17

9,S, ~ ~ •c14 ~ ~ (Sonui rocts abou t voc«>rs) I

(l) ..O~ c.,ah (Colllne1tr vector■):
,..,-.,r«a
'JflITTlll1I\ cwtl~ ~, C'NtJI~~~ ..Q1111fit• C:,~Jl4Clb CJSt, lit'! I
~.S '1t. Sr.I •• b,

'ltATJ. ~
AC . BD
~ 11Tf.si4"1i~hl ('8tl~
c, ~ \ll•Cll~II ~ I
cl - ~ ..a«11~ C18tJf 1$;1 ~
Ill~ ~91~~~ 9ITt'ff I a Jl~
j1
bfc /
ct:¥'"'~~ - ~ b ~ b ..a~ O r.'il!NU ~"lh~ I i
~ .,tC \It~ ah Clli'1NCIBI ~ J1lll ~ A1: Jl't CB
('lt,iitlltl 15i~'rf ~'fh 1
~ .lJ..it \ll<K'llb <:altml ~..a.~.~ 't1l1I ~ '1111.\Stil (like vectors).,,
~ IA .sm.1 \tt'lllt'lt'll'U'Y, n
~1'1IC•HI 'llfol~ ~ ~~ k = ➔
~i ~k~Jld~mfi'tllnl'llf•P lill.'oMII

I.Al
(li) ....._..,. . ~ ~ ~ (Null a nd unit vectors) :
~ I al"ll~C'1ll'511Nfil'tA Jl~~~B111if f~t"Pl111\?Cll m~~ ~ ~~'1311
~~ ~Ol'I ,..,
~ '1111•11ilf.\S!l '11$'1 f.l1"l ~'f-u ~ lAlll !{It~ lf1'6111 ~
~ c."5til'l,t9111"1Jflt'1!'~~,w 15i"'1'14>!1 ~ 1'c14ttf4>!1 C'&l'!lt♦ c<l-~ C'oltllll 'lcwt

~-_,11..,.-v... mlfllt ~~115i♦jq<f>!I

t ~- ➔
CIS°btcp O ("fO)ffl'! fflS!"! ~WI ~ ~ •
-15jcjij4qljl
cdl~1 'Gli01i0 'lqii'j ~~~(proper vectors) I
....._..... c:4tH ~ (Properties of null vector) :

(i) ct+ 0 = (t ; 15i-ft~~~ ft--mi-~115i<l>t(<1•1nfflClll11m:t'I,~ (tc.'&tlf

~ I

<ii>-: - er=-::~~-: ~151•1(411 ~ m 1 1 ~ - : "5k~ ,

.. ""0
(iii) \ - .=-0. ,, h -'="
; 'dl♦lq<lli:l(.IS~t♦ c~C'il'!T'l'l! 11-i("f{J~)ffll"l.'1~~'cWl<i~
~

Ce"tcff ~~ I
(1v)
. 0 (.... 0 ; ~ "5' ilt41 "fU ffl'1 1111 ~ "l.'W"l lb/<l5lq<IS'
o ) = -+ ., R C'5' Cll!I 'lliR "U I
~~, e;,al11c♦ ~1:11 ~ ~ 1~ "5tc-11-11 fltillfii-t 6 ~ f.lt4~~s w;u I C11 •
!ff ffl'I A c\S'ttl¥!115ifis1J.C"' ~ c.'Slllt• ~~w cJl1t a Clnl'l, t = a•A
y (lU) ~f.111-1111 .-.. ~ ~ ~ ~ (Ortho1onal unlt
vector or base vector):
lffl, f.at~!l ~-~ f.i~l!lC.H ~ 0 I X, Y ~ Z-""
J
cqfis13.CII( ~ ~ 11llfBlt'il i,j ~~ ii I 2.4 ~~ ~] I
.iil JI~ c.'Sh,ffetc• ~ -u~c,11.i,11
C\Sh~ ~ ~ 1 \lj~

(11- ~ ~ ~ ~ ~ ~ ~ -ei-. ~ ~ Vo'ttff

~(multiple)l ~~~~C'ollf A.Jffl~«iKI

S.201- 2
I II

A:
A
. A• ' '
.
AC I ♦
A', •
"',
.t, I ~
.
J t A, JI
-;(, • Ar h r J.' <1)1, A,.. A, tP:• A, r.s,r-~ ~ I

A. C'tltq 1n TOT ia iRl'lic>i I .4• 1 m I A' I •

9
JA~ +A! • A:
~ <""11 c- -irr C11R cr1 "'""~- ~ l atm~r.JC" a .
C"'O
c.,,'ltl,
~'1. ffe, • .fCl!lf tCf 'f:Cfi1,_. .trPf atf I
~JR • .t =ai +zi -2•
~ . I A' I • J,3' ~ -♦<-'Ir •./17 .

(Iv) <l4tli teftw1 (!qu.allty of veccon) :

'!f'C'6'1J "ii~-;: •~>fCPpic,~'""~~-o~~~ ,

➔ .... ,d--""1'1Rf
2.5~~ AB~CDMUC:"fl 1m~,6.x
....... 0M~
-, ➔
ON ca.,a.,...1'1"
[WI
....

B D 'iftll>jl!?J,l.~)~~'tlfis'f"'ftlf 1w.if.f ~ Ok•

"' • M L n ~ , ➔--
b co't?-HPRA WM~ lift;.~
... -,
a IA\ 1

,___ _ _tAu
c o
-, = KL
_ _ _ __, OM
1/
K
-+'11,......,c.
zc.111~ I
~~~~~u.-wi~ii!R 1C:1. ,,, .
.at'!fl;l r. 6'741~ ~ ~ . " t C ~ ~l~2,5~
OIT'CJ ~. AB
➔ -+
= co ;
➔ ➔
oN .. KL ; , ON ,
....
=, a1.
•

(v) ~ C'$ (Oppo•ite vecton) :

tit C'ofc-ll4 ~ 'fliR ~~ ~~ <t!I.~ 11111 G ~~ Cffl I
-, ➔ c;;..._-+ -,
a col tU~b Cet?ltJO(.,.~ U- a , 2.6-.it- LM ,lg\ NK r.'ick•UJ
..... -+ .-::1, ➔ .... ➔
~ C$ I 1f1t LM C4-!:t• a~ NK = - a ; ML = - a ; um
.... -+
KN =- a.
➔ ➔ ➔
'1'{111 ((lfn,I ClWJPl~ - (- a ) =....a <B<:\ I- a I =I a I
2.4. C-o1CUI 2'•fit<6if I
( i) C ~ ~ (Polar vector) : ~ ~ ~ fil<, ~ ~ c'e';hto ~ 9 ,
~ ~4 C ~ ~~c.q~'fl'J'l~ l f.ll"Zl,~ . ~'l.~ ~IX -~ ~ ~
~,p,:i,~. ~-4 ~ X-1'.H «1'81 @ 1~ ·y :a• <JsiWJI ~~~n :JRtillll,itl
~~ Y-~~~~ I
 19

(ii)~ ~ (Axial vrclor) :

~ ~~ .Q;T;J ~1t ~M 9fl<ill 1il'll' ~ -:.To{ <1 "?t"i;'i" ~ - a-.+,~ C'0"l7< '~
~ ' 1 . ~ l~nfibfl'fl~'I eslt>r., m C"ff'll1 Co~ .nJ • .tlClfi!I fiJl 1'!1?.1$1 ~ 1 ..er?":~ ~~
~ (axial vcct.orJ I
~ C4tml ~1111 'QiClllti•II ~1!1 71ri- I ~ (.'11if<f~ t;~~I 'iJ~ t"f'J ~,: ~ f;lm: m
"Im! I ~'11 C1!C~ffl ~ ~~~ ~ { J \l'ffr.'ITIJ "lfiM ~ .... ~I ii,
~ ~ Cll~ ~ ~ t ~ cllf<l<f. r.-:~-.r.1 ~'t ~~
'El~- Jltl ~ mf"1 :1tJ I
~ r-'ft~if ~11 C~ ,w, ~-411W,Ji f.lir.l (Lhumb ruleJ
!fC111~1 ~ ~ l~~~'f"ll"-~~ ~ >r-/l~ltdl ~
C3!C'lq:r,JJr,l}'l'.llltlJ!'l"fot~~. ~~-utr,p11'!ffel~~ t-011.!t-&
~ Clf'l111~ ~~m,rn~11m~.,,~~<fn 2.7) 1
.al<fimt'l ~ Cll~~ ro'h~1<11 1~ ~ ~ ~. 0
c<il'lf~ '?RI<!, (<$'If~ ~ ' I ~ ~ ~ I ~ C~ft?lsl
~~~.<ll~tt.1M:rc:iid~~ctret:'I<~~ !ni.1
~~) c l!fcll1t ~ ~ ~ 1R , ~'6!lr.,:i~ q.,(r,<t ZJR.®'l.t ~ ilftMFn ; , i ~ ~Ji::0
~~ ~ ~ ~ ~ Ce!cs1~1 af6,11A~~~ 1

2.5. C'4i'IFil !JRt C'51l ~ C1l1'51 (Addition of scalars and vectors} :

~1frlimfirc,,14 ~~Cll.~~11JP;.hl! hl ~ ,'31T?!l'~~W.W'.!lf'f>!.lt,~ ~
C111~cflm,1t1t01!1 >fllr-1 1c ; r . r - i , ~ ~ ~ l l ~ 5~. 1ozr:.~ 1s!!f.'~.~
c,nt l5s! = (5 + 10 + 15) = 30 Zi-1 I ~ m ~ . ~ "lff.J:'?Jq, ; ; i t ~ ~ ~ ~ V.1
1161111b'1 f.lc(;r~ ~~i'!I~ 5 'il~ l"l'i 71lj-s,e 1J' -i""'flJ~lif.i 3 r;<i;J ,,.,;; ;m:'l!, ~
~,tlt<BI ~ ClfC~~ (5 - 3) = 2 'ii~ I ~~1?:t-!. W'l c-0 .i.:ei.,1!,s: "1 ~~ ~ ~~ m'III
'<11111 1IR1 ~ I ~ '!t"it 1C>s1~ Clll'-1 ~ l!TV "II ~ C~ ~ 1:i~ ~ 'I q \ <l1l1 ; o1':R ,;cY.1
m~t
"41 lll?it.-,!I L<ll'SJ <JI ~ 7fl1 I
~ f f C<ll~ ~ ~~f.fil"ll ffi""1 ~ ~ 1~ -.;r,:i.;ji-~.:;~e
~f<1-c:q-5--i1 ~ m t ~ mfcrn Clll'I R•llll ~ -ii='<>'•"' ~ ,...c-t ~ : (i) rfflJ ':fl Oaw of
triangle), (ii) >ilSIIGfff ,pi (law of parallelogram)~ (iii) ~1:li (law of polygon) t
(i) ~ 'lil (Law of triangle) :
~ Gl'fCllll a;il-J.>IICI ~ ~ "1'{ i&-ff+.it® feii~ ~ Ccltc• 4\'r-1 E ffl ~
'ffll'I R"'lll&c;i ~1f-!hl ~ ~ <s ~ ~ c$ ~ c◄fl'l(ijii\ ~ i'l1iit c<FC1t◄i ~
cwllc:11Tt'1'1fii1'~~ 1
"IC'l~. A B - ~ (w.12.S) IA~~.re B •."l<l#i ~ert~1~J?.;~ ~ ,.~~ BC
c ~lf~H~C<l.~Qf<--:. B.!'i<:"~S:": C . .a~
,~1c11'!lC<11~f--~~m
A~ C-~~~ ABC li'~~~ I~ 1'.! 'l,'U
-,
-. -. _...__ ~ <=--- C
AB~ BC C-:ifi:hilr.¢ "11~ ~ ~ "'l ~,..,.,. 9".~ ~ A
ltO

11111ffl'lcN t
,., ,... fl't~ "'~ C.'fQII flfl.
t/ .. t/ "11~111 o' •
1•1
,,. I
. 1 //

If ,,_1"11) ,. ,.. JI (:A' ~11111 ~t, tl'I,

" (J

• t -, ' .,.
o, b ,l - o
'l'f~ra An , ,,,. • <:A u
t..rw I ~tif1 .,,. l'f( r.11,~'1 ft •ll!J ~-..<re•
1tfli 1/lt•I .n'1, ~,qlJC"I "1111~ ~

#lil1J"llrt ,Jlk H•k«n 10 Qr,h1r) ~ •rffl 'llt ~nat 'IJf~ 'l!l(ll "lJlll, ~r.4 .)r fit;lff) cntn-,,
1 t "I I
(II) ~fn ,,l
(I.aw or
puruHol oaitrum) 1
lj ..~ filtr<fl I 'lffit (1'fQl1 'llllflfjCfl r{f ,nslfiff ~ft \l1t-t
fife-• 'lf8 '"<fflt\1 "'- (I

,--~ - - - - -,- •om.1~41 ~.,-. fffl ~)llflt-w 41'1(

•

i
7I
'illt'il '41 fitr-• llfm C"fl~ 'lft 1'1f41f c,tlJ a.tllf "ti I

OA "11t (;;I "6li!U ..q~~ ~ 0 Ct.

~r.w ,1£1(.lfrt "'111 r-1-, 11QJ1a1C'Jl A ~.,t a,1fPIPlfff~
llQI 'fill,

,r ,. u i1Mt on ,~"1t.•rc-~ 'ffirf~ "'i tftlf f.mi OACB~

____n_,,;.;1;.;.;•11;..__ ___,J 'fil!Ctt Jt.A( <)(1 111i;;1-<s flfr.,r, tff11 c<itm.1' lt~fl'I ~ ,
- , ... ~ ♦ -➔
:, OA + 0 /J • OC '<M'II ,7 t- 7i • /l
~CffCflCl\ll't~lf. 11/ 1.. Ju~ ➔ b~+2ob cm•O" -'f'lt ~~~ too ♦ a ~
(Ill) '1tft11 ~ (Law 1>0lyicon) 1 or
'ftllf 'llfit• C'i111 c.tr.ttf C'ft'P,., 'li"f11 ~
"'';!fa
.,.-p ~'W1R'l 11Tlf 1~ , , ,nn
~ ?(CU '>flt fl:111 .D1P11"11• 1;,11arrt ~ Atf11 ~ ~l:"1. JI 'Pl~~ 1ITl ffl:ll'l'l •1i
1'\~ 'Jal -4\ ~ lfl •~ I

'11NIO'l-•11""11Cl
..c..-_ ...
a
... -, -t
I ' ti 2, 0 I I ~ U ,c "°' ~ -
f>T~ ~ f 1',Qfl l'R~o,'1 lllffl ~ l',,.. 2.lO(•)J~
rQ,...

,~~A,_O ,llfr• (i:1 2.HJ(b)J o;t 1 cs'h"'1•~ OA, =

... -+
t 1u 1A1 ~~t.\•
A 1"'~ cah .ri• 111~ A I1t2 • a2U 191.r@'IM ,\
Bo.1 ... ...
A2 ,~C-"11it~ma 3 ('¥'h~~ ~A3
I
• a➔3 " I ~ C l { a➔-- -+ I
4 <.11'1wtft A 3 A 4 ,:,th I
~ nt-1 'ff I o.CI~ ill(II calm "'1fit ~ 0 ,.q'lft

C"A~CRMllf~ A,.~'R t.alC'lfQ OA4

... <•>
C4ltff ;: J• <i2 ~ ~'1,fi111 Ctn•f'11'1 C'ITlVft1I L..-_ _ _ _ fn_i._1_0_ __ __,J

...,Ji -•
O ➔ .... ➔
...
,4, ■ a I + o 2 + a 3 +a• . '1'1'1Al'll81C'I
-t ... ... ...
...,"'''- • .. ' • " n = a I + a 2 + •••.•• + a 11 , ·"""-
0," ..,.11..,. ~
'?1:ICII 1'IJ9I ( ' l ' h ~ '"1(11( ffll !PJ1"1~18TJll~~ (polygon) ~<lltf I
• ••111•.cw~or.-.i~,
22 <1'1f'4•r,1

~ bq ~ ~ ~ C'8trnl ~~A l~~ c<Sl tllll c-1.ifii·~ A 2 ffi .a-41t ~

.... ~ 11
A 1A-i •t16"J213(n)] 1
1111
z.
~

0
(IA
.II
X

C1
i

.... ....
"J.'1'MC<I' XO ~ a .r1C9f ffl ~ 1
➔ ➔ 1 · -+ ➔
2.14 -r-m OB=
.... ....
OA.4'!:il~ OB= 4. 0A ; OC = 3 OA
2 0
➔ ➔
~ OE = 1.50A I D
fA2. 1◄
~ ~ -:.~ ~ ~:t~'Sl, ~qir(i ""c<S,,.h"' . C""<!' ~ ------J
~ llllC-tJl-<1<11 (reciprocal) ffll 11'1 <I'll! I ('ll'lr,i, "1 + x = l .i.
X
• C"lfh ~ ~ ( L a w s of vector algebra) :
1. "+ 1 = 6 + 71 .... e 1 1 1 ~ ~ ~ .
2. r/ + ( 6 + ? ) = (r/ + 6) + ? = (71 + ? ) + 6 -+ C(l1~111C<1lPIW11'11 f.wil I
3. m 71 = ctm -+ 1'1C.;if ~~ 1
4. m (n 71) = (mn) ct ➔ t'TC-HP!mt~ f.wq 1
5. (m + n) 71 = m 71 + n 71 ➔ <I~ (distributive) f.ml I
6· m( 7I + 6 ) = m ct + m 1 ➔ <1-t;i (distributive) f.ml 1
(Si>ftlmf '{Jli<I~ ~ ~ ~ qfffl 'lffil I <!l' ·1.~''-'" IW~"""'"'<l
la!~ ffT"1 ....-...,.~.,,,.,,
- ......_ ..., "'' "'" '
l.'el:!<
~ >il ffl
"!-ll4il'IC4i

r
 \

~~ "~•11t!l11 ~ ~ ~ ~ 1C't1Pl. 7J + 7J "' -t ~ .l'o!ll•n4t~n >1,-1r;,n ~

"4m 7/ =7 - 6 I
2.9. c<&IC1BI ~911(111 (Com ponen ts of a vector) l
(i) ll111r-.• c:itt"i lill<IN ~ ~ 7J ·~ 2.16 "~ $t;;ll ~ ClNtt<n ~ h'tcit~lll
At~'5Cl11 ~O~lilt~l'4"1(inltial point) tmr y
➔
OP~~~ 911fi't ff, 7J t1Jtc1111 ~ ~ P
(terminal point) ~ ~1-ilitjl _,.(11
(o.r, Oy- a) (o,. a,,. a,)
.p
~ i a.c, j a._,. ~ k a:. ~ -;/ (li!IICH cilflJ.tltil I
X·'lll"II', Y•'lll"II' ~ Z-'111"11'~ ~~("! Ci6f1I I
l
~~. i a.r• Ja1 ~ k a,.,~~ t1Jt11-,ffel11 i ,..X
;:;:.r+7.:+.\7 .,
~ Cl!i'IT -;/ qsrof'1-.
Z
··-·········;.;···· ..··········J
I a,
j a,
71 ~Rl!Am 1711 = Ja.!+a;+a; . fA t.16
~ 71 C'6hlt OX, OY~ oz~~Ct, p~( yc.1'1~ (n2.15), ~~

cos (X : %
- , cos p : "-
V
~ cos Y: '-z
a a a·
cos a , cos P<!!<fl. cos y~11r-!~ cl C'SltH ~ C♦i>dt,, (direction cosine} <11'11 U I
'1-Q.t-•~~ !l'lil1'1 ~ ~ cos2
.. a + cos2 p +
y = 1. cos 2
• ~-'1511 (Position vector) ;"Bet m, (Displacement vector):
C'IITCill~ (P)~llf (x,y, z)~. OW"{ ~ffllf,ft r -QJ<ll'll~~A"i_11~R-twh
(position vector)~ ~ -c.tl' (1'8dius vector).
7 = i .x + J.y + .\.z.......... (i)
y z

P(x,y.z)

,i.,::;=------x ~ - - ---Y

z
1n 2.10
➔ ➔ ➔ ....
P 1 ~P2 ~~~WIJ(.xpy 1,z1) ~ (x2, Y2, zz ) ~. OP1 = '1 ~ OP2 = '2

~ M"rQCilll ~ C15b (1A 2.16] t~ .

 14

..-.!111, n(7 - ;;) = m(;:;- 7)

_. ....
-..,.,,, ➔
r ~ +N'1 (..c
= -"------=- '11"1 m + n. ~ 0) ....•. (i"")
w
m+n .... -+
C11fiL4.B C3Nll'lf~~ u~ m = wa ~ ('Jlt«il 7 =
71 72
; •

S. l 0. 111' ~t11 iJI 'J."I""' (Product of two vectors) :

?it c:etcn ,rtilt'I~ (i)~~~~(ii).ait~ ~,9(t'l!9flfi1- .
•,ru ~~~'Gt~ (dot product)~ IM~ltt<!i ~ U ~ t"I ... ◄◄◄ iit
1f"1'11(cross product)~~ 'l'f-!~~ (.) ~Cek,rr-.c◄~IC\!5(x) ilf nu
~~,
~ ~ ~ 1J:'tlPI I
➔ ➔
A ~ B 1{ftC6,CiBlldl~al C◄ll't9~.~1{1t"6JcH ~,~=A.Bcoae;•
- _.,.
\i,yijl♦ tMICW 'li~C4ll c.l NI U A➔➔ ➔ ➔
•B -~'QA Gt B ) l(~ I

2.18 ~ fR lllli<!illl A.s = A.B. cos 8.

...
·- .,.-
..., >iQ.C.,ic~llTV
i. C11, A➔ ,B
➔ =➔
B . A➔ ~.._'1ict:!i ,ffef.fl
~ ""'"<llffl-. lffilQ.~ ~ 'J.'l11'1 ffl ~ 1~.nu ~
,<1...
~ Rf.ho, m (co mmutative law)~ 5'C1'1 I
~~ "fl Q.C\'5"~ 111ll' C1I 1{ft C:'6 ftll ~ "'1'1 8 ~ ,~'------X
➔.... • 0 A
coe 9 =A.B fQs.1a
AB
 25

A =0 ~ .1 = 0 ~ .8 =90° i 8 =90° '{(1'I ~ llTll C'ehtiG t'fll""'11.lBI ~ 1'l'"lt

\ti&i◄ "'Cll~'i"'1P'!~~I
(,.) ~ 8 = 0 ~t. C'6tiNG ~ . ~ ISTtlffl ~ 'l~ Cl6'tNCH 'ifi/1/lC-lf 11fil!Ti15
iJ."l+ciil "'11T-1-.til'I COB 0° =1 ~t. A.1 =A.B~. lJ.1t Cob vllt 4 lG!il >MO11l'1 ~l!Trnf
~ '1"""' c<eliNCH tffllif1t-lf , i ~ '>rllA ~ 1~"'In llfit = t B
~ ~~ cetNv
~U'm A.B = A.A = A2
('If) ~ Qc ,t.ilt. c'ilii4~.l!I~ ~ • • 1!-.r.; cos 8 =cos it =- I ;~ A.B =
- AB"!\ <af,11Jii4l1 C'¥tfttH ~ 'J.Cflfl'I~ "'lii"l"IIC-111 C-lt·1l.~'f'llt-l I
(11) ,.....if)tliiql\S ~ .ii 1ITTf ~. i , j ~ ic llfit ~ "iBl""'1fl♦ 1111leei~ ~ ~ ~
~ ~ ' m, i .i = j .i = ic.ic = 1 ~ . ~ ~ffl c.'61tli ◄(1..-rf 'JPltil l
(• ) ~ i, ] ~ i. 9fsl""'1Cilii'!f~i111\!lf9.(c1i) <c11,ffe~~~llnr l_j : ] j = k.i
= o~. C1' ~ttt~ C1lc11 c,pn 'J."1P'I -t-111
➔ • ➔
Examples : (1) Q,-4QflJ11TC-l'I-.U cdl A= ai - 2) + ic ~\ c.h B = 2a1
+ a) - 4k 9f~"""'tli~~? [C.U.2001)
➔ ➔ ➔➔
Ii A~ B ~C16'1119'1'""1lilii'l'N~~~"l.~"fU'U I :. A .B =O
~. (~t -2 ] +A ).(2a t +a] - 4k ) =O
~.2a2 (i.i)- 2a(j:j)- 4(k .k)= O [i _j = j.ic =~=OJ
~ . 2a2 - 2a - 4=0
~ . (a - 2)(a+ 1)=0
~.a = 2~ - l .
➔ • • ➔ • •
(Z) s-41 "'11t111m IF{J c-11•1,-,.., A = i + :c} + k ~ B = Si - 2 J - 2 k •9f«'"'""i..,""'ti i
12

11iJ$\◄ ~ f _ _ [C.U. 2007) [Ana. :L = 1)

'11 ~ ~ lcll-J.>t"'1 -♦ii" I
2.11. C'¥iilf "l,.llf1rll ~~ ~ ~ oC1f (Scalar product obeys
distributive law) I
➔,,.,....,... ➔ ➔
181T ~. p C'ti'ltH t9Ar Q ~ R c-e•tc-11-11 ~91 {projection) ll,q@lll OM~ MN
➔ ➔ ,,,...,...,,.. ➔ ➔
{1R 2.19] I ~ . Q ~ R C'ti'ttff ~ ( Q + R ) I R
➔
P "5ltiHI ~tff -'ll ~ ~ ONI
➔ ➔ ➔
~ . ~ p ~ ~ 06k ( Q + R) "fl'~
0 M N p
➔ ➔ ➔ !Aus
'J.'f.-r = P .(Q + R)
➔ ➔ ➔
= p .( Q + R) .ml' at'llfff
➔ ➔ ➔ ➔
= p xON= P(OM+MN)= P.OM. + P.MN
 . , r ""'
..
, ._,, A": r "'
'
...
l'IQ•
.
.-.1"f'11, " f ' f . t ~ ~ d
.. . ..
In· P~• PR
. __..

0 ft!ill "ti ('( ( a n ~ ~ 'If' l~ W'

-♦ ...
P , R • S) • Q ( ~ ..
·~. - ·.
~~~~a I P ♦ ijH • ~
...
Q\ • ~I
. ... • -4 --4

• P R • P , S • q It • ~
...... ..
i
e ... ...Gf'h ¥iltA9 1 1Ct-il ~QII.JN "'1tlt A9ffl TIii! 1 ' ~
-·,. " ....Q t'W•n1C• ~ M.♦ fice .-. """' ,.. u=. "111 «'I
"
f • I "• ♦
.
/ / '~
. ♦
.
• ~# ...._
.
I Q ♦ j (J, ♦ l 'J,
-·
Q r 11

,,. ,·
• I'I QI ; ; • P.~.. j ~ 1• Q.1 i, ,.. P.<J, j ; ~
I

• r.Q1 j .j • 1\Q.j 4 • P.Q~A ; + l'_<J, i, j • '•~ «'

- ~Q.•~~·~~ •
Ju
j •A • • l ~ ; j • j j, • • ~ " OJ
(

._...,1'
,' I •

c-.-.. ;

~ T""" ~ , i t . ~ 111jA,i11 '9fl("NCAiQ ... ...

'J"l'"'Q( ifl- "" 'Ii
.....
Atq rQ • l'.Qroae.

!_Q P.Q,. + P1 Q1 + P#Q•

~ . N»t• n• I
2
I
( P,.1 + PJ + P. ) 2 .(Q: + Q; + Q:)
2

• ~ i◄¥CG; ... "'4 I

'fl c.t<w ~ t,+cw,, >K'4k,. l lf!U'I ~., ~ f'5<111tf 11lfil OX ~
- Nlli--
➔
~ .., F . . dlit'CA {t4 -,y~ s V'
-f ~
m . ~ ___
,
flffllf = F.S cca ea, "ll'll~II

1fi'll' .,~ti ~ 9 H;'lllluti ~ '11f"" ~ ~ -.f. 'f"<III( ~ I

➔ ➔
Sf\'IWIC\ , ~ (P ) . , ~ ( F ) ~ ~ ' I t (V) "51t1BI c..-mi 'tl+lfllJIIIIII!.
➔➔
P = F.V

I .JI. cwtl-.t ~ 'r'*'• (Vector or Cross Produ ct) Z

➔ . •
~-
'1"' ~·· ·
' ....
A '41't 8 .a1J ~ 'ltf1Jt1'1lf 111-l :: I Jl.... )( B I '"' AB 81n 811 . . . . . ,

O~h 1 .c~ iJ ..-.v... oC'lbfflA111~W.ttJ1~ ,i~ ('ol«vC 'l \tc;-,, ~

~ . , . . ~ , , lf!:tlflf ~lt1U C""1 OJ f >< n (1P'll ,ll ,\► ~'>I fi) I 't X 8
t
r
f

27

llll~~ ~A JR\ B <.'8'1!iil!l ( V-~ ~ ~ 'di~· JP!\ A CIIW' i1 41'1 firtlfl ~

\SI-Mll<¥ill3!"1Tfflt'ltCtJ1ms1lll.<I ('l'\ei~spl ~ 2.20) I A• B
A XB Jl"ll\ B x1 t.~~ ~ '
Cl5'h ,r,«a,,i '1(111~ 1Sl'1'611.<1 "1'1ti116.ff ~ ON11llC<I
~'5Tffl~C'oh-C<ltW-111~~•1~~
'J{W C'olm "1~Hll.f (order} ~ '1\1111.cfI <.'8ICH
"ltfi@ii <l.,.,1(.11 \!JT-l•~t-JRI ~ ~ ~ 1 1 T l 1.
~ ~ ~ "I@. 1 x s =- 1 x 1. ¥ffi\,
B•A
J
~'l.~~All"ll~~-11 1
c.....
'ili.PNIV t C.) -a ➔ ➔
'll "I A x B =0181' ~
..,.
A =0 _ _ _ _fAuo
__ _ _ ____.
➔
ISf1IAt B = 0 ~ 8 = 0~1tm ;
8 = 0 ~ It l{t'f C16lll◄ II 'llllll111'1 ~ ~ '1111*1'1 M"lh~ 1'1f I ~ ;_t >\ilRSl1'1
c,sfc,1,1 t'ok ~ "t'O celtH lfllTOI 1111111A1 11.ft ~ 'lilR!llli1 :tt"I, ~ ~ 1l_Wll "f!J
m,
(-.f) C.'e'h◄II "'il"""ICM 1o1R;111 ~ . 8 = ~ ~\ sin 8 = 1 ; .t/t"Ril A x B = A.B
~ Cffli 'llTII' (:'q, A. B ~ (A 1 )~ ~ l'o'1i 911ll'"'1ffi♦ lblfis61'4 lfRP11<t!l (rig.ht
X
banded) ~ 'IT-I <Ila!' {fA 2.20] I
•
.. (11) i
~
, j ~ j, ...~.. "11'11""'1ffi♦
~ ... ...'dlfis.,._. l$IRII...~... ~ . ~9ft;mr
.. ... 0011001.. 1.~.. ~ '<lTV .
i )( J = - J X i =
k ,; j )( k = - k X j =
i ~ k X i =- i X k = j
(,r) '<II' loi,Fi"'l~ zrt:11111 ~ ~ f x f = j x j = k x k = 0, ffl'f ain 0° =0
➔ ➔ ➔ ~
C•) A x B = A . B sin 8 \'611T1f 9fit sin 8 = AlBB.
Example : ~ 9fwllt.tll c."1'0 QI ~ Q,Cttl ~

8
: A = si! B = s i~ c· [Burd. U. %005) ; [C.U. ZOO%, '06]
fi1ti,tit~~,
➔ ➔ ➔
Ii ~. A, B ~ C ~ ~ .q:~ ~ Ul'{Ctfll ~ .m MtFtio ~ [1A
➔ ➔ ➔
2.21] ~ A = B + C ...... (i)
➔
~9ftft '1~♦11t-111 ffl "1lt'( B c.'otlU ~ '1"f
➔ ➔ ➔ ➔ ➔
m. A x B = (B + C) x B .....
➔➔ ➔ ➔➔➔ ➔➔

-+
~ .A xB=BxB+CxB =Cx B
-+
.
[B X B = OJ
fA 1.11 ~"it-silt >tfrflti\ m 9f1T.1'1f'~-1 f<lt<lb-il ~ ~
➔ ➔ ➔➔
AB ain ( A .B ) = C.B sin (C.B )
 ~ 29

Exomple : C1fCt'O (11, A x (0 + C) + J.1 x( C + A) + °' (A 'iJ ) x

rc.u. 200-cJ
=0

➔ ➔ ➔ ➔➔➔ -t
\Sa C:8b ~'l,'ftil'lf ~~ 'fl ~ A >< ( 11 + C) = A x /J + A >< C,
-+ ➔ -+ -+ ➔ -+ ➔ ➔ ➔➔ ➔➔ -+ -+
8 >< ( C + A) + D x C ::: D x A "1'll~ C >< ( A >< 8 ) = C x A + C x D
-+ ➔ ➔ ➔ ➔➔ ➔➔ ➔➔ ➔ ➔
Ctl1'111R'C9'~, (A>< B + A >< C) + (8 x C +B >< A) + (C >< A + C .( B)
➔➔➔ ➔ -+-+ ➔ -+ ➔ -+-+-+
=( A xB+B x A )+(A >< C+ C><A)+ ( B xC+ Cx B)
➔➔➔ -+ -+ ➔➔➔ -+-+ -t ➔
=( A x B - A >< B) +( A >< C - A x C) + (8 xC + B >< C)
=O
➔ ➔ ➔ ➔ ➔ ➔ ➔ ➔ ➔
~ . A x ( 8 + C) + B x (C +A)+ C x (A + B) = 0.
e f.li'ltll 'IIINl•lt ~ ffl't ~ 'J."l'PC-,1 ~ I
(~ ~ <.'S'tt.1111 •t1<1¥1t<11 <.'Sf!Nt!Hl 'dilUl!ll♦lll ~"11Wt1 (rectangular componenta) 'l!t9ft'lf
• . ➔ ➔
A;i""fi\
0 81
~,o1,. .,l{!""C•"f !PIil"! ~ 11111': A "5'h ~ B (.ISiCiBI ~IJNll<llliil ~911~ f.r.i{~:
➔ ➔
A
!' -:-
= ' Az + J .A._, + k ~ ~~ ..
B = l• .B:x + J~ .By + -
k .Bz
➔ ➔ • ... ... - •
:. A X B = ( i .Al' + j .A._, + k Az) X ( i . Bx + j .By + h .Bz)
= (A,rB1) i x ij + (Ax82) i >< k
+ (A,rBy) i x

+ (A,,B) j )( ; + (A,B,) j X j + (A.,,.B,) j )( k

+ (Azll) k X i + <Az81) k x j + (Az.B1) k x k
= i (A,,Bz -AzH,) + j(A/J;r: - Az81 ) + k(Az81 -A,,BJ
[·: i x i = j x j = k x k = O~ i x j = - j x f = k ~
~~ .,.,,.,..., fiiottll~-OIC~'!I ~ firtlt'f ~ ~ Wl 'lit~ m' I

i J k
A x B = Ax Ay A 1
Bx B, B1
➔ ➔ ➔ ➔
['I: ( A x B)2 + (A . B)2 (A.. B sin 8)2 + (A..B cos 8)2= =A2.B2•
~'111~~ (identity)~ ~ I]
• eol'f ".l"f1,Ci1tl ~.. I
(i) ~ ~ ~ W (Moment of a force or a torque) :
➔
lfif, cf'f P ~~ A ~ mt ~ I ~ 0 ~ 'ITC~ lbll<I~ .. """' (A 2.24) I
~ fnt ~ t9P.f A~~-~ (position vector) = "1.
~ li'lf.t, 0 ~"ITC1'ft'lf 'q'(ll'!f $1flp ~ ~ 't = cf'f X Q.~ ~ ~ fnT CNl1f
• ) ➔ ➔
if't-~ =P ( r sm 8 = r >< p .