Vector ( सदिश ) @irfanullah_mehar.pdf sadish

Physics
 
Notes
 
IrfanullahMehar
 
V
[email protected]
 
o
@irfanullah_mehar
 
WorldofWisdom

2024
 
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A.
 
A
B

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afea
 
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n
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ne
 
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peint
 
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motio)
 
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far
 
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vectorswillbe
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toHead
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not
b
 
A
 
=
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66

9=
2
n
 
Combinatintoveetos(fas a)9)
 
theiresultt
or
 
Law
 
Swn
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()
 
tan
 
Tieptrs)
 
tan'.
 
Ban
 
(9-ro
 
tuole
 
Resultut
 
iRAftB2A
BCoso
 
Bsne
 
AtBos)
 
Resultnat)

Case
It5s
 
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0=1
 
AsAtB20Boso
 
AJA+B2+2AB
 
A-J(A+B)2
 
RA+B
 
MaximuY)
 
RmaxAB
 
tì)tanoka
 
tan
 
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AtBocG
 
Beno'
 
AB
CoS0
 
Note-Gk.
(
 
20
270360
 
RO 27036
 
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Range(o7)
 
OR
 
Cose

loseI|
 
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R4-)2
 
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R,
 
un
 
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(miinn)
 
A-8
 
8ne
 
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R0'
 
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ton
cbjhit
 
tcuno".T

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ase
 
o).R=A+2ABCoso
 
x
 
RA4B
 
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x?+t2%2
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R
2x2(1+ose)
 
i)
tan
 
tanx
 
A
 
B
&in
 
A+B09
 
3+Coso
 
(
 
Rs2x2:a
 
R2%0s

fslase
 
Note
-
 
)
 
(m)
 
x
 
R=
2xCos
 
Cas
 
2x
Ces
 
2X
 
OR
 
ABRX
 
o
2A2+2
+
ABCoss
 
X=Jx4X2422Coso
 
Sqie
RootCaucol
 
2x2CosO-X
 
4
 
2
 
2,
 
Cos
j20-h
 
R
(RGTG)
 
2-920

Q.5|
 
(9)0°,
 
(B)5N
 
(b)g0°
 
R
 
Ans(
 
(C)1N(D)
NT7NGE
 
(a)
N
 
(8)5N
 
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Q.3|
 
()
 
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M)
 
J202
 
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(B)I8.2()
S202
 
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tb)ta'
(Pd)()tasla/)
 
greta

tano
h
 
P+QCGA
 
Cioss
ma

Q.
s)
 
aA2+8'+9ABos
 
A(b)
A-+2AB(oto.
 
(4)
90
 
(b)120
 
A
At
B
 
A12N
 
RaA+B
 
125
 
A
 
(D)60°
 
=5N RF F?
 
-12-5itles
+s*JrHGt25
 
-]169
 
Re130
 
tail

11111
 
Ans
 
Rmax
-A+B=1+
 
Rmn
 
A=3x
 
R
99
 
A
 
A-13
.
 
385
 
R
A²42
 
2%2
 
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R'2R
 
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Ane-(
 
x
A

9.13
 
Ans
 
Cose
 
l)|20(3)50
 
ttnso'
 
14+BCge
 
2
 
24
 
R
 
1BSn9
 
A+13CosO
 
2
 
.A
 
8

4
 
Angs
 
A24B2-2A2
 
n2
+
B2
 
:20
 
3
 
Cose-J
 
'oe

aoloa)202y
 
R=
-
 
RA+B-
2AB
CoSO
 
X2+22x2,os
 
RaX(
Cose)
 
R2x in9/
 
Bubsractiom
o
Vetors
 
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2ABCosG
 
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0),er0:(ii)e.s6°T(i)
180"
 
RaA-BMeRJAYg2
 
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A+B
 
-1

(4)
 
(2)
 
()
 
tan,
Bsne.
 
A+B
Cos
 
(4)=0=)
R
At8
 
(G)
(86-
R
 
(D)(R)>B)
 
RJe2+gABCose
 
0fR
A-B
 
180°
R
A+B

Ans
 
Q.21|
 
Ans
 
A+B2ABOSe
 
(9)30
 
4A8,Cose
 
9-
0
 
(
 
(b)6o?
 
(Ani-)
 
(e)120-()450°
 
A B
R
 
12O
 
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Ans))Ae
 
A
8
 
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()
B0
 
fA+B
=1
 
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lA8)
 
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2ABCos
 
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Ce)(20°

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(A))0
 
60
 
dye
 
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cco
10J3dyne
 
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Ans
 
G.26
 
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-
 
(b)
 
Co1)
 
n~5,By,c-3
 
5-16+9+2y
Cose
 
()7
 
5 4
98
 
tno
Being
 
tana
 
-
 
do

tan
 
Method-T
 
Ans
 
A=534C3
 
A
 
5
 
Cot()
 
8ne
 
Beine
 
(Ans
)
 
-
 
(AJ
 
90
 
(B)

ACro
 
C=R
 
(A-JA+842nB)
 
B
J2A
 
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26)
 
92
 
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Re
 
(M)
 
(oimgamsticaliy)
 
b

Ans
 
(a)
g0
,
la5,135°
 
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PQ
Case
 
P
 
90°
 
allAQe 185
 
(b)90,45°,45°
 
()45°,
135°,135
 
I15
 
P
 
sin
 
PO)
 
P+
Plo)

Notei
 
F
 
(S
-0g)/(Sumn-olno)
 
(cTGEON)-01oj
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To
(Associautivelaw)
 
A+(B+)
=
(Ä+B)
+
 
B-(B)
 
120
 
(1E)-
 
90
 
(
St-d)
 
F
 
Fnet=0
 
)60
 
66

(5)E.
6T
 
)
 
(Eq
Wbaian)
 
A-B
 
<lc)
A
tB

(9)
 
lshiehofthe
 
(b)
 
(c)
 
(4)
 
(1)
 
4
 
Oa
Vectoin
 
4
 
Ax
 
15
 
203A
 
12.
 
1
 
-R
 
eomponents
 
(2
Dimendicnal)rrartay)
 
(a)
 
Ax
 
Vectors)
 
fty
 
R
 
i)
 
Cose
 
Ax=
ACase)

eolups
Sn
 
(9)3D
 
x
 
(5)gEaierc(vetial)
Componu
 
PrHtHy
(A
Cose)
+(Aeine)
 
Ay
 
Y
 
Ay
 
Ax
 
1
 
Ar
tAy Az
 
3D
 
(aivecton'cosine)
 
31
 
Ax/
 
As
 
A

A
 
Aces
 
cos-()
 
(
 
Az
 
Ax-
Acos
oX
-)
err t
 
Si=Sine
 
CusyAz
 
A
 
Az
-
Aces
 
Ssa-Co
sine,)
(tauged
 
()203
 
Cos'Coep+Csy|s
(

Note
 
NOTE
 
bapuA-JA+Ay
 
()
A
 
tai'"Ob)
 
C
Cosine)
 
(ine)
 
(D)AyR
Sh
 
Acose-0)
 
-
203R
 
()
 
Ay
 
Ano
 
3
 
T

Ex.
 
:
Ax
3
 
Ay
4
 
A=5
 
tan
 
19
 
tRstr
a(PositienVestos))
 
(X82)
 
Ax
 
-
XiYft2k
 
(3e83

Y
 
7-X,?+
Yý+Z,k
 
(F)-CohasVroifia83)g
 
(n)1+j+10k
 
(d)
2?+6k
 
(S600
 
farg
 
(facsa)
 
Ans
 
P(23,5)
 
CAnsc

Q.SN
 
Cosne)
 
Ans
 
1
 
A
 
(A)10N
 
A
 
5N
 
(B)
3N()4N(oj2:5
N
 
J95
 
Ax
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5
 
Ay
 
X%
 
tg(55N
 
fi(B)
 
(D)
u'

A
 
A
 
l8)e)n,(4)SaA
 
Coe
 
(9ange0
 
Ay A2
 
(e)

Q.Findthe
 
Ans-
 
5N
 
53
 
58)
 
5Cos53
 
-uftmo
 
Y
 
)57
 
5N
 
55n53)
 
SN
 
5Singg+5Sing3
 
5N
 
34°
 
51V
 
R
 
ResdtatR?2
 
X
 
scs53
 
5n53
 
5os32°
 
1N

Ans
 
Y
 
J36t69
 
R0

SN
 
Ans
 
R=
 
45
 
532J:5o&
 
(n)5m
 
^5N
 
5)2
 
220le5N
 
5V
 
5N
 
R=0

Ane
 
A
-
 
1
 
et 10sec

0)
 
(A)
 
Muttiph'ahibnoVeotore
 
mA
 
Ex
 
m
 
(DitProduct
 
+Azk(afG
 
mAx?t
m
 
m
 
(Al+Ayft
Azky
 
0203A
 
08|D4y
 
12152
Am
 
(B) CrOSshoduet

Acose
 
Lmatima
 
(soalar
pluuttipleabon)
 
A.An
os
 
Coso-1
 
Cose
 
Cos)80-1
 
mntmwm
 
1-4:1-
koR=1
 
1

E
i)
 
Note
 
=
AxBx
+
AyBytA2
Bz
 
(im)Flux(bodz)faT.
 
9209f
 
se
matPuchoRaste
Gha Ke,

Acose
 
ACoSe
 
A
 
1
 
-9to
 
12412,
 
T
 
AB

.31
 
Ans
 
B
 
B
 
4t53R
 
AB
1242ot|5
 
ACosG,
=
 
B
 
w
 
3k
 
AB
 
-R2-3

Q.32
 
.93
 
Ans
7
 
B)10
,
CC)20
 
A.B 42X+200
 
(d)
 
Sattu

.
35
 
Q.36
 
(P.&)
 
(b)30lc)-uscd)6o°
 
Coelg25-25

Ans
 
(a)Cos!
 
(a)()
Eoc)
 
J6+62+(3)2J4ty
 
co)
 
(54)
 
LJ9-y
 
Snos5/
 
6
 
Ani

Q99
 
(o)90
 
Cose
 
Coso
 
(4)4X3
 
(b)
o
 
AB
 
-60
 
(06X3
 
(dl6o
 
(d)4X6
 
/Ans

Ax-
A8onn
 
Cyoss
 
fetet
 
(Veckoxbultipiation)
 
Cas-I
 
AB

Case-I
 
Case

Caie-)
 
Ex•
 
fe-?
 
2
 
+90-926)
 
0URAtieloelewises
 
-i<

-Ans
 
3
 
3
 
()-iul
+34-16k
 
(7+9)R
+{1s+1)ft
(3-s5)
 
Case-I
J
 
Ang-l),
 
Rxi
AxBx(ix^)+
AxByltx3)+
AyBz(ixl+
 
Hy&x(3+i)
 
An
e()
1
A;Bz(-3)
+AyBa-k)
+Ay6a();
AzBx(3)+

Case
 
(AyBz-ABy)?
+(ABy
-Ay
Bz)tlAxyty)T
 
+(AxBy-
Ay&x)R
=ot
+of+ok
 
ti)
Ay
&2-AzBy
-b
 
AyBz
2
By
 
A
 
BA
 
B2
 
A
Bx Bz
 
Az
 
Bz
 
BX

Ans
 
CosAR
 
AB
 
-15R1
 
exABxt
AyBytAz
Bz
 
(9)Os-(b).-,2(c)
 
-yi-6-AK
 
AX Ay
 
By
 
(d)
4
 
d=-2
 
A
 
Bz

(9)
 
Arca +Ë)
 
Veateymulticaton):
 
eh
t
yovao
1XBI
 
A.
(Bx)o
 
L.
 
Ayea
 
(hiogenal)

.42
 
(a)3,
 
Cb)5,(c)-4,
(d)-8
 
,
imot
-
 
e.(ANE)
-o
 
(9+)k
+
(3-2)1+(I+G)f
 
Ax)=ioal
 
o1Sept224
 
(sl+p45R).(î3+5k)=0
 
3++p+25120
 
Te+28.0
 
-
28

-Dhd
Spethoc
Datn
Thathed
 
Ans
 
-a++s
 
(6)A+8
 
-
 
(f)
 
5
 
1(P-<)=o
 
20t6P+14
4Pp-6FO
 
98+7P=0
 
P--9
 
ta)(a+8As)
 
(A+82ANCost)

Q.
 
Ane
 
taun6S3
 
-
(A+8:
 
(a)
O
 
(ixE)
 
(92+8
+
AB)
 
(b)
n
(cxE(d)2(x
 
n°-R/
 
(&xR)4(ax-B)]+
(exA)
t(Ëyeo)

-
(i
18)(8
xÄ)
 
45
 
Anag
 
(4)
 
Ar
-
(d)
 
(4)

Q.
 
Ang
 
Ans
 
e)
 
3a)
G
(b)3Cc)(d)
o
 
B3
 
TS,)TESTeoe.
 
ABne
 
22
422
+
2.2.3Cos
 
2+3
+22,3Quso1

Ans
 
(a)
O
 
(9)
o
(b)
7
,
(c)
 
Lstgrade
 
Q.53
 
2o)R
 
(a)
1(B)
12(C)3(d)Zero
 
1
 
Ans

Ans
 
.
f
 
A=3
 
B4
 
R-1
 
R? +8242ABCosGJ
 
32os 24Cose
-2y
 
9203e
 
Cose-241
 
Ane
 
(68Ä)
 
(er373r1)+(4n3XSX0)-(9
xsX3X%-(6rs751

34t30e0s-25
 
(a)
e
 
30
CoS9
 
Cese=%o)
 
(6X8X3xi)+(3A5x
)-(9A5X8Xo)-(6r57e
 
)s4t18-31150
 
/2
 
-23
 
-78
=>-8)
-156
 
–118.5
 
X1)

Ans
 
-2c09(92hCose
2n2
 
oso
(--ntj(nt-1)
 
Cos0
 
Cos1

sing
 
d203v-
2
o)c
 
2
 
AB
 
2
 
2
 
theu ?
 
(20

2hdGrde,oe3
Sanskit)
 
Q.58)
 
2ndered2018|
 
5
 
(B)7(0ts)(e)(:0)
 
B

Ans
 
Fuet
 
(0g(5)
 
(i)
 
(c)ABsino
 
Ascwne(51/)
 
Fu
 
iv)

AX
0
 
(n)T2)
A
CC)zhovector
(ert
 
A-c)
 
Note
;-forety
.
 
2.2xt
3.2y20
 
.62
x
+y-ia fouf (1,1,1)
 
X2
 
A
=2xt2YÝ
-
1k
 
(11,4)
 
3
 
RO
 
J231221r

Q63|
 
Tao
Vectors
 
epoint.
 
DnoYeased
 
scalarn
 
game
 
becones
 
by
 
es th
 
tue'ytaile
 
te
 
the
auglelothenis
 
2o'thenthemagžudeof
 
produet
emains
same
 
-findtteeuhalagle
 
(o)60(B)70*(c)80()
90Tos
onpe
 
ABCos(o
+
20) 93Cos
 
codet0*
 
AL)egvoi
Gos{ot26)Fces(i80-0),
 
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co(i00-6)
 
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er
Rad
nmethod(mmom
Sense)
 
0-90
 
704)
 
0-0)
 
o
 
80()
 
(D)90-0
 
3203
A
 
90
 
30-120
 
241-2-Coso=2Jy?H3)2+21.1osi
 
JIt342J3
(0)
 
=2Ans

Q65|
 
Ang>
 
Aus
 
(A)
 
-5
 
2
 
(2-0)R+3+5)4+
(0+2)
 
-O200
 
cAsiw
 
5T
 
(o~2)to2)(2-o)

Q.67/
 
togve
 
92
 
dtferent
 
Ane
 
K
 
mn,no
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nagnutucle.
 
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5
 
28)3
5
 
6)4

WorldofWisdom
 
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