==êçï==Eçê==ÅçäìãåF=~êÉ=áåÅêÉ~ëÉÇ=Eçê= ÇÉÅêÉ~ëÉÇFÄó=Éèì~ä=ãìäíáéäÉë=çÑ=íÜÉ=ÅçêêÉëéçåÇáåÖ=ÉäÉãÉåíë= çÑ=~åó=çíÜÉê=êçï==Eçê=ÅçäìãåFI==íÜÉ=î~äìÉ=çÑ=íÜÉ=ÇÉíÉêãáå~åí= áë=ìåÅÜ~åÖÉÇK= ~N + âÄN ÄN = ~N ÄN = ~O + âÄO ÄO ~O ÄO = = = 5.3 Matrices = 525. aÉÑáåáíáçå= ^å= ã × å =ã~íêáñ=^=áë=~=êÉÅí~åÖìä~ê=~êê~ó=çÑ=ÉäÉãÉåíë=Eåìã- ÄÉêë=çê=ÑìåÅíáçåëF=ïáíÜ=ã=êçïë=~åÇ=å=ÅçäìãåëK==  ~NN ~NO K ~Nå    [ ]^ =~ áà=  ~ ON ~ OO K ~ Oå  == M M M ~ ãN  ~ãO K ~ ãå  = 526. pèì~êÉ=ã~íêáñ=áë=~=ã~íêáñ=çÑ=çêÇÉê= å × å K== = [ ]527. ^=ëèì~êÉ=ã~íêáñ== ~áà ==áë==ëóããÉíêáÅ==áÑ== ~áà = ~àá I==áKÉK==áí==áë= ëóããÉíêáÅ=~Äçìí=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äK== = [ ]528. ^=ëèì~êÉ=ã~íêáñ= ~áà =áë=ëâÉï-ëóããÉíêáÅ=áÑ= ~áà = −~àá K== = 110

CHAPTER 5. MATRICES AND DETERMINANTS 529. aá~Öçå~ä=ã~íêáñ==áë==~=ëèì~êÉ==ã~íêáñ=ïáíÜ=~ää==ÉäÉãÉåíë==òÉêç= ÉñÅÉéí=íÜçëÉ=çå=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äK== = 530. råáí=ã~íêáñ==áë==~=Çá~Öçå~ä==ã~íêáñ==áå=ïÜáÅÜ=íÜÉ=ÉäÉãÉåíë=çå= íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~ä=~êÉ=~ää=ìåáíóK=qÜÉ=ìåáí=ã~íêáñ=áë=========== ÇÉåçíÉÇ=Äó=fK== = 531. ^=åìää=ã~íêáñ=áë=çåÉ=ïÜçëÉ=ÉäÉãÉåíë=~êÉ=~ää=òÉêçK= = = = 5.4 Operations with Matrices = 532. qïç=ã~íêáÅÉë=^=~åÇ=_=~êÉ=Éèì~ä=áÑI=~åÇ=çåäó=áÑI=íÜÉó=~êÉ=ÄçíÜ= çÑ==íÜÉ==ë~ãÉ==ëÜ~éÉ== ã × å ==~åÇ=ÅçêêÉëéçåÇáåÖ=ÉäÉãÉåíë=~êÉ= Éèì~äK= = 533. qïç=ã~íêáÅÉë==^=~åÇ=_==Å~å=ÄÉ=~ÇÇÉÇ=Eçê=ëìÄíê~ÅíÉÇF=çÑI=~åÇ= çåäó=áÑI=íÜÉó=Ü~îÉ=íÜÉ=ë~ãÉ=ëÜ~éÉ= ã × å K=fÑ==  ~NN ~NO K ~Nå    [ ]^ =~ áà=  ~ ON ~ OO K ~ Oå  I== M M M ~ ãN  ~ãO K ~ ãå   ÄNN ÄNO K ÄNå    [ ]_ =Äáà=  ÄON ÄOO K ÄOå  I== M M M ÄãN  ÄãO K Äãå  = = = = = 111

CHAPTER 5. MATRICES AND DETERMINANTS íÜÉå==  ~NN + ÄNN ~NO + ÄNO K ~Nå + ÄNå    ^ + _ =  ~ ON + ÄON ~OO + ÄOO K ~Oå + ÄOå  K= M M M ~ãN + ÄãN  ~ãO + ÄãO K ~ ãå + Äãå  = [ ]534. fÑ=â=áë=~=ëÅ~ä~êI=~åÇ= ^ = ~áà =áë=~=ã~íêáñI=íÜÉå=  â~NN â~NO K â~Nå    [ ]â^ =  â~ ON â~ OO K â~ Oå  â~ áà = M K= M M â~ ãN  â~ ã O K â~ ãå  = 535. jìäíáéäáÅ~íáçå=çÑ=qïç=j~íêáÅÉë= qïç= ã~íêáÅÉë= Å~å= ÄÉ= ãìäíáéäáÉÇ= íçÖÉíÜÉê= çåäó= ïÜÉå= íÜÉ= åìãÄÉê= çÑ= Åçäìãåë= áå= íÜÉ= Ñáêëí= áë= Éèì~ä= íç= íÜÉ= åìãÄÉê= çÑ= êçïë=áå=íÜÉ=ëÉÅçåÇK== = fÑ=  ~NN ~NO K ~Nå    [ ]^ =~ áà =  ~ ON ~ OO K ~ Oå  I== M M M ~ ãN  ~ãO K ~ ãå  ÄNN ÄNO K ÄNâ    [ ]_ =Äáà =  ÄON ÄOO K ÄOâ  I= M M M ÄåN  ÄåO K Äåâ  = = = = = 112

CHAPTER 5. MATRICES AND DETERMINANTS íÜÉå==  ÅNN ÅNO K ÅNâ    ^_ = ` =  Å ON Å OO K Å Oâ  I== M M M ÄãN  ÅãO K Å ãâ  ïÜÉêÉ== å ∑Åáà = ~ ÄáN Nà + ~áOÄOà + K + ~áåÄåà = ~á λÄλ à = λ =N E á = NI OIKI ã X à = NI OIKI â FK== = qÜìë=áÑ= [ ] [ ]^ = ~NN ~NO ~  ÄN  ~ ON ~ OO ~  ÄO  ~ áà = NP  I= _ = Äá =  I== OP ÄP  íÜÉå== ^_ = ~NN ~NO ~NP  ⋅ ÄN  = ~NNÄN ~NOÄO ~NPÄP  K== ~ ON ~ OO ~ OP  ÄO  ~ ONÄN ~ OO ÄO ~ OP ÄP   ÄP    = 536. qê~åëéçëÉ=çÑ=~=j~íêáñ= fÑ=íÜÉ=êçïë=~åÇ=Åçäìãåë=çÑ=~=ã~íêáñ=~êÉ=áåíÉêÅÜ~åÖÉÇI=íÜÉå= íÜÉ=åÉï=ã~íêáñ=áë=Å~ääÉÇ=íÜÉ=íê~åëéçëÉ=çÑ=íÜÉ=çêáÖáå~ä=ã~íêáñK=== fÑ= ^= áë= íÜÉ= çêáÖáå~ä= ã~íêáñI= áíë= íê~åëéçëÉ= áë= ÇÉåçíÉÇ= ^q = çê= ^ú K== = 537. qÜÉ=ã~íêáñ=^=áë=çêíÜçÖçå~ä=áÑ= ^^q = f K== = 538. fÑ=íÜÉ=ã~íêáñ=éêçÇìÅí=^_=áë=ÇÉÑáåÉÇI=íÜÉå== (^_)q = _q^q K= = = 113

CHAPTER 5. MATRICES AND DETERMINANTS 539. ^Çàçáåí=çÑ=j~íêáñ= fÑ=^=áë=~=ëèì~êÉ= å × å ã~íêáñI=áíë=~ÇàçáåíI=ÇÉåçíÉÇ=Äó= ~Çà ^ I= áë=íÜÉ=íê~åëéçëÉ=çÑ=íÜÉ=ã~íêáñ=çÑ=ÅçÑ~Åíçêë= `áà =çÑ=^W= [ ]~Çà ^ = `áà q K== = 540. qê~ÅÉ=çÑ=~=j~íêáñ= fÑ= ^= áë= ~= ëèì~êÉ= å × å ã~íêáñI= áíë= íê~ÅÉI= ÇÉåçíÉÇ= Äó= íê ^ I= áë= ÇÉÑáåÉÇ=íç=ÄÉ==íÜÉ=ëìã=çÑ==íÜÉ=íÉêãë=çå=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äW= íê ^ = ~NN + ~OO + K + ~åå K= = 541. fåîÉêëÉ=çÑ=~=j~íêáñ= fÑ=^=áë=~=ëèì~êÉ= å × å ã~íêáñ=ïáíÜ=~=åçåëáåÖìä~ê=ÇÉíÉêãáå~åí= ÇÉí ^ I=íÜÉå=áíë=áåîÉêëÉ= ^−N =áë=ÖáîÉå=Äó= ^−N = ~Çà ^ K= ÇÉí ^ = 542. fÑ=íÜÉ=ã~íêáñ=éêçÇìÅí=^_=áë=ÇÉÑáåÉÇI=íÜÉå== (^_)−N = _−N^−N K= = 543. fÑ==^==áë=~=ëèì~êÉ=== å × å ==ã~íêáñI==íÜÉ==ÉáÖÉåîÉÅíçêë==u===ë~íáëÑó= íÜÉ=Éèì~íáçå= ^u = λu I== ïÜáäÉ=íÜÉ=ÉáÖÉåî~äìÉë= λ =ë~íáëÑó=íÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçå= ^ − λf = M K=== = = = 5.5 Systems of Linear Equations = = s~êá~ÄäÉëW=ñI=óI=òI= ñN I= ñO IK= oÉ~ä=åìãÄÉêëW= ~N I ~ O I ~P I ÄN I ~NN I ~NO IK= 114

CHAPTER 5. MATRICES AND DETERMINANTS aÉíÉêãáå~åíëW=aI= añ I= aó I= aò == j~íêáÅÉëW=^I=_I=u= = = 544. ~~NOññ + ÄNó = ÇN I== + ÄOó = ÇO ñ = añ I= ó = aó =E`ê~ãÉê∞ë=êìäÉFI== aa ïÜÉêÉ== a = ~N ÄN = ~NÄO − ~OÄN I== ~O ÄO añ = ÇN ÄN = ÇNÄO − ÇOÄN I== ÇO ÄO aó = ~N ÇN = ~NÇO − ~OÇN K== ~O ÇO = 545. fÑ= a ≠ M I=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë=~=ëáåÖäÉ=ëçäìíáçåW== ñ = añ I= ó = aó K= aa fÑ= a = M = ~åÇ= añ ≠ M Eçê= aó ≠ M FI= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= = åç== ëçäìíáçåK= fÑ= a = añ = aó = M I= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= = áåÑáåáíÉäó= = ã~åó== ëçäìíáçåëK= = 546. ~~NOññ + ÄNó + ÅNò = ÇN= I== + ÄOó + ÅOò = ÇO ~Pñ + ÄPó + ÅPò = ÇP ñ = añ I= ó = aó I= ò = aò =E`ê~ãÉê∞ë=êìäÉFI== a aa = 115

CHAPTER 5. MATRICES AND DETERMINANTS ïÜÉêÉ== ~N ÄN ÅN ÇN ÄN ÅN a = ~O ÄO ÅO I= añ = ÇO ÄO ÅO I= ~P ÄP ÅP ÇP ÄP ÅP ~N ÇN ÅN ~N ÄN ÇN aó = ~O ÇO ÅO I= aò = ~O ÄO ÇO K== ~P ÇP ÅP ~P ÄP ÇP = 547. fÑ= a ≠ M I=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë=~=ëáåÖäÉ=ëçäìíáçåW== ñ = añ I= ó = aó I= ò = aò K= a aa fÑ= a = M =~åÇ= añ ≠ M Eçê= aó ≠ M =çê= aò ≠ M FI=íÜÉå=íÜÉ=ëóëíÉã= Ü~ë=åç=ëçäìíáçåK= fÑ= a = añ = aó = aò = M I= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= áåÑáåáíÉäó= ã~åó=ëçäìíáçåëK= = 548. j~íêáñ=cçêã=çÑ=~=póëíÉã=çÑ=å=iáåÉ~ê=bèì~íáçåë=áå================= å=råâåçïåë= qÜÉ=ëÉí=çÑ=äáåÉ~ê=Éèì~íáçåë== ~NNñN + ~NOñ O + K + ~Nåñ å = ÄN K~ONKñNK+K~OOKñOK+KKK+K~OåKñKå =KÄO = ~åNñN + ~åOñ O + K + ~ååñ å = Äå Å~å=ÄÉ=ïêáííÉå=áå=ã~íêáñ=Ñçêã=  ~NN ~NO K ~Nå   ñN   ÄN   ~ ON ~ OO K ~ Oå   ñO   ÄO    ⋅   =   I==  M M M   M   M  ~ åN ~ åO K ~ åå ñå Äå áKÉK== ^ ⋅ u = _ I== 116

CHAPTER 5. MATRICES AND DETERMINANTS ïÜÉêÉ==  ~NN ~NO K ~Nå   ñN   ÄN   ~ ON ~ OO K ~ Oå   ñO   ÄO  ^ =  K  I= u =   I= _ =   K==  M M M   M   M  ~ åN ~åO ~ åå ñå Äå = 549. pçäìíáçå=çÑ=~=pÉí=çÑ=iáåÉ~ê=bèì~íáçåë= å × å = u = ^−N ⋅ _ I== ïÜÉêÉ= ^−N =áë=íÜÉ=áåîÉêëÉ=çÑ=^K= = = 117

Chapter 6 Vectors = = = = sÉÅíçêëW= ìr I= îr I= ìrïr I= rêîr → I=£= sÉÅíçê=äÉåÖíÜW= I= I= ^_ I=£= kråìáääí==îîÉÉÅÅííççêêWë=WMr= rá= I= rà I= âr = `ççêÇáå~íÉë=çÑ=îÉÅíçê= ìîrr W= uN I vNI wN = W= uO I vO I wO = `ççêÇáå~íÉë=çÑ=îÉÅíçê= pÅ~ä~êëW= λ I µ = aáêÉÅíáçå=ÅçëáåÉëW= Åçë α I= Åçëβ I= Åçë γ = ^åÖäÉ=ÄÉíïÉÉå=íïç=îÉÅíçêëW= θ = = = 6.1 Vector Coordinates = 550. rârrráà ==å=á(((íNMM=IsIIMNMÉIIIÅMMNí))ç)IIIê===ë= rá = rà = âr = NK= = rá rà âr 551. rê = → = (ñ N − ñ M ) + (ó N − ó M ) + (òN − òM ) = ^_ = 118

CHAPTER 6. VECTORS ======= = = = Figure 73. 552. rê = → = (ñN − )ñM O + (óN − )óM O + (òN − òM )O = ^_ = 553. fÑ= → = rê I=íÜÉå= → = − rê K= ^_ _^ = = = 554. u = rê Åçëα I= Figure 74. v = rê Åçëβ I= w = rê Åçë γ K= = 119

CHAPTER 6. VECTORS ===== = = Figure 75. 555. = rê (uI v I w) = rêN (uN I vN I wN ) I=íÜÉå== fÑ= u = uNI= v = vN I= w = wN K== == = 6.2 Vector Addition 556. =ïr = ìr + îr = = == = = Figure 76. 120

CHAPTER 6. VECTORS == = = Figure 77. 557. =ïr = ìrN + ìr O + ìrP +K + ìr å = = == = = Figure 78. = 558. `ìr ç+ãîrã=ìîrí+~íìárî=É=i~ï= = 559. ^(ìrëë+çîrÅá)~+íáïîrÉ==iìr~ï+=(îr + ïr )= 560. =ìr + îr = (uN + uO I vN + vO I wN + wO )= = = = = = = 121

CHAPTER 6. VECTORS 6.3 Vector Subtraction 561. =ïr = ìr − îr =áÑ= îr + ïr = ìr K= = = = Figure 79. = == = = Figure 80. 562. =ìr − îr = ìr + (− îr )= 563. =ìr − ìr = Mr = (MI MI M)= 564. = Mr = M = 565. =ìr − îr = (uN − uO I vN − vO I wN − wO )I== = = = 6.4 Scaling Vectors 566. =ïr = λìr = 122

CHAPTER 6. VECTORS = = 567. = ïr = λ ⋅ ìr = Figure 81. 568. =λìr = (λuI λvI λw) = 569. =λìr = ìrλ = 570. = + µ) ìr = λìr + µìr = (λ 571. =λ(µìr ) = µ(λìr ) = (λµ)ìr = 572. =λ(ìr + îr ) = λìr + λîr = = = = 6.5 Scalar Product 573. = ìr =~åÇ îr = pìrÅ⋅~îrä~=ê=ìmrê⋅çîrÇì⋅ ÅÅçí=ëçθÑ=sI==ÉÅíçêë= ïÜÉêÉ= θ =áë=íÜÉ=~åÖäÉ=ÄÉíïÉÉå=îÉÅíçêë= ìr =~åÇ îr K==== = 123

CHAPTER 6. VECTORS = == Figure 82. = 574. pfìrÑÅ=⋅~ìîrrä~==ê=(umuNêNuçIÇOvì+NIÅvwí=NNávå)OI==`+îrçw=çNê(wÇuOáOåK=I~víÉO I=cwçOê)ãI=í=ÜÉå== = 575. f^Ñå= ìrÖä=É=(_uÉNíIïvÉNÉIåw=Nq)ïI=çîr=s=É(uÅíOçIêvë=O=I wO ) I=íÜÉå== Åçëθ = uNuO + vNvO + wNwO K= uNO + vNO + wNO u O + vOO + wOO O = 576. `ìr ç⋅ îrã=ãîrì⋅íìr~í=áîÉ=mêçéÉêíó= = 577. ^(λëìrëç)⋅Å(áµ~îírá)î=É=mλµêçìré⋅Éîrêí=ó= = 578. aìr ⋅áë(îírê+áÄïìrí)á=îÉìr=m⋅êîrç+éÉìrê⋅íïór= = = 579. ìr ⋅ îr = M =áÑ= ìr I îr =~êÉ=çêíÜçÖçå~ä=E θ = π FK= O = 580. ìr ⋅ îr > M =áÑ= M < θ < π K= O = 124

CHAPTER 6. VECTORS 581. ìr ⋅ îr < M =áÑ= π < θ < π K= O 582. =ìr ⋅ îr ≤ ìr ⋅ îr = 583. =ìr ⋅ îr = ìr ⋅ îr =áÑ= ìr I îr =~êÉ=é~ê~ääÉä=E θ = M FK= 584. = ìr = (uN I vN I wN ) I=íÜÉå== fÑ= ìr ⋅ ìr = ìr O = ìr O = uNO + vNO + wNO K= 585. =rá ⋅ rá = rà ⋅ rà = âr ⋅ âr = N= 586. =rá ⋅ rà = rà ⋅ âr = âr ⋅ rá = M = = = = 6.6 Vector Product 587. = ìr =~åÇ îr = sìr ×ÉÅîríç=ê=ïmr êIç=ïÇÜìÉÅêíÉ=ç==Ñ=sÉÅíçêë= • ïr = ìr ⋅ îr ⋅I=ëîrá~ååI=θÇïr=I=ïïr=ÑçÜ⊥êÉãêîrÉ=X=~=M=ê≤áÖθÜ≤í-ÜπO~Xå= ÇÉÇ=ëÅêÉïK= • ïr ⊥ ìr = ìr • =sÉÅíçêë= = 125

CHAPTER 6. VECTORS ======= = = Figure 83. = rá rà âr 588. ïr = ìr × îr = uN vN wN = uO vO wO = 589. ïr = ìr × îr =  vN wN I − uN wN I uN vN  = vO wO uO wO uO vO 590. = = ìr × îr = ìr ⋅ îr ⋅ ëáå θ =EcáÖKUPF= p = 591. ^ëáååÖθäÉ==_ììrrÉí×⋅ïîîrrÉÉ=å=qïç=sÉÅíçêë=EcáÖKUPF= = 592. kìr ×çåîrÅ=çã−(ãîr ×ìíìr~)íá==îÉ=mêçéÉêíó= = 593. ^(λëìrëç)×Åá(~µíîárî)É==mλêµçìérÉ×êîíró== = = 126

CHAPTER 6. VECTORS 594. aìr ×áë(íîrêá+Äìïríá)î=É=ìrm×êçîré+Éêìríó×= ïr = 595. =ìr × îr = Mr =áÑ= ìr =~åÇ= îr =~êÉ=é~ê~ääÉä=E θ = M FK= 596. =rá × rá = rà × rà = âr × âr = Mr = 597. =rá × rà = âr I= rà × âr = rá I= âr × rá = rà = = = = 6.7 Triple Product = 598. [pìrÅîr~ïär~]ê==qìrêá⋅é(äîrÉ×=mïêrç)Ç=ìîrÅí⋅=(ïr × ìr ) = ïr ⋅ (ìr × îr ) = 599. =[ìrîrïr ]= [ïr ìrîr]= [îrïr ìr]= −[îrìrïr ]= −[ïr îrìr]= −[ìrïr îr]= 600. =âìr ⋅(îr × ïr ) = â[ìrîrïr ]= = 601. pÅ~ä~ê=qêáéäÉ=mêçÇìÅí=áå=`ççêÇáå~íÉ=cçêã= ìr ⋅ (îr × ïr ) = uN vN wN uO vO wO I== uP vP wP ïìr Ü=É(êuÉ=N=I vNI wN)I= îr = (uO I vO I wO ) I= ïr = (uP I vP I wP ) K== = 602. ssç=äììrã⋅É(îr=ç×Ñ=ïmr~)ê=~ääÉäÉéáéÉÇ= = 127

CHAPTER 6. VECTORS ============ = = Figure 84. = 603. sçäìãÉ=çÑ=móê~ãáÇ= s = N ìr ⋅(îr × ïr ) = S = = = Figure 85. 604. =fÇÑÉ==éìrÉ⋅å(ÇîrÉ×åïír=I)=ë=çM= ïrI=í=ÜÉλåìr=í+Üɵ=îrîÉ=ÑÅçíçê=êëëç==ãìrÉI==ëîrÅ~I=ä~~åêÇë==λïr=~=~åêÇÉ==äµáåK=É= ~êäó= 605. = ìr ⋅ (îr × ïr ) ≠ M I=íÜÉå=íÜÉ=îÉÅíçêë== ìr I= îr I=~åÇ= ïr =~êÉ=äáåÉ~êäó= fÑ== áåÇÉéÉåÇÉåíK= = 128

CHAPTER 6. VECTORS 606. sìr ×ÉÅ(íîrç×ê=qïrê)á=éä(Éìr=m⋅êïrç)Çîrì−Åí(=ìr ⋅ îr )ïr == = = = = = = = = 129

Chapter 7 Analytic Geometry = = = = 7.1 One-Dimensional Coordinate System = mçáåí=ÅççêÇáå~íÉëW= ñM I= ñN I= ñO I= óM I= óN I= óO = oÉ~ä=åìãÄÉêW= λ == aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç= = = 607. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë= Ç = ^_ = ñO − ñN = ñN − ñO = = = = Figure 86. = 608. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç= λ = ñM = ñN + λñO I= λ = ^` I= λ ≠ −NK= N+ λ `_ = ======== = = Figure 87. 130

CHAPTER 7. ANALYTIC GEOMETRY 609. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí= ñM = ñN + ñO I= λ = N K= O = = = 7.2 Two-Dimensional Coordinate System = mçáåí=ÅççêÇáå~íÉëW= ñM I= ñN I= ñO I= óM I= óN I= óO = mçä~ê=ÅççêÇáå~íÉëW= êI ϕ = oÉ~ä=åìãÄÉêW= λ == mçëáíáîÉ=êÉ~ä=åìãÄÉêëW=~I=ÄI=ÅI== aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç= ^êÉ~W=p= = = 610. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë= Ç = ^_ = (ñO − ñN )O + (óO − óN )O = = = = Figure 88. 131

CHAPTER 7. ANALYTIC GEOMETRY 611. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç= λ = ñM = ñN + λñO I= óM = óN + λóO I== N+ λ N+ λ λ = ^` I= λ ≠ −NK= `_ = ======= = = Figure 89. = = 132

CHAPTER 7. ANALYTIC GEOMETRY ======= = = Figure 90. = 612. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí= ñM = ñN + ñO I= óM = óN + óO I= λ = N K= O O = 613. `ÉåíêçáÇ=EfåíÉêëÉÅíáçå=çÑ=jÉÇá~åëF=çÑ=~=qêá~åÖäÉ= ñM = ñN + ñO + ñP I= óM = óN + óO + óP I== P P ïÜÉêÉ== ^(ñNIóN)I== _(ñO IóO )I==~åÇ== `(ñP I óP )==~êÉ=îÉêíáÅÉë=çÑ= íÜÉ=íêá~åÖäÉ= ^_` K= = = 133

CHAPTER 7. ANALYTIC GEOMETRY ========= = = Figure 91. = 614. fåÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^åÖäÉ=_áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ= ñM = ~ñN + Äñ O + ÅñP I= óM = ~óN + ÄóO + ÅóP I== ~+Ä+Å ~+Ä+Å ïÜÉêÉ= ~ = _` I= Ä = `^ I= Å = ^_ K== = ======== = = Figure 92. 134

CHAPTER 7. ANALYTIC GEOMETRY 615. `áêÅìãÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=íÜÉ=páÇÉ=mÉêéÉåÇáÅìä~ê====================== _áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ= ñ O + ó O óN N ñN ñ O + óNO N N N N ñ O + ó O óO N ñO ñ O + ó O N O O O O ñM = ñ O + ó O óP N ñP ñ O + ó O N P P óN N I= óM = P P N= ñN ñN óN O ñO óO N O ñO óO N ñP óP N ñP óP N = ======== = == Figure 93. = = = = = = = 135

CHAPTER 7. ANALYTIC GEOMETRY 616. lêíÜçÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^äíáíìÇÉëF=çÑ=~=qêá~åÖäÉ= óN ñ OñP + óNO N ñNO + óOóP ñN N N óO ñPñN + ó O N ñ O + óPóN ñO N O O = ñM = óP ñNñ O + ó O N ñ O + óNóO ñP P I= óM = P N ñN óN N ñN óN ñO óO N ñO óO N ñP óP N ñP óP N = ====== = = Figure 94. = 617. ^êÉ~=çÑ=~=qêá~åÖäÉ= N ñN óN N N ñO − ñN óO − óN = O ñO óO O ñP − ñN óP − óN p = (±) ñP óP N = (±) N = = = 136

CHAPTER 7. ANALYTIC GEOMETRY 618. ^êÉ~=çÑ=~=nì~Çêáä~íÉê~ä= p = (±) N [(ñN − ñ O )(óN + ó O ) + (ñ O − ñ P )(ó O + óP ) + = O + (ñP − ñQ )(óP + óQ )+ (ñQ − ñN)(óQ + óN)] = = === = = Figure 95. = kçíÉW=få=Ñçêãìä~ë=SNTI=SNU=ïÉ=ÅÜççëÉ=íÜÉ=ëáÖå=EHF=çê=E¥F=ëç= íÜ~í=íç=ÖÉí=~=éçëáíáîÉ=~åëïÉê=Ñçê=~êÉ~K== = 619. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë=áå=mçä~ê=`ççêÇáå~íÉë= Ç = ^_ = êNO + êOO − OêNêO Åçë(ϕO − ϕN ) = = 137

CHAPTER 7. ANALYTIC GEOMETRY = = Figure 96. = 620. `çåîÉêíáåÖ=oÉÅí~åÖìä~ê=`ççêÇáå~íÉë=íç=mçä~ê=`ççêÇáå~íÉë= ñ = ê Åçë ϕ I= ó = ê ëáå ϕ K= = = = Figure 97. = 621. `çåîÉêíáåÖ=mçä~ê=`ççêÇáå~íÉë=íç=oÉÅí~åÖìä~ê=`ççêÇáå~íÉë= ê = ñO + óO I= í~å ϕ = ó K= ñ 138

CHAPTER 7. ANALYTIC GEOMETRY 7.3 Straight Line in Plane = mçáåí=ÅççêÇáå~íÉëW=uI=vI=ñI= ñM I= ñN I== óM I= óN I= ~N I= ~O I=£== oÉ~ä=åìãÄÉêëW=âI=~I=ÄI=éI=íI=^I=_I=`I= ^N I= ^O I=£= ^åÖäÉëW= α I= β = km^çåçëÖêáãäíÉáç=~ÄåäÉ=î=íîïÉÉÅÉÅííÉççåêê=Wíë=ïåWr= çrê= =Iä=á~råIÉ=ëÄrW==ϕ = = = 622. dÉåÉê~ä=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ= ^ñ + _ó + ` = M = = 623. kqÜçÉê=ãîÉ~Åäí=sçêÉ=Ååríç(^ê=Ií_ç=)~==ápëí=åê~çáêÖãÜí~=äi=íáçå=Éí=ÜÉ=äáåÉ= ^ñ + _ó + ` = M K= = = = Figure 98. = 624. bñéäáÅáí=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ=EpäçéÉ-fåíÉêÅÉéí=cçêãF= ó = âñ + Ä K== 139

CHAPTER 7. ANALYTIC GEOMETRY qÜÉ=Öê~ÇáÉåí=çÑ=íÜÉ=äáåÉ=áë= â = í~å α K= = = = = Figure 99. 625. dê~ÇáÉåí=çÑ=~=iáåÉ== â = í~å α = óO − óN = ñO − ñN = = = Figure 100. 140