1300 Math Formulas(MATHS 4 ALL)

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