NEBOSH INTERNATIONAL DIPLOMA IN OCCUPATIONAL SAFETY AND HEALTHZain Shah
This is the highest level of qualification in Occupational Safety and health Industry.This is for those who have around 20 years of experience in safety industry.
NEBOSH INTERNATIONAL DIPLOMA IN OCCUPATIONAL SAFETY AND HEALTHZain Shah
This is the highest level of qualification in Occupational Safety and health Industry.This is for those who have around 20 years of experience in safety industry.
Aku Bersyukur Pada-Mu merupakan ungkapan Doa kita bagi Tuhan. Karena setiap manusia telah mendapatkan kelimpahan rejeki, rumah, pakaian dan lain sebagainya. Maka sudah selayaknya manusia Bersyukur kepada Tuhan, Sang Pemilik Dunia.
Lataif e ashrafi malfoozat e syed makhdoom ashraf 30Aale Rasool Ahmad
لطائف اشرفی حصہ اول
یا
ملفوظات سید مخدوم اشرف جہانگیرسمنانی کچھوچھوی
مترجم
حضرت شمس بریلوی
نظر ثانی
ڈاکٹر خضر نوشاہی
مدیر و ناشر
نذر اشرف شیخ محمد ہاشم رضا اشرفی
خلیفہ مجاز مخدوم المشائخ غوث الوقت سید مختار اشرف اشرف جیلانی سرکارکلاں کچھوچھو شریف
یہ کتاب تارک السلطنت غوث العالم محبوب یزدانی سلطان اوحدالدین قدوۃ الکبریٰ مخدوم سیداشرف جہانیاں جہانگیر سمنانی رضی اللہ تعالیٰ عنہ کے ملفوظات اور ارشادات و کمالات و فضائل پر مبنی ہے جسے انکے مرید خاص ابوالفضائل شیخ الاسلام والمسلمین حضرت نظام یمنی المعروف نظام حاجی غریب یمنی قدس سرہ النورانی نے مرتب کیا ۔ وہ آپکے خدمت میں تیس سال رہے۔
اپلود آن ارچیو
دیوانۂ مجمع البحرین حاجی الحرمین الشریفین اعلیٰ حضرت قدسی منزلت مخدوم الاولیاء مرشدالعالم محبوب ربانی ہم شبیہ غوث الاعظم حضرت سید شاہ ابواحمد المدعومحمد علی حسین اشرف اشرؔفی میاں الحسنی الحسینی قدس سرہ النوران
الحاج شیخ آلِ رسول احمد الصدیقی الاشرفی القادری کٹیہاری
This document discusses 10 methods for making money online: list building, social media, article writing, blogging, mobile marketing, apps, YouTube, paid traffic, affiliate marketing, and selling products online. It provides details on each method, including how they work and their benefits. List building involves building an email list of subscribers that can be marketed to. Social media marketing utilizes platforms like Facebook, Twitter, blogs, and Instagram to generate awareness and sales. Article writing can involve freelance writing, paid reviews, or article marketing. Blogging makes money through ads, sponsorships, and paid writing for other blogs. Mobile marketing capitalizes on mobile use through SMS, MMS, mobile apps, and mobile web.
The document profiles Jason and Lindy Pieratt, missionaries with Children's Relief International who help run various projects in 6 countries and 9 locations to provide aid to needy children. These projects focus on access to clean water, education, medical care, food and sharing the gospel. The document encourages readers that anyone who is willing to help those in need through acts of service, sacrifice and sharing God's love can be considered a missionary.
This document provides an introduction to programming by outlining what programming is and the process involved. It explains that programming involves breaking tasks down into small, precise steps that a computer can understand. This is because computers are very limited in their abilities and need complete, detailed instructions. The document also discusses who might be suited to programming, noting it helps to be logical, patient, and enjoy intellectual challenges. It concludes by describing the typical programming process of writing code, compiling it, running it, debugging any issues, and repeating until the program is complete.
This document provides 70 tips for generating more traffic and sales for a website. Some key tips include setting up a blog using WordPress and posting regularly to keep it fresh. Videos can also drive traffic when embedded in blog posts and uploaded to multiple hosting sites. Squeeze pages asking for email addresses and exit popups with offers can grow email lists. Giveaways, contests and adding free extras to purchases can attract people who love free items. Overall the tips focus on content creation, social media engagement, advertising and promotions to bring more visitors to a site.
Aku Bersyukur Pada-Mu merupakan ungkapan Doa kita bagi Tuhan. Karena setiap manusia telah mendapatkan kelimpahan rejeki, rumah, pakaian dan lain sebagainya. Maka sudah selayaknya manusia Bersyukur kepada Tuhan, Sang Pemilik Dunia.
Lataif e ashrafi malfoozat e syed makhdoom ashraf 30Aale Rasool Ahmad
لطائف اشرفی حصہ اول
یا
ملفوظات سید مخدوم اشرف جہانگیرسمنانی کچھوچھوی
مترجم
حضرت شمس بریلوی
نظر ثانی
ڈاکٹر خضر نوشاہی
مدیر و ناشر
نذر اشرف شیخ محمد ہاشم رضا اشرفی
خلیفہ مجاز مخدوم المشائخ غوث الوقت سید مختار اشرف اشرف جیلانی سرکارکلاں کچھوچھو شریف
یہ کتاب تارک السلطنت غوث العالم محبوب یزدانی سلطان اوحدالدین قدوۃ الکبریٰ مخدوم سیداشرف جہانیاں جہانگیر سمنانی رضی اللہ تعالیٰ عنہ کے ملفوظات اور ارشادات و کمالات و فضائل پر مبنی ہے جسے انکے مرید خاص ابوالفضائل شیخ الاسلام والمسلمین حضرت نظام یمنی المعروف نظام حاجی غریب یمنی قدس سرہ النورانی نے مرتب کیا ۔ وہ آپکے خدمت میں تیس سال رہے۔
اپلود آن ارچیو
دیوانۂ مجمع البحرین حاجی الحرمین الشریفین اعلیٰ حضرت قدسی منزلت مخدوم الاولیاء مرشدالعالم محبوب ربانی ہم شبیہ غوث الاعظم حضرت سید شاہ ابواحمد المدعومحمد علی حسین اشرف اشرؔفی میاں الحسنی الحسینی قدس سرہ النوران
الحاج شیخ آلِ رسول احمد الصدیقی الاشرفی القادری کٹیہاری
This document discusses 10 methods for making money online: list building, social media, article writing, blogging, mobile marketing, apps, YouTube, paid traffic, affiliate marketing, and selling products online. It provides details on each method, including how they work and their benefits. List building involves building an email list of subscribers that can be marketed to. Social media marketing utilizes platforms like Facebook, Twitter, blogs, and Instagram to generate awareness and sales. Article writing can involve freelance writing, paid reviews, or article marketing. Blogging makes money through ads, sponsorships, and paid writing for other blogs. Mobile marketing capitalizes on mobile use through SMS, MMS, mobile apps, and mobile web.
The document profiles Jason and Lindy Pieratt, missionaries with Children's Relief International who help run various projects in 6 countries and 9 locations to provide aid to needy children. These projects focus on access to clean water, education, medical care, food and sharing the gospel. The document encourages readers that anyone who is willing to help those in need through acts of service, sacrifice and sharing God's love can be considered a missionary.
This document provides an introduction to programming by outlining what programming is and the process involved. It explains that programming involves breaking tasks down into small, precise steps that a computer can understand. This is because computers are very limited in their abilities and need complete, detailed instructions. The document also discusses who might be suited to programming, noting it helps to be logical, patient, and enjoy intellectual challenges. It concludes by describing the typical programming process of writing code, compiling it, running it, debugging any issues, and repeating until the program is complete.
This document provides 70 tips for generating more traffic and sales for a website. Some key tips include setting up a blog using WordPress and posting regularly to keep it fresh. Videos can also drive traffic when embedded in blog posts and uploaded to multiple hosting sites. Squeeze pages asking for email addresses and exit popups with offers can grow email lists. Giveaways, contests and adding free extras to purchases can attract people who love free items. Overall the tips focus on content creation, social media engagement, advertising and promotions to bring more visitors to a site.
This document contains 9 grammar and vocabulary questions with multiple choice answers. The questions cover topics such as identifying complete and simple sentences, parts of speech, synonyms and antonyms.
The document provides guidance for traders on developing a successful trading strategy and overcoming psychological obstacles. It discusses that trading rules are easy to learn but difficult to apply in live markets. Traders must determine their individual weaknesses and find ways to execute strategies properly despite psychological and external conflicts. Maintaining a suitable strategy for market conditions is key to avoiding losses from improper strategy application. The document emphasizes trading mechanically without trying to predict the future or dwelling on missed opportunities.
14. CHAPTER 1. NUMBER SETS
=
=====
=
Figure 3.
=
19.
_ y ^ = _ y (^ ∩ _ )
=
20.
_ y ^ = _ ∩ ^′
21.
^y^=∅
22.
^ y _ = ^ =áÑ= ^ ∩ _ = ∅ .
=
=
=
=====
=
Figure 4.
=
23.
(^ y _) ∩ ` = (^ ∩ `) y (_ ∩ `)
24.
^′ = f y ^
25.
`~êíÉëá~å=mêçÇìÅí
` = ^ × _ = {(ñ I ó ) ö ñ ∈ ^ ~åÇ ó ∈ _}
=
=
4
=
15. CHAPTER 1. NUMBER SETS
1.2 Sets of Numbers
=
26.
27.
=
28.
=
29.
=
30.
k~íìê~ä=åìãÄÉêëW=k=
tÜçäÉ=åìãÄÉêëW= kM =
fåíÉÖÉêëW=w=
mçëáíáîÉ=áåíÉÖÉêëW= w + =
kÉÖ~íáîÉ=áåíÉÖÉêëW= w − =
o~íáçå~ä=åìãÄÉêëW=n=
oÉ~ä=åìãÄÉêëW=o==
`çãéäÉñ=åìãÄÉêëW=`==
=
=
k~íìê~ä=kìãÄÉêë
`çìåíáåÖ=åìãÄÉêëW k = {NI OI PI K} K=
tÜçäÉ=kìãÄÉêë
`çìåíáåÖ=åìãÄÉêë=~åÇ=òÉêçW= k M = {MI NI OI PI K} K=
fåíÉÖÉêë
tÜçäÉ=åìãÄÉêë=~åÇ=íÜÉáê=çééçëáíÉë=~åÇ=òÉêçW=
w + = k = {NI OI PI K}I=
w − = {KI − PI − OI − N} I=
w = w − ∪ {M} ∪ w + = {KI − PI − OI − NI MI NI OI PI K} K=
o~íáçå~ä=kìãÄÉêë
oÉéÉ~íáåÖ=çê=íÉêãáå~íáåÖ=ÇÉÅáã~äëW==
~
n = ñ ö ñ = ~åÇ ~ ∈ w ~åÇ Ä ∈ w ~åÇ Ä ≠ M K=
Ä
fêê~íáçå~ä=kìãÄÉêë
kçåêÉéÉ~íáåÖ=~åÇ=åçåíÉêãáå~íáåÖ=ÇÉÅáã~äëK
=
5
18. CHAPTER 1. NUMBER SETS
43.
^ëëçÅá~íáîÉ=çÑ=jìäíáéäáÅ~íáçå=
(~ ⋅ Ä)⋅ Å = ~ ⋅ (Ä ⋅ Å )
=
aáëíêáÄìíáîÉ=i~ï=
~ (Ä + Å ) = ~Ä + ~Å =
44.
=
45.
aÉÑáåáíáçå=çÑ=aáîáëáçå=
~
N
= ~⋅ =
Ä
Ä
=
=
=
1.4 Complex Numbers
=
k~íìê~ä=åìãÄÉêW=å=
fã~Öáå~êó=ìåáíW=á=
`çãéäÉñ=åìãÄÉêW=ò=
oÉ~ä=é~êíW=~I=Å=
fã~Öáå~êó=é~êíW=ÄáI=Çá=
jçÇìäìë=çÑ=~=ÅçãéäÉñ=åìãÄÉêW=êI= êN I= êO =
^êÖìãÉåí=çÑ=~=ÅçãéäÉñ=åìãÄÉêW= ϕ I= ϕN I= ϕO =
=
=
46.
=
47.
=
48.
áN = á =
á O = −N =
á P = −á =
áQ = N=
áR = á =
á S = −N =
á T = −á =
áU = N =
á Q å +N = á =
á Q å+ O = −N =
á Q å + P = −á =
á Qå = N =
ò = ~ + Äá =
`çãéäÉñ=mä~åÉ=
=
8
20. CHAPTER 1. NUMBER SETS
=
=
Figure 7.
55.
=
56.
=
mçä~ê=mêÉëÉåí~íáçå=çÑ=`çãéäÉñ=kìãÄÉêë=
~ + Äá = ê(Åçë ϕ + á ëáå ϕ) =
jçÇìäìë=~åÇ=^êÖìãÉåí=çÑ=~=`çãéäÉñ=kìãÄÉê=
fÑ= ~ + Äá =áë=~=ÅçãéäÉñ=åìãÄÉêI=íÜÉå=
ê = ~ O + ÄO =EãçÇìäìëFI==
Ä
ϕ = ~êÅí~å =E~êÖìãÉåíFK=
~
=
57.
=
58.
mêçÇìÅí=áå=mçä~ê=oÉéêÉëÉåí~íáçå=
ò N ⋅ ò O = êN (Åçë ϕN + á ëáå ϕN ) ⋅ êO (Åçë ϕO + á ëáå ϕO ) =
= êNêO [Åçë(ϕN + ϕO ) + á ëáå(ϕN + ϕO )] =
`çåàìÖ~íÉ=kìãÄÉêë=áå=mçä~ê=oÉéêÉëÉåí~íáçå=
|||||||||||||||||||||
ê(Åçë ϕ + á ëáå ϕ) = ê[Åçë(− ϕ) + á ëáå(− ϕ)] =
=
59.
fåîÉêëÉ=çÑ=~=`çãéäÉñ=kìãÄÉê=áå=mçä~ê=oÉéêÉëÉåí~íáçå=
N
N
= [Åçë(− ϕ) + á ëáå(− ϕ)] =
ê(Åçë ϕ + á ëáå ϕ) ê
10
21. CHAPTER 1. NUMBER SETS
60.
=
61.
=
62.
=
63.
=
64.
nìçíáÉåí=áå=mçä~ê=oÉéêÉëÉåí~íáçå=
ò N êN (Åçë ϕN + á ëáå ϕN ) êN
= [Åçë(ϕN − ϕO ) + á ëáå(ϕN − ϕO )] =
=
ò O êO (Åçë ϕO + á ëáå ϕO ) êO
mçïÉê=çÑ=~=`çãéäÉñ=kìãÄÉê=
å
ò å = [ê(Åçë ϕ + á ëáå ϕ)] = ê å [Åçë(åϕ) + á ëáå(åϕ)] =
cçêãìä~=±aÉ=jçáîêÉ≤=
(Åçë ϕ + á ëáå ϕ)å = Åçë(åϕ) + á ëáå(åϕ) =
kíÜ=oççí=çÑ=~=`çãéäÉñ=kìãÄÉê=
ϕ + Oπâ
ϕ + Oπâ
å
ò = å ê(Åçë ϕ + á ëáå ϕ) = å ê Åçë
+ á ëáå
I==
å
å
ïÜÉêÉ==
â = MI NI OI KI å − N K==
bìäÉê∞ë=cçêãìä~=
É áñ = Åçë ñ + á ëáå ñ =
=
=
11
22. Chapter 2
Algebra
=
=
=
=
2.1 Factoring Formulas
=
oÉ~ä=åìãÄÉêëW=~I=ÄI=Å==
k~íìê~ä=åìãÄÉêW=å=
=
=
65.
=
66.
=
67.
=
68.
=
69.
=
70.
=
71.
=
72.
~ O − ÄO = (~ + Ä)(~ − Ä) =
~ P − ÄP = (~ − Ä)(~ O + ~Ä + ÄO ) =
~ P + ÄP = (~ + Ä)(~ O − ~Ä + ÄO ) =
~ Q − ÄQ = (~ O − ÄO )(~ O + ÄO ) = (~ − Ä)(~ + Ä)(~ O + ÄO ) =
~ R − ÄR = (~ − Ä)(~ Q + ~ P Ä + ~ O ÄO + ~ÄP + ÄQ ) =
~ R + ÄR = (~ + Ä)(~ Q − ~ P Ä + ~ O ÄO − ~ÄP + ÄQ ) =
fÑ=å=áë=çÇÇI=íÜÉå=
~ å + Äå = (~ + Ä)(~ å−N − ~ å −O Ä + ~ å −P ÄO − K − ~Äå −O + Äå −N ) K==
fÑ=å=áë=ÉîÉåI=íÜÉå==
~ å − Äå = (~ − Ä)(~ å −N + ~ å −O Ä + ~ å −P ÄO + K + ~Äå−O + Äå −N ) I==
12
23. CHAPTER 2. ALGEBRA
~ å + Äå = (~ + Ä)(~ å−N − ~ å −O Ä + ~ å −P ÄO − K + ~Äå−O − Äå −N ) K=
=
=
=
2.2 Product Formulas
73.
=
74.
=
75.
=
76.
=
77.
=
78.
=
79.
=
80.
=
81.
oÉ~ä=åìãÄÉêëW=~I=ÄI=Å==
tÜçäÉ=åìãÄÉêëW=åI=â=
=
=
(~ − Ä)O = ~ O − O~Ä + ÄO =
(~ + Ä)O = ~ O + O~Ä + ÄO =
(~ − Ä)P = ~ P − P~ O Ä + P~ÄO − ÄP =
(~ + Ä)P = ~ P + P~ OÄ + P~ÄO + ÄP =
(~ − Ä)Q = ~ Q − Q~ P Ä + S~ O ÄO − Q~ÄP + ÄQ =
(~ + Ä)Q = ~ Q + Q~ P Ä + S~ OÄO + Q~ÄP + ÄQ =
_áåçãá~ä=cçêãìä~=
(~ + Ä)å = å` M~ å + å`N~ å−NÄ + å` O~ å−OÄO + K + å` å−N~Äå−N + å` å Äå I
å>
ïÜÉêÉ= å ` â =
=~êÉ=íÜÉ=Äáåçãá~ä=ÅçÉÑÑáÅáÉåíëK=
â> (å − â )>
(~ + Ä + Å )O = ~ O + ÄO + Å O + O~Ä + O~Å + OÄÅ =
(~ + Ä + Å + K + ì + î )O = ~ O + ÄO + Å O + K + ì O + î O + =
+ O(~Ä + ~Å + K + ~ì + ~î + ÄÅ + K + Äì + Äî + K + ìî ) =
13
25. CHAPTER 2. ALGEBRA
2.4 Roots
=
91.
=
_~ëÉëW=~I=Ä==
mçïÉêë=Eê~íáçå~ä=åìãÄÉêëFW=åI=ã=
~ I Ä ≥ M =Ñçê=ÉîÉå=êççíë=E å = Oâ I= â ∈ k F=
=
=
å
~Ä = å ~ å Ä =
92.
=
å
~ ã Ä = åã ~ ã Äå =
93.
å
~ å~
=
I= Ä ≠ M =
Ä åÄ
=
94.
=
95.
=
96.
=
~ åã ~ ã åã ~ ã
I= Ä ≠ M K=
=
=
ã
Äå
Ä åã Äå
å
(~ )
å
ã
( ~)
å
å
é
= å ~ ãé =
=~=
åé
97.
=
å
~ã =
98.
=
å
~ =~ =
99.
=
ã å
100.
=
ã
å
ã
~ = ãå ~ =
( ~)
å
~ ãé =
ã
= å ~ã =
15
26. CHAPTER 2. ALGEBRA
N å ~ å −N
=
I= ~ ≠ M K=
å
~
~
101.
=
~± Ä =
102.
~ + ~O − Ä
~ − ~O − Ä
±
=
O
O
=
N
~m Ä
=
=
~−Ä
~± Ä
103.
=
=
=
2.5 Logarithms
=
104.
105.
106.
107.
108.
109.
mçëáíáîÉ=êÉ~ä=åìãÄÉêëW=ñI=óI=~I=ÅI=â=
k~íìê~ä=åìãÄÉêW=å==
=
=
aÉÑáåáíáçå=çÑ=içÖ~êáíÜã=
ó = äçÖ ~ ñ =áÑ=~åÇ=çåäó=áÑ= ñ = ~ ó I= ~ > M I= ~ ≠ N K=
=
äçÖ ~ N = M =
=
äçÖ ~ ~ = N =
=
− ∞ áÑ ~ > N
äçÖ ~ M =
=
+ ∞ áÑ ~ < N
=
äçÖ ~ (ñó ) = äçÖ ~ ñ + äçÖ ~ ó =
=
ñ
äçÖ ~ = äçÖ ~ ñ − äçÖ ~ ó =
ó
16
27. CHAPTER 2. ALGEBRA
110. äçÖ ~ (ñ å ) = å äçÖ ~ ñ =
=
N
111. äçÖ ~ å ñ = äçÖ ~ ñ =
å
=
äçÖ Å ñ
112. äçÖ ~ ñ =
= äçÖ Å ñ ⋅ äçÖ ~ Å I= Å > M I= Å ≠ N K=
äçÖ Å ~
=
N
113. äçÖ ~ Å =
=
äçÖ Å ~
=
114. ñ = ~ äçÖ ~ ñ =
=
115. içÖ~êáíÜã=íç=_~ëÉ=NM=
äçÖ NM ñ = äçÖ ñ =
=
116. k~íìê~ä=içÖ~êáíÜã=
äçÖ É ñ = äå ñ I==
â
N
ïÜÉêÉ= É = äáã N + = OKTNUOUNUOUK =
â →∞
â
=
N
117. äçÖ ñ =
äå ñ = MKQPQOVQ äå ñ =
äå NM
=
N
118. äå ñ =
äçÖ ñ = OKPMORUR äçÖ ñ =
äçÖ É
=
=
=
=
=
17
28. CHAPTER 2. ALGEBRA
2.6 Equations
=
oÉ~ä=åìãÄÉêëW=~I=ÄI=ÅI=éI=èI=ìI=î=
pçäìíáçåëW= ñ N I= ñ O I= ó N I= ó O I= ó P =
=
=
119. iáåÉ~ê=bèì~íáçå=áå=låÉ=s~êá~ÄäÉ=
Ä
~ñ + Ä = M I= ñ = − K==
~
=
120. nì~Çê~íáÅ=bèì~íáçå=
− Ä ± ÄO − Q~Å
~ñ + Äñ + Å = M I= ñ NI O =
K=
O~
=
121. aáëÅêáãáå~åí=
a = ÄO − Q~Å =
=
122. sáÉíÉ∞ë=cçêãìä~ë=
fÑ= ñ O + éñ + è = M I=íÜÉå==
ñ N + ñ O = −é
K=
ñ Nñ O = è
=
Ä
123. ~ñ O + Äñ = M I= ñ N = M I= ñ O = − K=
~
=
Å
124. ~ñ O + Å = M I= ñ NI O = ± − K=
~
=
125. `ìÄáÅ=bèì~íáçåK=`~êÇ~åç∞ë=cçêãìä~K==
ó P + éó + è = M I==
O
18
29. CHAPTER 2. ALGEBRA
ó N = ì + î I= ó OI P = −
N
(ì + î ) ± P (ì + î ) á I==
O
O
ïÜÉêÉ==
O
ì=P −
O
O
O
è
è
è é
è é
+ + I= î = P − − + K==
O
O
O P
O P
=
=
2.7 Inequalities
s~êá~ÄäÉëW=ñI=óI=ò=
~ I ÄI ÅI Ç
oÉ~ä=åìãÄÉêëW=
I=ãI=å=
~N I ~ O I ~ P I KI ~ å
aÉíÉêãáå~åíëW=aI= añ I= aó I= aò ==
=
=
126. fåÉèì~äáíáÉëI=fåíÉêî~ä=kçí~íáçåë=~åÇ=dê~éÜë==
=
fåÉèì~äáíó=
fåíÉêî~ä=kçí~íáçå=
dê~éÜ=
[~I Ä]=
~ ≤ ñ ≤ Ä=
~ < ñ ≤ Ä=
(~I Ä] =
=
~ ≤ ñ < Ä=
[~I Ä) =
=
~ < ñ < Ä=
(~I Ä) =
=
− ∞ < ñ ≤ Ä I=
ñ≤Ä=
− ∞ < ñ < Ä I=
ñ<Ä=
~ ≤ ñ < ∞ I=
ñ≥~=
~ < ñ < ∞ I=
ñ >~=
(− ∞I Ä] =
=
=
(− ∞I Ä) =
=
[~I ∞ ) =
=
(~I ∞ ) =
=
19
30. CHAPTER 2. ALGEBRA
127.
=
128.
=
129.
=
130.
=
131.
=
132.
=
133.
=
fÑ= ~ > Ä I=íÜÉå= Ä < ~ K=
fÑ= ~ > Ä I=íÜÉå= ~ − Ä > M =çê= Ä − ~ < M K=
fÑ= ~ > Ä I=íÜÉå= ~ + Å > Ä + Å K=
fÑ= ~ > Ä I=íÜÉå= ~ − Å > Ä − Å K=
fÑ= ~ > Ä =~åÇ= Å > Ç I=íÜÉå= ~ + Å > Ä + Ç K=
fÑ= ~ > Ä =~åÇ= Å > Ç I=íÜÉå= ~ − Ç > Ä − Å K=
fÑ= ~ > Ä =~åÇ= ã > M I=íÜÉå= ã~ > ãÄ K=
134. fÑ= ~ > Ä =~åÇ= ã > M I=íÜÉå=
~ Ä
> K=
ã ã
=
135. fÑ= ~ > Ä =~åÇ= ã < M I=íÜÉå= ã~ < ãÄ K=
=
~ Ä
136. fÑ= ~ > Ä =~åÇ= ã < M I=íÜÉå= < K=
ã ã
=
137. fÑ= M < ~ < Ä =~åÇ= å > M I=íÜÉå= ~ å < Äå K=
=
138. fÑ= M < ~ < Ä =~åÇ= å < M I=íÜÉå= ~ å > Äå K=
=
139. fÑ= M < ~ < Ä I=íÜÉå= å ~ < å Ä K=
=
~+Ä
I==
140.
~Ä ≤
O
ïÜÉêÉ= ~ > M =I= Ä > M X=~å=Éèì~äáíó=áë=î~äáÇ=çåäó=áÑ= ~ = Ä K==
=
N
141. ~ + ≥ O I=ïÜÉêÉ= ~ > M X=~å=Éèì~äáíó=í~âÉë=éä~ÅÉ=çåäó=~í= ~ = N K=
~
20
31. CHAPTER 2. ALGEBRA
142.
å
~N~ O K~ å ≤
~N + ~ O + K + ~ å
I=ïÜÉêÉ= ~N I ~ O I KI ~ å > M K=
å
=
Ä
143. fÑ= ~ñ + Ä > M =~åÇ= ~ > M I=íÜÉå= ñ > − K=
~
=
Ä
144. fÑ= ~ñ + Ä > M =~åÇ= ~ < M I=íÜÉå= ñ < − K==
~
=
145. ~ñ O + Äñ + Å > M =
=
=
~ > M=
=
=
=
=
a>M=
=
=
=
a=M=
=
=
=
a<M=
=
ñ < ñ N I= ñ > ñ O =
=
ñ N < ñ I= ñ > ñ N =
=
=
−∞< ñ <∞=
=
21
~ <M=
=
=
ñN < ñ < ñ O =
=
ñ ∈∅ =
=
=
ñ ∈∅ =
=
=
=
36. CHAPTER 3. GEOMETRY
165. Ü O = ÑÖ I===
ïÜÉêÉ=Ü=áë=íÜÉ=~äíáíìÇÉ=Ñêçã=íÜÉ=êáÖÜí=~åÖäÉK==
=
O
O
~
Ä
166. ã O = ÄO − I= ã O = ~ O − I===
~
Ä
Q
Q
ïÜÉêÉ= ã ~ =~åÇ= ã Ä =~êÉ=íÜÉ=ãÉÇá~åë=íç=íÜÉ=äÉÖë=~=~åÇ=ÄK==
=
=
=
Figure 10.
=
Å
167. ã Å = I==
O
ïÜÉêÉ= ã Å =áë=íÜÉ=ãÉÇá~å=íç=íÜÉ=ÜóéçíÉåìëÉ=ÅK=
=
Å
168. o = = ã Å =
O
=
~ +Ä−Å
~Ä
=
=
169. ê =
O
~ +Ä+Å
=
170. ~Ä = ÅÜ =
=
=
26
37. CHAPTER 3. GEOMETRY
171. p =
~Ä ÅÜ
=
=
O
O
=
=
=
3.2 Isosceles Triangle
=
_~ëÉW=~=
iÉÖëW=Ä=
_~ëÉ=~åÖäÉW= β =
sÉêíÉñ=~åÖäÉW= α =
^äíáíìÇÉ=íç=íÜÉ=Ä~ëÉW=Ü=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
Figure 11.
=
172. β = VM° −
α
=
O
=
173. Ü O = ÄO −
O
~
=
Q
27
38. CHAPTER 3. GEOMETRY
174. i = ~ + OÄ =
=
175. p =
O
~Ü Ä
= ëáå α =
O
O
=
=
=
3.3 Equilateral Triangle
=
páÇÉ=çÑ=~=Éèìáä~íÉê~ä=íêá~åÖäÉW=~=
^äíáíìÇÉW=Ü=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
Figure 12.
=
176. Ü =
~ P
=
O
=
28
39. CHAPTER 3. GEOMETRY
O
~ P
=
177. o = Ü =
P
P
=
N
~ P o
= =
178. ê = Ü =
P
S
O
=
179. i = P~ =
=
180. p =
O
~Ü ~ P
=
=
O
Q
=
=
=
3.4 Scalene Triangle
E^=íêá~åÖäÉ=ïáíÜ=åç=íïç=ëáÇÉë=Éèì~äF=
=
=
páÇÉë=çÑ=~=íêá~åÖäÉW=~I=ÄI=Å=
~ +Ä+Å
==
pÉãáéÉêáãÉíÉêW= é =
O
^åÖäÉë=çÑ=~=íêá~åÖäÉW= αI βI γ =
^äíáíìÇÉë=íç=íÜÉ=ëáÇÉë=~I=ÄI=ÅW= Ü ~ I Ü Ä I Ü Å =
jÉÇá~åë=íç=íÜÉ=ëáÇÉë=~I=ÄI=ÅW= ã ~ I ã Ä I ã Å =
_áëÉÅíçêë=çÑ=íÜÉ=~åÖäÉë= αI βI γ W= í ~ I í Ä I í Å =
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
^êÉ~W=p=
=
=
29
40. CHAPTER 3. GEOMETRY
=
=====
=
Figure 13.
=
181. α + β + γ = NUM° =
182. ~ + Ä > Å I==
Ä + Å > ~ I==
~ + Å > Ä K=
=
183. ~ − Ä < Å I==
Ä − Å < ~ I==
~ − Å < Ä K=
=
=
184. jáÇäáåÉ=
~
è = I= è öö ~ K=
O
=
=
=
=====
Figure 14.
=
30
41. CHAPTER 3. GEOMETRY
185. i~ï=çÑ=`çëáåÉë=
~ O = ÄO + Å O − OÄÅ Åçë α I=
ÄO = ~ O + Å O − O~Å Åçë β I=
Å O = ~ O + ÄO − O~Ä Åçë γ K=
=
186. i~ï=çÑ=páåÉë=
~
Ä
Å
=
=
= Oo I==
ëáå α ëáå β ëáå γ
ïÜÉêÉ=o=áë=íÜÉ=ê~Çáìë=çÑ=íÜÉ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉK==
=
~
Ä
Å
ÄÅ
~Å
~Ä ~ÄÅ
=
=
=
=
=
=
187. o =
=
O ëáå α O ëáå β O ëáå γ OÜ ~ OÜ Ä OÜ Å Qp
=
(é − ~ )(é − Ä)(é − Å ) I==
188. ê O =
é
N N
N
N
= +
+ K=
ê Ü~ ÜÄ ÜÅ
=
(é − Ä)(é − Å ) I=
α
189. ëáå =
O
ÄÅ
Åçë
α
é(é − ~ )
I=
=
O
ÄÅ
í~å
α
=
O
(é − Ä)(é − Å ) K=
é(é − ~ )
=
O
190. Ü ~ =
é(é − ~ )(é − Ä)(é − Å ) I=
~
O
é(é − ~ )(é − Ä)(é − Å ) I=
ÜÄ =
Ä
O
ÜÅ =
é(é − ~ )(é − Ä)(é − Å ) K=
Å
31
42. CHAPTER 3. GEOMETRY
191. Ü ~ = Ä ëáå γ = Å ëáå β I=
Ü Ä = ~ ëáå γ = Å ëáå α I=
Ü Å = ~ ëáå β = Ä ëáå α K=
=
Ä +Å ~
− I==
O
Q
O
O
~ + Å ÄO
ãO =
− I==
Ä
O
Q
O
O
~ + Ä ÅO
O
ãÅ =
− K=
O
Q
192. ã O =
~
O
O
O
=
=
=
=====
Figure 15.
=
O
O
O
193. ^j = ã ~ I= _j = ã Ä I= `j = ã Å =EcáÖKNRFK=
P
P
P
=
QÄÅé(é − ~ )
194. í O =
I==
~
(Ä + Å )O
Q~Åé(é − Ä)
íO =
I==
Ä
(~ + Å )O
Q~Äé(é − Å )
íO =
K=
Å
(~ + Ä)O
=
32
43. CHAPTER 3. GEOMETRY
~Ü ~ ÄÜ Ä ÅÜ Å
=
=
I==
O
O
O
~Ä ëáå γ ~Å ëáå β ÄÅ ëáå α
I==
p=
=
=
O
O
O
p = é(é − ~ )(é − Ä)(é − Å ) =EeÉêçå∞ë=cçêãìä~FI=
p = éê I==
~ÄÅ
p=
I=
Qo
p = Oo O ëáå α ëáå β ëáå γ I=
α
β
γ
p = éO í~å í~å í~å K=
O
O
O
195. p =
=
=
=
3.5 Square
páÇÉ=çÑ=~=ëèì~êÉW=~=
aá~Öçå~äW=Ç=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
Figure 16.
33
44. CHAPTER 3. GEOMETRY
196. Ç = ~ O ==
=
197. o =
Ç ~ O
=
=
O
O
=
~
198. ê = =
O
199. i = Q~ =
=
=
200. p = ~ =
=
=
=
O
3.6 Rectangle
=
páÇÉë=çÑ=~=êÉÅí~åÖäÉW=~I=Ä=
aá~Öçå~äW=Ç=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
Figure 17.
=
201. Ç = ~ O + ÄO ==
34
45. CHAPTER 3. GEOMETRY
202. o =
Ç
=
O
=
203. i = O(~ + Ä) =
=
204. p = ~Ä =
=
=
=
3.7 Parallelogram
=
páÇÉë=çÑ=~=é~ê~ääÉäçÖê~ãW=~I=Ä=
aá~Öçå~äëW= ÇN I Ç O =
`çåëÉÅìíáîÉ=~åÖäÉëW= αI β =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW= ϕ =
^äíáíìÇÉW=Ü==
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=====
=
Figure 18.
=
205. α + β = NUM° =
206. Ç + Ç = O(~ + Ä ) =
O
N
O
O
O
=
O
=
35
46. CHAPTER 3. GEOMETRY
207. Ü = Ä ëáå α = Ä ëáå β =
208. i = O(~ + Ä) =
209. p = ~Ü = ~Ä ëáå α I==
N
p = ÇNÇ O ëáå ϕ K=
O
=
=
=
=
=
3.8 Rhombus
=
páÇÉ=çÑ=~=êÜçãÄìëW=~=
aá~Öçå~äëW= ÇN I Ç O =
`çåëÉÅìíáîÉ=~åÖäÉëW= αI β =
^äíáíìÇÉW=e=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
=====
Figure 19.
=
36
47. CHAPTER 3. GEOMETRY
210. α + β = NUM° =
=
211. Ç + Ç = Q~ =
O
N
O
O
O
=
212. Ü = ~ ëáå α =
ÇNÇ O
=
O~
=
Ü ÇÇ
~ ëáå α
213. ê = = N O =
=
O
Q~
O
=
214. i = Q~ =
=
215. p = ~Ü = ~ ëáå α I==
N
p = ÇNÇ O K=
O
=
=
=
O
3.9 Trapezoid
=
_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=
jáÇäáåÉW=è=
^äíáíìÇÉW=Ü=
^êÉ~W=p=
=
=
37
48. CHAPTER 3. GEOMETRY
=
=
Figure 20.
=
216. è =
217. p =
~+Ä
=
O
~+Ä
⋅ Ü = èÜ =
O
=
=
=
=
3.10 Isosceles Trapezoid
=
_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=
iÉÖW=Å=
jáÇäáåÉW=è=
^äíáíìÇÉW=Ü=
aá~Öçå~äW=Ç=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
^êÉ~W=p=
=
=
38
49. CHAPTER 3. GEOMETRY
=
=
Figure 21.
=
218. è =
~+Ä
=
O
=
219. Ç = ~Ä + Å =
=
N
O
220. Ü = Å O − (Ä − ~ ) =
Q
O
=
Å ~Ä + Å O
=
(OÅ − ~ + Ä)(OÅ + ~ − Ä)
=
~+Ä
222. p =
⋅ Ü = èÜ =
O
=
=
=
=
=
=
221. o =
39
50. CHAPTER 3. GEOMETRY
3.11 Isosceles Trapezoid with
Inscribed Circle
=
_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=
iÉÖW=Å=
jáÇäáåÉW=è=
^äíáíìÇÉW=Ü=
aá~Öçå~äW=Ç=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=o=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=ê=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
Figure 22.
=
223. ~ + Ä = OÅ =
=
~+Ä
224. è =
=Å=
O
=
225. Ç = Ü + Å =
O
O
O
=
40
51. CHAPTER 3. GEOMETRY
226. ê =
Ü
~Ä
=
=
O
O
=
Ä
ÅÇ ÅÇ Å
Å
Å
~+Ä ~
N+
ÜO + Å O =
=
=
=
+S+ =
OÜ Qê O
~Ä OÜ
U
Ä
~
=
228. i = O(~ + Ä) = QÅ =
=
(~ + Ä) ~Ä = èÜ = ÅÜ = iê ==
~+Ä
⋅Ü =
229. p =
O
O
O
=
=
=
227. o =
O
3.12 Trapezoid with Inscribed Circle
=
_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=
i~íÉê~ä=ëáÇÉëW=ÅI=Ç=
jáÇäáåÉW=è=
^äíáíìÇÉW=Ü=
aá~Öçå~äëW= ÇN I Ç O =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW= ϕ =
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
41
52. CHAPTER 3. GEOMETRY
=
=
Figure 23.
=
230. ~ + Ä = Å + Ç =
~+Ä Å+Ç
=
=
231. è =
O
O
232. i = O(~ + Ä) = O(Å + Ç ) =
=
=
=
~+Ä
Å+Ç
⋅Ü =
⋅ Ü = èÜ I==
O
O
N
p = ÇNÇ O ëáå ϕ K=
O
233. p =
=
=
=
3.13 Kite
=
páÇÉë=çÑ=~=âáíÉW=~I=Ä=
aá~Öçå~äëW= ÇN I Ç O =
^åÖäÉëW= αI βI γ =
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
42
53. CHAPTER 3. GEOMETRY
=
=
Figure 24.
=
234. α + β + Oγ = PSM° =
235. i = O(~ + Ä) =
=
=
236. p =
ÇNÇ O
=
O
=
=
=
3.14 Cyclic Quadrilateral
páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=
aá~Öçå~äëW= ÇN I Ç O =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW= ϕ =
fåíÉêå~ä=~åÖäÉëW= αI βI γ I δ =
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
pÉãáéÉêáãÉíÉêW=é==
^êÉ~W=p=
43
54. CHAPTER 3. GEOMETRY
=
=
Figure 25.
=
237. α + γ = β + δ = NUM° =
=
238. míçäÉãó∞ë=qÜÉçêÉã=
~Å + ÄÇ = ÇNÇ O =
239. i = ~ + Ä + Å + Ç =
=
=
N (~Å + ÄÇ )(~Ç + ÄÅ )(~Ä + ÅÇ )
I==
240. o =
Q (é − ~ )(é − Ä)(é − Å )(é − Ç )
i
ïÜÉêÉ= é = K=
O
=
N
241. p = ÇNÇ O ëáå ϕ I==
O
p = (é − ~ )(é − Ä)(é − Å )(é − Ç ) I==
i
ïÜÉêÉ= é = K=
O
=
=
=
44
55. CHAPTER 3. GEOMETRY
3.15 Tangential Quadrilateral
=
páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=
aá~Öçå~äëW= ÇN I Ç O =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW= ϕ =
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
mÉêáãÉíÉêW=i=
pÉãáéÉêáãÉíÉêW=é==
^êÉ~W=p=
=
=
=
=
Figure 26.
=
242. ~ + Å = Ä + Ç =
=
243. i = ~ + Ä + Å + Ç = O(~ + Å ) = O(Ä + Ç ) =
=
O
ÇN Ç O − (~ − Ä) (~ + Ä − é )
O
I==
Oé
i
ïÜÉêÉ= é = K==
O
=
O
O
244. ê =
45
56. CHAPTER 3. GEOMETRY
N
245. p = éê = ÇNÇ O ëáå ϕ =
O
=
=
=
3.16 General Quadrilateral
=
páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=
aá~Öçå~äëW= ÇN I Ç O =
^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW= ϕ =
fåíÉêå~ä=~åÖäÉëW= αI βI γ I δ =
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
=======
Figure 27.
=
246. α + β + γ + δ = PSM° =
247. i = ~ + Ä + Å + Ç =
=
=
46
57. CHAPTER 3. GEOMETRY
N
248. p = ÇNÇ O ëáå ϕ =
O
=
=
=
3.17 Regular Hexagon
=
páÇÉW=~=
fåíÉêå~ä=~åÖäÉW= α =
pä~åí=ÜÉáÖÜíW=ã=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
pÉãáéÉêáãÉíÉêW=é==
^êÉ~W=p=
=
=
=
=
Figure 28.
=
249. α = NOM° =
=
250. ê = ã =
~ P
=
O
47
58. CHAPTER 3. GEOMETRY
251. o = ~ =
=
252. i = S~ =
=
O
~ P P
I==
O
i
ïÜÉêÉ= é = K=
O
=
=
=
253. p = éê =
3.18 Regular Polygon
=
páÇÉW=~=
kìãÄÉê=çÑ=ëáÇÉëW=å=
fåíÉêå~ä=~åÖäÉW= α =
pä~åí=ÜÉáÖÜíW=ã=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
mÉêáãÉíÉêW=i=
pÉãáéÉêáãÉíÉêW=é==
^êÉ~W=p=
=
=
48
59. CHAPTER 3. GEOMETRY
=
=
Figure 29.
=
254. α =
255. α =
å−O
⋅ NUM° =
O
=
å−O
⋅ NUM° =
O
=
256. o =
~
π
O ëáå
å
=
=
257. ê = ã =
~
O í~å
π
å
= oO −
~O
=
Q
=
258. i = å~ =
=
259. p =
åo
Oπ
ëáå I==
O
å
O
p = éê = é o O −
~O
I==
Q
49
60. CHAPTER 3. GEOMETRY
ïÜÉêÉ= é =
i
K==
O
=
=
=
3.19 Circle
=
o~ÇáìëW=o=
aá~ãÉíÉêW=Ç=
`ÜçêÇW=~=
pÉÅ~åí=ëÉÖãÉåíëW=ÉI=Ñ=
q~åÖÉåí=ëÉÖãÉåíW=Ö=
`Éåíê~ä=~åÖäÉW= α =
fåëÅêáÄÉÇ=~åÖäÉW= β =
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
α
260. ~ = Oo ëáå =
O
=
=
=
Figure 30.
=
50
63. CHAPTER 3. GEOMETRY
3.20 Sector of a Circle
=
o~Çáìë=çÑ=~=ÅáêÅäÉW=o=
^êÅ=äÉåÖíÜW=ë=
`Éåíê~ä=~åÖäÉ=Eáå=ê~Çá~åëFW=ñ=
`Éåíê~ä=~åÖäÉ=Eáå=ÇÉÖêÉÉëFW= α =
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
Figure 35.
=
267. ë = oñ =
268. ë =
=
πoα
=
NUM°
=
269. i = ë + Oo =
=
270. p =
oë o ñ πo α
=
=
==
O
O
PSM°
O
O
=
=
53
64. CHAPTER 3. GEOMETRY
3.21 Segment of a Circle
=
o~Çáìë=çÑ=~=ÅáêÅäÉW=o=
^êÅ=äÉåÖíÜW=ë=
`ÜçêÇW=~=
`Éåíê~ä=~åÖäÉ=Eáå=ê~Çá~åëFW=ñ=
`Éåíê~ä=~åÖäÉ=Eáå=ÇÉÖêÉÉëFW= α =
eÉáÖÜí=çÑ=íÜÉ=ëÉÖãÉåíW=Ü=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
=
=
Figure 36.
=
271. ~ = O OÜo − Ü O =
=
N
272. Ü = o −
Qo O − ~ O I= Ü < o =
O
=
273. i = ë + ~ =
=
54
65. CHAPTER 3. GEOMETRY
O
O
N
[ëo − ~(o − Ü )] = o απ − ëáå α = o (ñ − ëáå ñ ) I==
O
O NUM°
O
O
p ≈ Ü~ K=
P
274. p =
=
=
=
3.22 Cube
=
bÇÖÉW=~==
aá~Öçå~äW=Ç=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ëéÜÉêÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ëéÜÉêÉW=ê=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
===
Figure 37.
=
275. Ç = ~ P =
=
~
276. ê = =
O
=
55
66. CHAPTER 3. GEOMETRY
277. o =
~ P
=
O
=
278. p = S~ =
O
=
279. s = ~ ==
=
=
=
P
3.23 Rectangular Parallelepiped
=
bÇÖÉëW=~I=ÄI=Å==
aá~Öçå~äW=Ç=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
=====
Figure 38.
=
280. Ç = ~ O + ÄO + Å O =
281. p = O(~Ä + ~Å + ÄÅ ) =
282. s = ~ÄÅ ==
=
=
56
67. CHAPTER 3. GEOMETRY
3.24 Prism
=
i~íÉê~ä=ÉÇÖÉW=ä=
eÉáÖÜíW=Ü=
i~íÉê~ä=~êÉ~W= p i =
^êÉ~=çÑ=Ä~ëÉW= p_ =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
=====
Figure 39.
=
283. p = p i + Op_ K==
=
284. i~íÉê~ä=^êÉ~=çÑ=~=oáÖÜí=mêáëã=
p i = (~ N + ~ O + ~ P + K + ~ å )ä =
=
285. i~íÉê~ä=^êÉ~=çÑ=~å=lÄäáèìÉ=mêáëã=
p i = éä I==
ïÜÉêÉ=é=áë=íÜÉ=éÉêáãÉíÉê=çÑ=íÜÉ=Åêçëë=ëÉÅíáçåK=
=
57
68. CHAPTER 3. GEOMETRY
286. s = p_ Ü =
=
287. `~î~äáÉêáDë=mêáåÅáéäÉ==
dáîÉå=íïç=ëçäáÇë=áåÅäìÇÉÇ=ÄÉíïÉÉå=é~ê~ääÉä=éä~åÉëK=fÑ=ÉîÉêó=
éä~åÉ=Åêçëë=ëÉÅíáçå=é~ê~ääÉä=íç=íÜÉ=ÖáîÉå=éä~åÉë=Ü~ë=íÜÉ=ë~ãÉ=
~êÉ~=áå=ÄçíÜ=ëçäáÇëI=íÜÉå=íÜÉ=îçäìãÉë=çÑ=íÜÉ=ëçäáÇë=~êÉ=Éèì~äK=
=
=
=
3.25 Regular Tetrahedron
=
qêá~åÖäÉ=ëáÇÉ=äÉåÖíÜW=~=
eÉáÖÜíW=Ü=
^êÉ~=çÑ=Ä~ëÉW= p_ =
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
Figure 40.
=
288. Ü =
O
~=
P
=
58
69. CHAPTER 3. GEOMETRY
289. p_ =
P~ O
=
Q
=
290. p = P~ =
=
N
~P
291. s = p_ Ü =
K==
P
S O
=
=
=
O
3.26 Regular Pyramid
=
páÇÉ=çÑ=Ä~ëÉW=~=
i~íÉê~ä=ÉÇÖÉW=Ä=
eÉáÖÜíW=Ü=
pä~åí=ÜÉáÖÜíW=ã==
kìãÄÉê=çÑ=ëáÇÉëW=å==
pÉãáéÉêáãÉíÉê=çÑ=Ä~ëÉW=é=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ëéÜÉêÉ=çÑ=Ä~ëÉW=ê=
^êÉ~=çÑ=Ä~ëÉW= p_ =
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= pi =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
59
70. CHAPTER 3. GEOMETRY
=
=
Figure 41.
=
292. ã = ÄO −
~O
=
Q
=
293. Ü =
π O
−~
å
=
π
O ëáå
å
QÄO ëáå O
=
N
N
294. p i = å~ã = å~ QÄO − ~ O = éã =
O
Q
=
295. p_ = éê =
=
296. p = p_ + p i =
=
N
N
297. s = p_ Ü = éêÜ ==
P
P
=
=
=
60
71. CHAPTER 3. GEOMETRY
3.27 Frustum of a Regular Pyramid
=
~N I ~ O I ~ P IKI ~ å
=
_~ëÉ=~åÇ=íçé=ëáÇÉ=äÉåÖíÜëW=
ÄN I ÄO I ÄP IKI Äå
eÉáÖÜíW=Ü=
pä~åí=ÜÉáÖÜíW=ã==
^êÉ~=çÑ=Ä~ëÉëW= pN I= pO =
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= p i =
mÉêáãÉíÉê=çÑ=Ä~ëÉëW= mN I= mO =
pÅ~äÉ=Ñ~ÅíçêW=â=
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
Figure 42.
=
298.
ÄN ÄO ÄP
Ä
Ä
= = =K= å = = â =
~N ~ O ~ P
~å ~
=
61
72. CHAPTER 3. GEOMETRY
299.
pO
= âO =
pN
=
ã(mN + mO )
=
300. p i =
O
=
301. p = p i + pN + pO =
=
Ü
302. s = pN + pNpO + pO =
P
=
O
Üp Ä Ä Üp
303. s = N N + + = N N + â + â O =
P ~ ~ P
=
=
=
(
)
[
]
3.28 Rectangular Right Wedge
=
páÇÉë=çÑ=Ä~ëÉW=~I=Ä=
qçé=ÉÇÖÉW=Å=
eÉáÖÜíW=Ü=
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= p i =
^êÉ~=çÑ=Ä~ëÉW= p_ =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
62
73. CHAPTER 3. GEOMETRY
=
=
Figure 43.
=
N
(~ + Å ) QÜO + ÄO + Ä ÜO + (~ − Å )O =
O
=
305. p_ = ~Ä =
=
306. p = p_ + p i =
=
ÄÜ
(O~ + Å ) =
307. s =
S
=
=
=
304. p i =
3.29 Platonic Solids
=
bÇÖÉW=~=
o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
63
75. CHAPTER 3. GEOMETRY
311. p = O~ O P =
=
~P O
312. s =
=
P
=
=
Icosahedron
=
=
=
Figure 45.
=
313. ê =
(
=
314. o =
)
~ P P+ R
=
NO
(
)
~
O R+ R =
Q
=
315. p = R~ O P =
=
R~ P P + R
316. s =
=
NO
=
=
(
)
65
76. CHAPTER 3. GEOMETRY
Dodecahedron
=
=
=
Figure 46.
317. ê =
(
~ NM OR + NN R
=
O
=
318. o =
)
=
(
)
~ P N+ R
=
Q
=
(
)
319. p = P~ O R R + O R =
=
~ P NR + T R
320. s =
=
Q
=
=
=
(
)
3.30 Right Circular Cylinder
=
o~Çáìë=çÑ=Ä~ëÉW=o=
aá~ãÉíÉê=çÑ=Ä~ëÉW=Ç=
66
77. CHAPTER 3. GEOMETRY
eÉáÖÜíW=e=
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= p i =
^êÉ~=çÑ=Ä~ëÉW= p_ =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=====
=
Figure 47.
=
321. p i = Oπoe =
=
Ç
322. p = p i + Op_ = Oπo(e + o ) = πÇ e + =
O
=
323. s = p_ e = πo O e =
=
=
=
67
78. CHAPTER 3. GEOMETRY
3.31 Right Circular Cylinder with
an Oblique Plane Face
=
o~Çáìë=çÑ=Ä~ëÉW=o=
qÜÉ=ÖêÉ~íÉëí=ÜÉáÖÜí=çÑ=~=ëáÇÉW= ÜN =
qÜÉ=ëÜçêíÉëí=ÜÉáÖÜí=çÑ=~=ëáÇÉW= Ü O =
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= p i =
^êÉ~=çÑ=éä~åÉ=ÉåÇ=Ñ~ÅÉëW= p_ =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
Figure 48.
=
324. p i = πo(ÜN + Ü O ) =
=
O
Ü − ÜO
325. p_ = πo + πo o + N
=
O
=
O
O
68
79. CHAPTER 3. GEOMETRY
O
ÜN − Ü O
O
326. p = p i + p_ = πo ÜN + Ü O + o + o +
=
O
=
πo O
(ÜN + ÜO ) =
327. s =
O
=
=
=
3.32 Right Circular Cone
o~Çáìë=çÑ=Ä~ëÉW=o=
aá~ãÉíÉê=çÑ=Ä~ëÉW=Ç=
eÉáÖÜíW=e=
pä~åí=ÜÉáÖÜíW=ã=
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= pi =
^êÉ~=çÑ=Ä~ëÉW= p_ =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
=
Figure 49.
69
80. CHAPTER 3. GEOMETRY
328. e = ã O − o O =
=
πãÇ
329. p i = πoã =
=
O
=
330. p_ = πo O =
=
N
Ç
331. p = p i + p_ = πo (ã + o ) = πÇ ã + =
O
O
=
N
N
332. s = p_ e = πo O e =
P
P
=
=
=
3.33 Frustum of a Right Circular Cone
=
o~Çáìë=çÑ=Ä~ëÉëW=oI=ê=
eÉáÖÜíW=e=
pä~åí=ÜÉáÖÜíW=ã=
pÅ~äÉ=Ñ~ÅíçêW=â=
^êÉ~=çÑ=Ä~ëÉëW= pN I= pO =
i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= p i =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
70
81. CHAPTER 3. GEOMETRY
=
=
Figure 50.
=
333. e = ã O − (o − ê ) =
=
o
334.
=â=
ê
=
p oO
335. O = O = â O =
pN ê
=
336. p i = πã(o + ê ) =
=
337. p = pN + pO + p i = π o O + ê O + ã(o + ê ) =
=
Ü
338. s = pN + pNpO + pO =
P
=
O
ÜpN o o ÜpN
339. s =
N+ â + âO =
N + + =
P ê ê P
=
=
=
O
[
(
]
)
[
71
]
82. CHAPTER 3. GEOMETRY
3.34 Sphere
=
o~ÇáìëW=o=
aá~ãÉíÉêW=Ç=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
=
Figure 51.
=
340. p = Qπo O =
=
Q
N
N
341. s = πo P e = πÇ P = po =
P
S
P
=
=
=
3.35 Spherical Cap
o~Çáìë=çÑ=ëéÜÉêÉW=o=
o~Çáìë=çÑ=Ä~ëÉW=ê=
eÉáÖÜíW=Ü=
^êÉ~=çÑ=éä~åÉ=Ñ~ÅÉW= p_ =
^êÉ~=çÑ=ëéÜÉêáÅ~ä=Å~éW= p` =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
72
83. CHAPTER 3. GEOMETRY
=
=
Figure 52.
=
342. o =
ê O + ÜO
=
OÜ
=
343. p_ = πê O =
=
344. p` = π(Ü O + ê O )=
=
345. p = p_ + p` = π(Ü O + Oê O ) = π(OoÜ + ê O ) =
=
π
π
346. s = Ü O (Po − Ü ) = Ü(Pê O + Ü O ) =
S
S
=
=
=
3.36 Spherical Sector
=
o~Çáìë=çÑ=ëéÜÉêÉW=o=
o~Çáìë=çÑ=Ä~ëÉ=çÑ=ëéÜÉêáÅ~ä=Å~éW=ê=
eÉáÖÜíW=Ü=
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
73
84. CHAPTER 3. GEOMETRY
======
=
===
=
Figure 53.
=
347. p = πo(OÜ + ê ) =
=
O
348. s = πo O Ü =
P
=
kçíÉW= qÜÉ= ÖáîÉå= Ñçêãìä~ë= ~êÉ= ÅçêêÉÅí= ÄçíÜ= Ñçê= ±çéÉå≤= ~åÇ=
±ÅäçëÉÇ≤=ëéÜÉêáÅ~ä=ëÉÅíçêK=
=
=
=
3.37 Spherical Segment
=
o~Çáìë=çÑ=ëéÜÉêÉW=o=
o~Çáìë=çÑ=Ä~ëÉëW= êN I= êO =
eÉáÖÜíW=Ü=
^êÉ~=çÑ=ëéÜÉêáÅ~ä=ëìêÑ~ÅÉW= pp =
^êÉ~=çÑ=éä~åÉ=ÉåÇ=Ñ~ÅÉëW= pN I= pO =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
74
85. CHAPTER 3. GEOMETRY
=
=====
=
Figure 54.
=
349. pp = OπoÜ =
=
350. p = pp + pN + pO = π(OoÜ + êNO + êOO ) =
=
N
351. s = πÜ(PêNO + PêOO + Ü O )=
S
=
=
=
3.38 Spherical Wedge
=
o~ÇáìëW=o=
aáÜÉÇê~ä=~åÖäÉ=áå=ÇÉÖêÉÉëW=ñ=
aáÜÉÇê~ä=~åÖäÉ=áå=ê~Çá~åëW= α =
^êÉ~=çÑ=ëéÜÉêáÅ~ä=äìåÉW= p i =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
75
86. CHAPTER 3. GEOMETRY
=
=
Figure 55.
=
352. p i =
πo O
α = Oo O ñ =
VM
=
353. p = πo O +
πo O
α = πo O + Oo O ñ =
VM
=
354. s =
πoP
O
α = oP ñ =
OTM
P
=
=
=
3.39 Ellipsoid
=
pÉãá-~ñÉëW=~I=ÄI=Å=
sçäìãÉW=s=
76
87. CHAPTER 3. GEOMETRY
=
=======
=
Figure 56.
=
Q
355. s = π~ÄÅ =
P
=
=
=
Prolate Spheroid
=
pÉãá-~ñÉëW=~I=ÄI=Ä=E ~ > Ä F=
pìêÑ~ÅÉ=~êÉ~W=p=
sçäìãÉW=s=
=
=
~ ~êÅëáå É
356. p = OπÄ Ä +
I==
É
ïÜÉêÉ= É =
~ O − ÄO
K=
~
=
Q
357. s = πÄO~ =
P
=
77
97. CHAPTER 4. TRIGONOMETRY
4.4 Trigonometric Functions of Common
Angles
381. =
α° = α ê~Ç =
M=
M=
π
=
PM=
S
π
=
QR=
Q
π
=
SM=
P
π
=
VM=
O
Oπ
=
NOM=
P
NUM=
π=
Pπ
=
OTM=
O
PSM= Oπ =
=
=
=
=
=
=
=
=
=
=
=
=
=
O
=
O
P
=
O
Åçë α =
N=
P
=
O
O
=
O
N
=
O
N=
M=
P
=
O
M=
N
− =
O
− N=
− N=
M=
ëáå α =
M=
N
=
O
í~å α = Åçí α
M=
∞=
N
=
P=
P
ëÉÅ α =
N=
O
=
P
ÅçëÉÅ α =
∞=
O=
N=
N=
P=
N
=
P
O=
O
=
P
M=
∞=
N=
∞=
O=
O=
M=
N
P
∞=
− N=
O
=
P
∞=
M=
∞=
M=
∞=
− N=
N=
M=
∞=
N=
∞=
− P=
87
−
−O=
98. CHAPTER 4. TRIGONOMETRY
382. =
α° = α ê~Ç =
π
=
NR=
NO
ëáå α =
Åçë α =
í~å α =
Åçí α =
S− O
=
Q
S+ O
=
Q
O− P =
O+ P =
R−O R
=
R
R+O R =
NU=
π
=
NM
R −N
=
Q
NM + O R
Q
PS=
π
=
R
NM − O R
Q
R +N
=
Q
RQ=
Pπ
=
NM
R +N
=
Q
NM − O R
Q
TO=
Oπ
=
R
NM + O R
Q
R −N
=
Q
TR=
Rπ
=
NO
S+ O
=
Q
S− O
=
Q
=
=
=
4.5 Most Important Formulas
=
383. ëáå O α + Åçë O α = N =
=
384. ëÉÅ O α − í~å O α = N =
=
385. ÅëÅ O α − Åçí O α = N =
=
ëáå α
=
386. í~å α =
Åçë α
88
NM − O R
R +N
R +N
NM − O R
R +N
NM − O R
=
NM − O R
R +N
=
R+O R =
R−O R
R
=
O+ P =
O− P =
100. CHAPTER 4. TRIGONOMETRY
4.7 Periodicity of Trigonometric Functions
=
392. ëáå(α ± Oπå ) = ëáå α I=éÉêáçÇ= Oπ =çê= PSM° K=
=
393. Åçë(α ± Oπå ) = Åçë α I=éÉêáçÇ= Oπ =çê= PSM° K=
=
394. í~å(α ± πå ) = í~å α I=éÉêáçÇ= π =çê= NUM° K=
=
395. Åçí(α ± πå ) = Åçí α I=éÉêáçÇ= π =çê= NUM° K=
=
=
=
4.8 Relations between Trigonometric
Functions
=
396. ëáå α = ± N − Åçë O α = ±
α
O =
=
α
N + í~å O
O
N
(N − Åçë Oα ) = O Åçë O α − π − N =
O
O Q
O í~å
=
=
397. Åçë α = ± N − ëáå O α = ±
α
O=
=
α
N + í~å O
O
N
(N + Åçë Oα ) = O Åçë O α − N =
O
O
N − í~å O
=
=
398. í~å α =
ëáå α
ëáå Oα
N − Åçë Oα
= ± ëÉÅ O α − N =
=
=
Åçë α
N + Åçë Oα
ëáå Oα
90
101. CHAPTER 4. TRIGONOMETRY
α
N − Åçë Oα
O =
=±
=
N + Åçë Oα
O α
N + í~å
O
O í~å
=
=
Åçë α
N + Åçë Oα
ëáå Oα
= ± ÅëÅ O α − N =
=
=
ëáå α
ëáå Oα
N − Åçë Oα
α
N − í~å O
N + Åçë Oα
O=
=
= =±
α
N − Åçë Oα
O í~å
O
399. Åçí α =
=
α
N
O=
400. ëÉÅ α =
= ± N + í~å O α =
α
Åçë α
N − í~å O
O
=
α
N + í~å O
N
O=
401. ÅëÅ α =
= ± N + Åçí O α =
α
ëáå α
O í~å
O
=
=
=
N + í~å O
4.9 Addition and Subtraction Formulas
=
402. ëáå(α + β) = ëáå α Åçë β + ëáå β Åçë α =
=
403. ëáå(α − ó ) = ëáå α Åçë β − ëáå β Åçë α =
=
404. Åçë(α + β ) = Åçë α Åçë β − ëáå α ëáå β =
=
405. Åçë(α − β ) = Åçë α Åçë β + ëáå α ëáå β =
91
102. CHAPTER 4. TRIGONOMETRY
406. í~å(α + β ) =
=
407. í~å(α − β ) =
=
408. Åçí(α + β) =
=
409. Åçí(α − β) =
í~å α + í~å β
=
N − í~å α í~å β
í~å α − í~å β
=
N + í~å α í~å β
N − í~å α í~å β
=
í~å α + í~å β
N + í~å α í~å β
=
í~å α − í~å β
=
=
=
4.10 Double Angle Formulas
=
410. ëáå Oα = O ëáå α ⋅ Åçë α =
=
411. Åçë Oα = Åçë O α − ëáå O α = N − O ëáå O α = O Åçë O α − N =
=
O í~å α
O
412. í~å Oα =
=
=
O
N − í~å α Åçí α − í~å α
=
Åçí O α − N Åçí α − í~å α
=
=
413. Åçí Oα =
O Åçí α
O
=
=
=
=
=
=
92
103. CHAPTER 4. TRIGONOMETRY
4.11 Multiple Angle Formulas
=
414. ëáå Pα = P ëáå α − Q ëáå P α = P Åçë O α ⋅ ëáå α − ëáåP α =
=
415. ëáå Qα = Q ëáå α ⋅ Åçë α − U ëáå P α ⋅ Åçë α =
=
416. ëáå Rα = R ëáå α − OM ëáå P α + NS ëáå R α =
=
417. Åçë Pα = Q ÅçëP α − P Åçë α = Åçë P α − P Åçë α ⋅ ëáå O α =
=
418. Åçë Qα = U Åçë Q α − U Åçë O α + N =
=
419. Åçë Rα = NS Åçë R α − OM Åçë P α + R Åçë α =
=
P í~å α − í~å P α
420. í~å Pα =
=
N − P í~å O α
=
Q í~å α − Q í~å P α
=
421. í~å Qα =
N − S í~å O α + í~å Q α
=
í~å R α − NM í~å P α + R í~å α
=
422. í~å Rα =
N − NM í~å O α + R í~å Q α
=
Åçí P α − P Åçí α
423. Åçí Pα =
=
P Åçí O α − N
=
N − S í~å O α + í~å Q α
==
424. Åçí Qα =
Q í~å α − Q í~å P α
=
93
104. CHAPTER 4. TRIGONOMETRY
425. Åçí Rα =
N − NM í~å O α + R í~å Q α
=
í~å R α − NM í~å P α + R í~å α
=
=
=
4.12 Half Angle Formulas
=
426. ëáå
α
N − Åçë α
=
=±
O
O
=
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
=
α
O =
430. ëáå α =
α
N + í~å O
O
=
O í~å
94
105. CHAPTER 4. TRIGONOMETRY
α
O=
431. Åçë α =
O α
N + í~å
O
=
α
O í~å
O =
432. í~å α =
α
N − í~å O
O
=
α
N − í~å O
O=
433. Åçí α =
α
O í~å
O
=
=
=
N − í~å O
4.14 Transforming of Trigonometric
Expressions to Product
=
434. ëáå α + ëáå β = O ëáå
=
435. ëáå α − ëáå β = O Åçë
α+β
α −β
=
Åçë
O
O
α +β
α −β
=
ëáå
O
O
=
436. Åçë α + Åçë β = O Åçë
α+β
α −β
=
Åçë
O
O
=
437. Åçë α − Åçë β = −O ëáå
α +β
α −β
=
ëáå
O
O
=
95
117. Chapter 5
Matrices and Determinants
=
=
=
=
j~íêáÅÉëW=^I=_I=`=
bäÉãÉåíë=çÑ=~=ã~íêáñW= ~ á I= Äá I= ~ áà I= Äáà I= Å áà =
aÉíÉêãáå~åí=çÑ=~=ã~íêáñW= ÇÉí ^ =
jáåçê=çÑ=~å=ÉäÉãÉåí= ~ áà W= j áà =
`çÑ~Åíçê=çÑ=~å=ÉäÉãÉåí= ~ áà W= ` áà =
ú
qê~åëéçëÉ=çÑ=~=ã~íêáñW= ^ q I= ^ =
^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= ñ á =
k~íìê~ä=åìãÄÉêëW=ãI=å===
=
=
5.1 Determinants
=
513. pÉÅçåÇ=lêÇÉê=aÉíÉêãáå~åí=
~ ÄN
ÇÉí ^ = N
= ~ N Ä O − ~ O ÄN =
~ O ÄO
=
=
=
=
=
107
118. 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
~ ON ~ OO K ~ O à
K K K K
ÇÉí ^ =
~ áN ~ á O K ~ áà
K K K K
~ åN ~ å O K ~ åà
K ~Nå
K ~ Oå
K K
K ~ áå
=
K K
K ~ åå
=
517. jáåçê=
qÜÉ=ãáåçê= j áà =~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
119. CHAPTER 5. MATRICES AND DETERMINANTS
518. `çÑ~Åíçê=
á +à
` áà = (− N) j áà =
=
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 OI KI å K=
à=N
i~éä~ÅÉ=Éñé~åëáçå=Äó=ÉäÉãÉåíë=çÑ=íÜÉ=à-íÜ=Åçäìãå=
å
ÇÉí ^ = ∑ ~ áà` áà I= à = NI OI KI å 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=
~ ~ O ~N ÄN
=
==
= N
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
~ ÄO
=− O
=
~ 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
120. 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
~ ÄN
=â N
=
~ 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=EåìãÄÉêë=çê=ÑìåÅíáçåëF=ïáíÜ=ã=êçïë=~åÇ=å=ÅçäìãåëK==
~ NN ~ NO K ~ Nå
~
~ OO K ~ Oå
==
ON
^ = ~ áà =
M
M
M
~ ãN ~ ã O 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
121. 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
~
~ OO K ~ Oå
I==
^ = ~ áà = ON
M
M
M
~ ãN ~ ã O K ~ ãå
ÄNN ÄNO K ÄNå
Ä
ÄOO K ÄOå
I==
_ = Äáà = ON
M
M
M
ÄãN Äã O K Äãå
=
=
=
=
=
[ ]
[ ]
111
122. CHAPTER 5. MATRICES AND DETERMINANTS
íÜÉå==
~NO + ÄNO K ~Nå + ÄNå
~NN + ÄNN
~ +Ä
~ OO + ÄOO K ~ Oå + ÄOå
K=
ON ON
^+_=
M
M
M
~ ãN + ÄãN ~ ã O + Äã O 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~ OO K â~ Oå
K=
ON
â^ = â~ áà =
M
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
~
~ OO K ~ Oå
I==
^ = ~ áà = ON
M
M
M
~ ãN ~ ã O K ~ ãå
ÄNN ÄNO K ÄNâ
Ä
ÄOO K ÄO â
I=
_ = Äáà = ON
M
M
M
ÄåN Äå O K Äåâ
=
=
=
=
=
[ ]
[ ]
[ ]
112
123. CHAPTER 5. MATRICES AND DETERMINANTS
íÜÉå==
ÅNN ÅNO K ÅNâ
Å
Å OO K Å O â
I==
ON
^_ = ` =
M
M
M
Ä ãN Å ã O K Å ãâ
ïÜÉêÉ==
å
Å áà = ~ áNÄNà + ~ á O ÄO à + K + ~ áå Äåà = ∑ ~ á λ Äλ à =
E á = NI OI KI ã X à = NI OI KI â FK==
=
qÜìë=áÑ=
[ ]
~ NN
^ = ~ áà =
~ ON
~ NO
~ OO
λ =N
ÄN
~ NP
I= _ = [Ä á ] = Ä O I==
~ OP
ÄP
íÜÉå==
~ NN ~ NO
^_ =
~ ON ~ OO
Ä
~ NP N ~ NNÄN
⋅ Ä =
~ OP O ~ ONÄN
Ä
P
~ NO Ä O
~ OO Ä O
~ NP ÄP
K==
~ OP 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
124. 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 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=
~Çà ^
^ −N =
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===
=
=
=
q
5.5 Systems of Linear Equations
=
=
s~êá~ÄäÉëW=ñI=óI=òI= ñ N I= ñ O I K =
oÉ~ä=åìãÄÉêëW= ~ N I ~ O I ~ P I ÄN I ~ NN I ~ NO I K =
114
125. CHAPTER 5. MATRICES AND DETERMINANTS
aÉíÉêãáå~åíëW=aI= añ I= aó I= aò ==
j~íêáÅÉëW=^I=_I=u=
=
=
~ ñ + ÄNó = ÇN
I==
544. N
~ O ñ + ÄO ó = Ç O
aó
a
=E`ê~ãÉê∞ë=êìäÉFI==
ñ = ñ I= ó =
a
a
ïÜÉêÉ==
~ ÄN
a= N
= ~NÄO − ~ O ÄN I==
~ O ÄO
Ç ÄN
añ = N
= ÇNÄO − Ç O ÄN I==
Ç O ÄO
~ ÇN
aó = N
= ~NÇ O − ~ OÇN K==
~ O ÇO
=
545. fÑ= a ≠ M I=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë=~=ëáåÖäÉ=ëçäìíáçåW==
aó
a
K=
ñ = ñ I= ó =
a
a
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=
=
~Nñ + ÄNó + ÅNò = ÇN=
546. ~ O ñ + ÄO ó + Å Oò = Ç O I==
~ ñ + Ä ó + Å ò = Ç
P
P
P
P
ñ=
aó
añ
a
I= ó =
I= ò = ò =E`ê~ãÉê∞ë=êìäÉFI==
a
a
a
=
115
126. CHAPTER 5. MATRICES AND DETERMINANTS
ïÜÉêÉ==
~N ÄN
a = ~ O ÄO
~ P ÄP
ÅN
ÇN
ÄN
ÅN
Å O I= añ = Ç O
ÄO
Å O I=
ÅP
ÄP
ÅP
ÇP
~N
ÇN
ÅN
~N
ÄN
ÇN
aó = ~ O
~P
ÇO
ÇP
Å O I= aò = ~ O
ÅP
~P
ÄO
ÄP
Ç O K==
ÇP
=
547. fÑ= a ≠ M I=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë=~=ëáåÖäÉ=ëçäìíáçåW==
aó
a
a
I= ò = ò K=
ñ = ñ I= ó =
a
a
a
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==
~NNñ N + ~ NO ñ O + K + ~ Nå ñ å = ÄN
~ ñ + ~ ñ + K + ~ ñ = Ä
ON N OO O
Oå å
O
=
KKKKKKKKKKKK
~ åNñ N + ~ å O ñ O + K + ~ åå ñ å = Äå
Å~å=ÄÉ=ïêáííÉå=áå=ã~íêáñ=Ñçêã=
~ NN ~ NO K ~ Nå ñ N ÄN
~ ON ~ OO K ~ Oå ñ O Ä O
I==
=
⋅
M
M
M M M
~
åN ~ å O K ~ åå ñ å Ä å
áKÉK==
^ ⋅ u = _ I==
116
127. CHAPTER 5. MATRICES AND DETERMINANTS
ïÜÉêÉ==
~ NN
~
^ = ON
M
~
åN
~ NO K ~ Nå
ñN
ÄN
~ OO K ~ Oå
ñO
Ä
I= u = I= _ = O K==
M
M
M
M
ñ
Ä
~ å O K ~ åå
å
å
=
549. pçäìíáçå=çÑ=~=pÉí=çÑ=iáåÉ~ê=bèì~íáçåë= å × å =
u = ^ −N ⋅ _ I==
ïÜÉêÉ= ^ −N =áë=íÜÉ=áåîÉêëÉ=çÑ=^K=
=
=
117
128. Chapter 6
Vectors
=
=
=
=
r r r r →
sÉÅíçêëW= ì I= î I= ï I= ê I= ^_ I=£=
r r
sÉÅíçê=äÉåÖíÜW= ì I= î I=£=
r r r
råáí=îÉÅíçêëW= á I= à I= â =
r
kìää=îÉÅíçêW= M =
r
`ççêÇáå~íÉë=çÑ=îÉÅíçê= ì W= uN I vN I wN =
r
`ççêÇáå~íÉë=çÑ=îÉÅíçê= î W= u O I vO I wO =
pÅ~ä~êëW= λ I µ =
aáêÉÅíáçå=ÅçëáåÉëW= Åçë α I= Åçë β I= Åçë γ =
^åÖäÉ=ÄÉíïÉÉå=íïç=îÉÅíçêëW= θ =
=
=
6.1 Vector Coordinates
=
550. råáí=sÉÅíçêë=
r
á = (NI MI M) I=
r
à = (MI NI M) I=
r
â = (MI MI N) I=
r r r
á = à = â = N K=
=
r
r
r
r →
551. ê = ^_ = (ñ N − ñ M ) á + (ó N − ó M ) à + (ò N − ò M ) â =
=
118
129. CHAPTER 6. VECTORS
=======
=
=
Figure 73.
=
→
r
ê = ^_ =
552.
(ñ N − ñ M )O + (óN − ó M )O + (òN − ò M )O =
=
→
→
r
r
553. fÑ= ^_ = ê I=íÜÉå= _^ = − ê K=
=
=
=
Figure 74.
r
554. u = ê Åçë α I=
r
v = ê Åçë β I=
r
w = ê Åçë γ K=
=
119
130. CHAPTER 6. VECTORS
=
=====
=
Figure 75.
=
r
r
555. fÑ= ê (uI v I w ) = êN (uN I vN I wN ) I=íÜÉå==
u = uN I= v = vN I= w = wN K==
==
=
6.2 Vector Addition
=
r r r
556. ï = ì + î =
=
=
==
=
Figure 76.
120
131. CHAPTER 6. VECTORS
=
==
=
Figure 77.
=
r
r r r r
557. ï = ìN + ì O + ìP + K + ì å =
=
=
=
==
Figure 78.
=
558. `çããìí~íáîÉ=i~ï=
r r r r
ì+ î =î+ì=
=
559. ^ëëçÅá~íáîÉ=i~ï=
r r r r r r
(ì + î ) + ï = ì + (î + ï ) =
=
r r
560. ì + î = (uN + u O I vN + vO I wN + wO ) =
=
=
=
=
=
=
121
132. CHAPTER 6. VECTORS
6.3 Vector Subtraction
=
r r r r r r
561. ï = ì − î =áÑ= î + ï = ì K=
=
=
=
Figure 79.
=
=
==
=
Figure 80.
=
r r r
r
562. ì − î = ì + (− î ) =
=
r r r
563. ì − ì = M = (MI MI M ) =
=
r
564. M = M =
=
r r
565. ì − î = (uN − u O I vN − vO I wN − w O ) I==
=
=
=
6.4 Scaling Vectors
=
r
r
566. ï = λì =
122
133. CHAPTER 6. VECTORS
=
=
Figure 81.
=
567.
r
r
ï = λ⋅ì=
=
r
568. λì = (λuI λv I λw ) =
=
r r
569. λì = ìλ =
=
r
r
r
570. (λ + µ ) ì = λì + µì =
=
r
r
r
571. λ(µì ) = µ(λì ) = (λµ )ì =
=
r r
r
r
572. λ(ì + î ) = λì + λî =
=
=
=
6.5 Scalar Product
=
r
r
573. pÅ~ä~ê=mêçÇìÅí=çÑ=sÉÅíçêë= ì =~åÇ î =
r r r r
ì ⋅ î = ì ⋅ î ⋅ Åçë θ I==
r
r
ïÜÉêÉ= θ =áë=íÜÉ=~åÖäÉ=ÄÉíïÉÉå=îÉÅíçêë= ì =~åÇ î K====
=
123
134. CHAPTER 6. VECTORS
=
=
=
Figure 82.
=
574. pÅ~ä~ê=mêçÇìÅí=áå=`ççêÇáå~íÉ=cçêã=
r
r
fÑ= ì = (uN I vN I wN ) I= î = (u O I vO I w O ) I=íÜÉå==
r r
ì ⋅ î = uNu O + vNvO + wNwO K=
=
575. ^åÖäÉ=_ÉíïÉÉå=qïç=sÉÅíçêë==
r
r
fÑ= ì = (uN I vN I wN ) I= î = (u O I vO I w O ) I=íÜÉå==
uNu O + vNvO + wNw O
K=
Åçë θ =
O
O
O
O
O
O
uN + vN + wN u O + vO + w O
=
576. `çããìí~íáîÉ=mêçéÉêíó=
r r r r
ì⋅î = î ⋅ì=
=
577. ^ëëçÅá~íáîÉ=mêçéÉêíó=
r
r
r r
(λì ) ⋅ (µî ) = λµì ⋅ î =
=
578. aáëíêáÄìíáîÉ=mêçéÉêíó=
r r r r r r r
ì ⋅ (î + ï ) = ì ⋅ î + ì ⋅ ï =
=
π
r r
r r
579. ì ⋅ î = M =áÑ= ì I î =~êÉ=çêíÜçÖçå~ä=E θ = FK=
O
=
π
r r
580. ì ⋅ î > M =áÑ= M < θ < K=
O
=
124
135. CHAPTER 6. VECTORS
π
r r
581. ì ⋅ î < M =áÑ= < θ < π K=
O
=
r r r r
582. ì ⋅ î ≤ ì ⋅ î =
=
r r r r
r r
583. ì ⋅ î = ì ⋅ î =áÑ= ì I î =~êÉ=é~ê~ääÉä=E θ = M FK=
=
r
584. fÑ= ì = (uN I vN I wN ) I=íÜÉå==
r r r
rO
O
O
O
ì ⋅ ì = ì O = ì = uN + vN + wN K=
=
r r r r r r
585. á ⋅ á = à ⋅ à = â ⋅ â = N =
=
r r r r r r
586. á ⋅ à = à ⋅ â = â ⋅ á = M =
=
=
=
6.6 Vector Product
=
r
r
587. sÉÅíçê=mêçÇìÅí=çÑ=sÉÅíçêë= ì =~åÇ î =
r r r
ì × î = ï I=ïÜÉêÉ==
π
r r r
•
ï = ì ⋅ î ⋅ ëáå θ I=ïÜÉêÉ= M ≤ θ ≤ X=
O
r r
r r
•
ï ⊥ì=
~åÇ= ï ⊥ î X=
r r r
• =sÉÅíçêë= ì I= î I= ï =Ñçêã=~=êáÖÜí-Ü~åÇÉÇ=ëÅêÉïK=
=
125
136. CHAPTER 6. VECTORS
=
=======
=
Figure 83.
=
r
á
r r r
588. ï = ì × î = u N
uO
r
à
vN
vO
r
â
wN =
wO
=
uN wN uN vN
r r r v wN
=
589. ï = ì × î = N
I−
I
v w
u O w O u O vO
O
O
=
r r r r
590. p = ì × î = ì ⋅ î ⋅ ëáå θ =EcáÖKUPF=
=
591. ^åÖäÉ=_ÉíïÉÉå=qïç=sÉÅíçêë=EcáÖKUPF=
r r
ì× î
ëáå θ = r r =
ì⋅î
=
592. kçåÅçããìí~íáîÉ=mêçéÉêíó=
r r
r r
ì × î = −(î × ì ) ==
=
593. ^ëëçÅá~íáîÉ=mêçéÉêíó=
r
r
r r
(λì )× (µî ) = λµì × î =
=
=
126
137. CHAPTER 6. VECTORS
594. aáëíêáÄìíáîÉ=mêçéÉêíó=
r r r r r r r
ì × (î + ï ) = ì × î + ì × ï =
=
r r r r
r
595. ì × î = M =áÑ= ì =~åÇ= î =~êÉ=é~ê~ääÉä=E θ = M FK=
=
r r r r r r r
596. á × á = à × à = â × â = M =
=
r r r r r r r r r
597. á × à = â I= à × â = á I= â × á = à =
=
=
=
6.7 Triple Product
598.
599.
600.
601.
=
pÅ~ä~ê=qêáéäÉ=mêçÇìÅí=
rr r r r r r r r r r r
[ìîï ] = ì ⋅ (î × ï ) = î ⋅ (ï × ì ) = ï ⋅ (ì × î ) =
=
rr r
r rr
rr r
rr r
r rr
rrr
[ìîï ] = [ïìî ] = [îïì] = −[îìï ] = −[ïîì] = −[ìïî ] =
=
r r r
rr r
âì ⋅ (î × ï ) = â[ìîï ] =
=
pÅ~ä~ê=qêáéäÉ=mêçÇìÅí=áå=`ççêÇáå~íÉ=cçêã=
uN vN wN
r r r
ì ⋅ (î × ï ) = u O vO w O I==
uP vP wP
ïÜÉêÉ==
r
r
r
ì = (uN I vN I wN ) I= î = (u O I vO I w O ) I= ï = (uP I vP I wP ) K==
=
602. sçäìãÉ=çÑ=m~ê~ääÉäÉéáéÉÇ=
r r r
s = ì ⋅ (î × ï ) =
=
127
138. CHAPTER 6. VECTORS
=
============
=
Figure 84.
=
603. sçäìãÉ=çÑ=móê~ãáÇ=
Nr r r
s = ì ⋅ (î × ï ) =
S
=
=
=
Figure 85.
=
r r r
r r
r
604. fÑ== ì ⋅ (î × ï ) = M I=íÜÉå=íÜÉ=îÉÅíçêë== ì I= î I=~åÇ= ï =~êÉ=äáåÉ~êäó=
r
r
r
ÇÉéÉåÇÉåí=I=ëç= ï = λì + µî =Ñçê=ëçãÉ=ëÅ~ä~êë= λ =~åÇ= µ K==
=
r r r
r r
r
605. fÑ== ì ⋅ (î × ï ) ≠ M I=íÜÉå=íÜÉ=îÉÅíçêë== ì I= î I=~åÇ= ï =~êÉ=äáåÉ~êäó=
áåÇÉéÉåÇÉåíK=
=
128
139. CHAPTER 6. VECTORS
606. sÉÅíçê=qêáéäÉ=mêçÇìÅí=
r r r
r r r r r r
ì × (î × ï ) = (ì ⋅ ï )î − (ì ⋅ î )ï ==
=
=
=
=
=
=
=
=
129
140. Chapter 7
Analytic Geometry
=
=
=
=
7.1 One-Dimensional Coordinate System
=
mçáåí=ÅççêÇáå~íÉëW= ñ M I= ñ N I= ñ O I= ó M I= ó N I= ó O =
oÉ~ä=åìãÄÉêW= λ ==
aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç=
=
=
607. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë=
Ç = ^_ = ñ O − ñ N = ñ N − ñ O =
=
=
=
Figure 86.
=
608. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç= λ =
ñ + λñ O
^`
I= λ =
ñM = N
I= λ ≠ −N K=
N+ λ
`_
=
=
========
Figure 87.
130
=
141. CHAPTER 7. ANALYTIC GEOMETRY
609. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí=
ñ + ñO
ñM = N
I= λ = N K=
O
=
=
=
7.2 Two-Dimensional Coordinate System
=
mçáåí=ÅççêÇáå~íÉëW= ñ M I= ñ N I= ñ O I= ó M I= ó N I= ó O =
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 = ^_ =
=
(ñ O − ñ N )O + (ó O − óN )O =
=
=
Figure 88.
131
142. CHAPTER 7. ANALYTIC GEOMETRY
611. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç= λ =
ñ + λñ O
ó + λó O
ñM = N
I= ó M = N
I==
N+ λ
N+ λ
^`
λ=
I= λ ≠ −N K=
`_
=
=======
=
=
Figure 89.
=
=
132
143. CHAPTER 7. ANALYTIC GEOMETRY
=======
=
=
Figure 90.
=
612. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí=
ñ + ñO
ó + óO
I= ó M = N
I= λ = N K=
ñM = N
O
O
=
613. `ÉåíêçáÇ=EfåíÉêëÉÅíáçå=çÑ=jÉÇá~åëF=çÑ=~=qêá~åÖäÉ=
ñ + ñ O + ñP
ó + óO + óP
I= ó M = N
ñM = N
I==
P
P
ïÜÉêÉ== ^(ñ N I ó N ) I== _(ñ O I ó O ) I==~åÇ== `(ñ P I ó P ) ==~êÉ=îÉêíáÅÉë=çÑ=
íÜÉ=íêá~åÖäÉ= ^_` K=
=
=
133
144. CHAPTER 7. ANALYTIC GEOMETRY
=========
=
=
Figure 91.
=
614. fåÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^åÖäÉ=_áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ=
~ñ + Äñ O + Åñ P
~ó + Äó O + Åó P
I= ó M = N
ñM = N
I==
~ +Ä+Å
~ +Ä+Å
ïÜÉêÉ= ~ = _` I= Ä = `^ I= Å = ^_ K==
=
========
=
=
Figure 92.
134
145. CHAPTER 7. ANALYTIC GEOMETRY
615. `áêÅìãÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=íÜÉ=páÇÉ=mÉêéÉåÇáÅìä~ê======================
_áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ=
O
O
O
O
ñN + óN óN N
ñN ñN + óN N
ñO + óO óO N
ñO ñO + óO N
O
O
O
O
O
O
O
O
ñP + óP óP N
ñP ñP + óP N
ñM =
I= ó M =
=
ñN óN N
ñN óN N
O ñO
ñP
óO N
óP N
O ñO
ñP
óO N
óP N
=
=
========
==
Figure 93.
=
=
=
=
=
=
=
135
146. CHAPTER 7. ANALYTIC GEOMETRY
616. lêíÜçÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^äíáíìÇÉëF=çÑ=~=qêá~åÖäÉ=
O
O
óN ñ O ñ P + óN N
ñN + ó OóP ñN N
ó O ñPñN + ó O N
ñ O + ó P óN ñ O N
O
O
O
O
ó P ñ Nñ O + ó P N
ñ P + ó Nó O ñ P N
I= ó M =
=
ñM =
ñN óN N
ñN óN N
ñO óO N
ñO óO N
ñP óP N
ñP óP N
=
=
======
=
Figure 94.
=
617. ^êÉ~=çÑ=~=qêá~åÖäÉ=
ñ N óN N
N
N ñ O − ñN
p = (± ) ñ O ó O N = (± )
O
O ñ P − ñN
ñP óP N
=
=
=
136
ó O − óN
ó P − óN
=
147. CHAPTER 7. ANALYTIC GEOMETRY
618. ^êÉ~=çÑ=~=nì~Çêáä~íÉê~ä=
N
p = (± ) [(ñ N − ñ O )(ó N + ó O ) + (ñ O − ñ P )(ó O + ó P ) + =
O
+ (ñ P − ñ Q )(ó P + ó Q ) + (ñ Q − ñ N )(ó Q + ó N )] =
=
===
=
=
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(ϕ O − ϕN ) =
=
137
148. 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=
ó
ê = ñ O + ó O I= í~å ϕ = K=
ñ
138
149. CHAPTER 7. ANALYTIC GEOMETRY
7.3 Straight Line in Plane
=
mçáåí=ÅççêÇáå~íÉëW=uI=vI=ñI= ñ M I= ñ N I== ó M I= ó N I= ~N I= ~ O I=£==
oÉ~ä=åìãÄÉêëW=âI=~I=ÄI=éI=íI=^I=_I=`I= ^N I= ^ O I=£=
^åÖäÉëW= α I= β =
^åÖäÉ=ÄÉíïÉÉå=íïç=äáåÉëW= ϕ =
r
kçêã~ä=îÉÅíçêW= å =
r r r
mçëáíáçå=îÉÅíçêëW= ê I= ~ I= Ä =
=
=
622. dÉåÉê~ä=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ=
^ñ + _ó + ` = M =
=
623. kçêã~ä=sÉÅíçê=íç=~=píê~áÖÜí=iáåÉ=
r
qÜÉ=îÉÅíçê= å(^I _ ) =áë=åçêã~ä=íç=íÜÉ=äáåÉ= ^ñ + _ó + ` = M K=
=
=
=
Figure 98.
=
624. bñéäáÅáí=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ=EpäçéÉ-fåíÉêÅÉéí=cçêãF=
ó = âñ + Ä K==
139
150. CHAPTER 7. ANALYTIC GEOMETRY
qÜÉ=Öê~ÇáÉåí=çÑ=íÜÉ=äáåÉ=áë= â = í~å α K=
=
=
=
Figure 99.
=
625. dê~ÇáÉåí=çÑ=~=iáåÉ==
ó − óN
â = í~å α = O
=
ñ O − ñN
=
=
=
Figure 100.
140
151. CHAPTER 7. ANALYTIC GEOMETRY
626. bèì~íáçå=çÑ=~=iáåÉ=dáîÉå=~=mçáåí=~åÇ=íÜÉ=dê~ÇáÉåí=
ó = ó M + â (ñ − ñ M ) I==
ïÜÉêÉ=â=áë=íÜÉ=Öê~ÇáÉåíI= m(ñ M I ó M ) =áë=~=éçáåí=çå=íÜÉ=äáåÉK=
=
=
=
Figure 101.
=
627. bèì~íáçå=çÑ=~=iáåÉ=qÜ~í=m~ëëÉë=qÜêçìÖÜ=qïç=mçáåíë=
ó − óN
ñ − ñN
=
==
ó O − óN ñ O − ñN
çê=
ñ ó N
ñ N ó N N = M K=
ñO óO N
=
141
153. CHAPTER 7. ANALYTIC GEOMETRY
629. kçêã~ä=cçêã=
ñ Åçë β + ó ëáå β − é = M =
=
=
=
Figure 104.
=
630. mçáåí=aáêÉÅíáçå=cçêã=
ñ − ñ N ó − óN
=
I==
u
v
ïÜÉêÉ= (uI v ) = áë= íÜÉ= ÇáêÉÅíáçå= çÑ= íÜÉ= äáåÉ= ~åÇ= mN (ñ N I ó N ) = äáÉë=
çå=íÜÉ=äáåÉK=
=
143
154. CHAPTER 7. ANALYTIC GEOMETRY
=
=
Figure 105.
=
631. sÉêíáÅ~ä=iáåÉ=
ñ =~=
=
632. eçêáòçåí~ä=iáåÉ=
ó=Ä=
=
633. sÉÅíçê=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ=
r r r
ê = ~ + íÄ I==
ïÜÉêÉ==
l=áë=íÜÉ=çêáÖáå=çÑ=íÜÉ=ÅççêÇáå~íÉëI=
u=áë=~åó=î~êá~ÄäÉ=éçáåí=çå=íÜÉ=äáåÉI==
r
~ =áë=íÜÉ=éçëáíáçå=îÉÅíçê=çÑ=~=âåçïå=éçáåí=^=çå=íÜÉ=äáåÉ=I=
r
Ä =áë=~=âåçïå=îÉÅíçê=çÑ=ÇáêÉÅíáçåI=é~ê~ääÉä=íç=íÜÉ=äáåÉI==
í=áë=~=é~ê~ãÉíÉêI==
r →
ê = lu =áë=íÜÉ=éçëáíáçå=îÉÅíçê=çÑ=~åó=éçáåí=u=çå=íÜÉ=äáåÉK==
=
144
155. CHAPTER 7. ANALYTIC GEOMETRY
=
=
Figure 106.
=
634. píê~áÖÜí=iáåÉ=áå=m~ê~ãÉíêáÅ=cçêã=
ñ = ~N + íÄN
I==
ó = ~ O + íÄO
ïÜÉêÉ==
(ñ I ó ) ~êÉ=íÜÉ=ÅççêÇáå~íÉë=çÑ=~åó=ìåâåçïå=éçáåí=çå=íÜÉ=äáåÉI==
(~N I ~ O ) =~êÉ=íÜÉ=ÅççêÇáå~íÉë=çÑ=~=âåçïå=éçáåí=çå=íÜÉ=äáåÉI==
(ÄN I ÄO ) =~êÉ=íÜÉ=ÅççêÇáå~íÉë=çÑ=~=îÉÅíçê=é~ê~ääÉä=íç=íÜÉ=äáåÉI==
í=áë=~=é~ê~ãÉíÉêK=
=
145
157. CHAPTER 7. ANALYTIC GEOMETRY
636. m~ê~ääÉä=iáåÉë=
qïç=äáåÉë= ó = â Nñ + ÄN =~åÇ= ó = â O ñ + ÄO =~êÉ=é~ê~ääÉä=áÑ==
â N = â O K=
qïç= äáåÉë= ^Nñ + _Nó + `N = M = ~åÇ= ^ O ñ + _O ó + ` O = M = ~êÉ=
é~ê~ääÉä=áÑ=
^N _N
=
K=
^ O _O
=
=
=
Figure 109.
=
637. mÉêéÉåÇáÅìä~ê=iáåÉë=
qïç=äáåÉë= ó = â Nñ + ÄN =~åÇ= ó = â O ñ + ÄO =~êÉ=éÉêéÉåÇáÅìä~ê=áÑ==
N
â O = − =çêI=Éèìáî~äÉåíäóI= â Nâ O = −N K=
âN
qïç= äáåÉë= ^Nñ + _Nó + `N = M = ~åÇ= ^ O ñ + _ O ó + ` O = M = ~êÉ=
éÉêéÉåÇáÅìä~ê=áÑ=
^N^ O + _N_ O = M K=
=
147
158. CHAPTER 7. ANALYTIC GEOMETRY
=
=
Figure 110.
=
638. ^åÖäÉ=_ÉíïÉÉå=qïç=iáåÉë=
â − âN
í~å ϕ = O
I==
N + â Nâ O
^N^ O + _N_ O
Åçë ϕ =
K=
O
O
^N + _N ⋅ ^ O + _ O
O
O
=
148
159. CHAPTER 7. ANALYTIC GEOMETRY
=
=
Figure 111.
=
639. fåíÉêëÉÅíáçå=çÑ=qïç=iáåÉë=
fÑ=íïç=äáåÉë= ^Nñ + _Nó + `N = M =~åÇ= ^ O ñ + _ O ó + ` O = M =áåíÉêëÉÅíI=íÜÉ=áåíÉêëÉÅíáçå=éçáåí=Ü~ë=ÅççêÇáå~íÉë=
− `N_ O + ` O_N
− ^N` O + ^ O`N
ñM =
I= ó M =
K=
^N_ O − ^ O_N
^N_ O − ^ O_N
=
=
=
7.4 Circle
=
o~ÇáìëW=o=
`ÉåíÉê=çÑ=ÅáêÅäÉW= (~ I Ä) =
mçáåí=ÅççêÇáå~íÉëW=ñI=óI= ñ N I= ó N I=£=
oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=í=
149
160. CHAPTER 7. ANALYTIC GEOMETRY
640. bèì~íáçå=çÑ=~=`áêÅäÉ=`ÉåíÉêÉÇ=~í=íÜÉ=lêáÖáå=Epí~åÇ~êÇ=
cçêãF=
ñ O + ó O = oO =
======
=
=
Figure 112.
=
641. bèì~íáçå=çÑ=~=`áêÅäÉ=`ÉåíÉêÉÇ=~í=^åó=mçáåí= (~I Ä)
(ñ − ~ )O + (ó − Ä)O = o O
Figure 113.
150
161. CHAPTER 7. ANALYTIC GEOMETRY
642. qÜêÉÉ=mçáåí=cçêã
ñO + óO ñ ó N
O
O
ñN + óN ñN óN N
=M
ñO + óO ñO óO N
O
O
O
O
ñP + óP ñP óP N
=
=
=
Figure 114.
=
643. m~ê~ãÉíêáÅ=cçêã
ñ = o Åçë í
I= M ≤ í ≤ Oπ K
ó = o ëáå í
=
644. dÉåÉê~ä=cçêã
^ñ O + ^ó O + añ + bó + c = M =E^=åçåòÉêçI= aO + b O > Q ^c FK==
qÜÉ=ÅÉåíÉê=çÑ=íÜÉ=ÅáêÅäÉ=Ü~ë=ÅççêÇáå~íÉë= (~ I Ä) I=ïÜÉêÉ==
a
b
~=−
I= Ä = −
K=
O^
O^
qÜÉ=ê~Çáìë=çÑ=íÜÉ=ÅáêÅäÉ=áë
151
162. CHAPTER 7. ANALYTIC GEOMETRY
o=
aO + b O − Q ^c
K
O^
=
=
=
7.5 Ellipse
=
pÉãáã~àçê=~ñáëW=~=
pÉãáãáåçê=~ñáëW=Ä=
cçÅáW= cN (− ÅI M) I= cO (ÅI M) =
aáëí~åÅÉ=ÄÉíïÉÉå=íÜÉ=ÑçÅáW=OÅ= =
bÅÅÉåíêáÅáíóW=É==
oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=í=
mÉêáãÉíÉêW=i=
^êÉ~W=p=
=
=
645. bèì~íáçå=çÑ=~å=bääáéëÉ=Epí~åÇ~êÇ=cçêãF
ñO óO
+ =N
~ O ÄO
=
=
Figure 115.
152
163. CHAPTER 7. ANALYTIC GEOMETRY
646. êN + êO = O~ I=
ïÜÉêÉ== êN I== êO ==~êÉ==Çáëí~åÅÉë==Ñêçã==~åó==éçáåí== m(ñ I ó ) ==çå=
íÜÉ=ÉääáéëÉ=íç=íÜÉ=íïç=ÑçÅáK=
=
=
=
Figure 116.
=
647. ~ O = ÄO + Å O
=
648. bÅÅÉåíêáÅáíó
Å
É = <N=
~
=
649. bèì~íáçåë=çÑ=aáêÉÅíêáÅÉë
~
~O
ñ=± =± =
É
Å
=
650. m~ê~ãÉíêáÅ=cçêã
ñ = ~ Åçë í
I= M ≤ í ≤ Oπ K
ó = Ä ëáå í
=
=
153
164. CHAPTER 7. ANALYTIC GEOMETRY
651. dÉåÉê~ä=cçêã
^ñ O + _ñó + `ó O + añ + bó + c = M I==
ïÜÉêÉ= _ O − Q ^` < M K=
=
652. dÉåÉê~ä=cçêã=ïáíÜ=^ñÉë=m~ê~ääÉä=íç=íÜÉ=`ççêÇáå~íÉ=^ñÉë
^ñ O + `ó O + añ + bó + c = M I==
ïÜÉêÉ= ^` > M K
=
653. `áêÅìãÑÉêÉåÅÉ
i = Q~b(É ) I==
ïÜÉêÉ==íÜÉ==ÑìåÅíáçå=b==áë==íÜÉ=ÅçãéäÉíÉ==ÉääáéíáÅ=áåíÉÖê~ä==çÑ=
íÜÉ=ëÉÅçåÇ=âáåÇK==
=
654. ^ééêçñáã~íÉ=cçêãìä~ë=çÑ=íÜÉ=`áêÅìãÑÉêÉåÅÉ
i = π NKR(~ + Ä) − ~Ä I==
(
i = π O(~ O + ÄO ) K=
=
655. p = π~Ä =
=
=
=
)
7.6 Hyperbola
=
qê~åëîÉêëÉ=~ñáëW=~=
`çåàìÖ~íÉ=~ñáëW=Ä=
cçÅáW= cN (− ÅI M) I= cO (ÅI M) =
aáëí~åÅÉ=ÄÉíïÉÉå=íÜÉ=ÑçÅáW=OÅ= =
bÅÅÉåíêáÅáíóW=É==
^ëóãéíçíÉëW=ëI=í=
oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=íI=â=
=
=
=
154
165. CHAPTER 7. ANALYTIC GEOMETRY
656. bèì~íáçå=çÑ=~=eóéÉêÄçä~=Epí~åÇ~êÇ=cçêãF=
ñO óO
− = N=
~ O ÄO
=
=
=
Figure 117.
=
657.
êN − êO = O~ I=
ïÜÉêÉ== êN I== êO ==~êÉ==Çáëí~åÅÉë==Ñêçã==~åó=éçáåí== m(ñ I ó ) ==çå=
íÜÉ=ÜóéÉêÄçä~=íç=íÜÉ=íïç=ÑçÅáK=
=
155
166. CHAPTER 7. ANALYTIC GEOMETRY
=
=
Figure 118.
658.
659.
660.
661.
=
bèì~íáçåë=çÑ=^ëóãéíçíÉë=
Ä
ó=± ñ=
~
=
Å O = ~ O + ÄO =
=
bÅÅÉåíêáÅáíó
Å
É = > N=
~
=
bèì~íáçåë=çÑ=aáêÉÅíêáÅÉë
~
~O
ñ=± =± =
É
Å
=
=
=
156
167. CHAPTER 7. ANALYTIC GEOMETRY
662. m~ê~ãÉíêáÅ=bèì~íáçåë=çÑ=íÜÉ=oáÖÜí=_ê~åÅÜ=çÑ=~=eóéÉêÄçä~=
ñ = ~ ÅçëÜ í
I= M ≤ í ≤ Oπ K
ó = Ä ëáåÜ í
=
663. dÉåÉê~ä=cçêã
^ñ O + _ñó + `ó O + añ + bó + c = M I==
ïÜÉêÉ= _ O − Q ^` > M K=
=
664. dÉåÉê~ä=cçêã=ïáíÜ=^ñÉë=m~ê~ääÉä=íç=íÜÉ=`ççêÇáå~íÉ=^ñÉë
^ñ O + `ó O + añ + bó + c = M I==
ïÜÉêÉ= ^` < M K=
665. ^ëóãéíçíáÅ=cçêã=
ÉO
ñó = I==
Q
çê==
ÉO
â
ó = I=ïÜÉêÉ= â = K=
ñ
Q
få= íÜáë= Å~ëÉ= I= íÜÉ= ~ëóãéíçíÉë= Ü~îÉ= Éèì~íáçåë= ñ = M = ~åÇ=
ó = M K==
=
157
168. CHAPTER 7. ANALYTIC GEOMETRY
=
=
Figure 119.
=
=
=
7.7 Parabola
=
cçÅ~ä=é~ê~ãÉíÉêW=é=
cçÅìëW=c=
sÉêíÉñW= j(ñ M I ó M ) =
oÉ~ä=åìãÄÉêëW=^I=_I=`I=aI=bI=cI=éI=~I=ÄI=Å=
=
=
666. bèì~íáçå=çÑ=~=m~ê~Äçä~=Epí~åÇ~êÇ=cçêãF
ó O = Oéñ
=
158
169. CHAPTER 7. ANALYTIC GEOMETRY
=
=
Figure 120.
=
bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ
é
ñ = − I=
O
`ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë=
é
c I M I=
O
`ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ=
j(MI M) K=
=
667. dÉåÉê~ä=cçêã
^ñ O + _ñó + `ó O + añ + bó + c = M I==
ïÜÉêÉ= _ O − Q ^` = M K=
=
N
668. ó = ~ñ O I= é = K=
O~
bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ
159
170. CHAPTER 7. ANALYTIC GEOMETRY
é
ó = − I=
O
`ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë=
é
c MI I=
O
`ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ=
j(MI M) K=
=
=
=
Figure 121.
=
669. dÉåÉê~ä=cçêãI=^ñáë=m~ê~ääÉä=íç=íÜÉ=ó-~ñáë==
^ñ O + añ + bó + c = M =E^I=b=åçåòÉêçFI==
N
ó = ~ñ O + Äñ + Å I= é = K==
O~
bèì~íáçå=çÑ=íÜÉ=ÇáêÉÅíêáñ
é
ó = ó M − I=
O
`ççêÇáå~íÉë=çÑ=íÜÉ=ÑçÅìë=
160
171. CHAPTER 7. ANALYTIC GEOMETRY
é
c ñ M I ó M + I=
O
`ççêÇáå~íÉë=çÑ=íÜÉ=îÉêíÉñ=
Ä
Q~Å − ÄO
K=
ñ M = − I= ó M = ~ñ O + Äñ M + Å =
M
O~
Q~
=
=
=
Figure 122.
=
=
=
7.8 Three-Dimensional Coordinate System
=
mçáåí=ÅççêÇáå~íÉëW= ñ M I= ó M I= ò M I= ñ N I= ó N I= ò N I=£=
oÉ~ä=åìãÄÉêW= λ ==
aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç=
^êÉ~W=p=
sçäìãÉW=s=
=
161
172. CHAPTER 7. ANALYTIC GEOMETRY
670. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë=
Ç = ^_ =
=
=
(ñ O − ñ N )O + (ó O − óN )O + (ò O − òN )O =
=
===
Figure 123.
=
671. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç= λ =
ñ + λñ O
ó + λó O
ò + λò O
ñM = N
I= ó M = N
I= ò M = N
I==
N+ λ
N+ λ
N+ λ
ïÜÉêÉ=
^`
λ=
I= λ ≠ −N K=
`_
=
162