mathongo.com-NCERT-Solutions-Class-12-Maths-Chapter-10-Vector-Algebra.pdf

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1. > Bigas vtr OR ay)

eek that JOA = 48 tos)

of vector OR malos an engl
30 JRR North be Earth
odes
Nets, = Beuth of West > A
vector in South-West quadrant
making on angle ef oth Wart
2. Ohesk di following monsunen a scalar sod veste
30mg GH) 2 molars morch-reet du) dan
Wo) 40 de (6) 7M coum er
Bok () 20 gi mosca man snd tro meer.
(CE kan ne mean, Hs magn a
A A af vlasty (Le, hae
mmagalcodo wed diexion bith) und hades I a vos
(id) 400 i a monture of xagli’-c. 19 magnitud erly ond,
For. diront
A A hes ne
Serie) and tarado, à eta
() 19-19 coulea ia a mioanure of last ange (Le, le
ruagaícad el) nd helos, à mal.
(oh 20 men a meosuro al oeelration Le, a a mesure of
ato of change fac amd bonos ia a verter
3. Canet the fllowiog as scala and vector quantitlan,
(O tine period) distanse CE
(da) reisen, (o) werk dane.
Sel (0 Timescsioe (4) Distance sons (Fame tee

2. (D À and are collinear.
GD) Two collinear vectors are abvays equal la magnitude.
(di) Two vectors haviog sarna magaitade aro collivear.
ds) Two coller vectors having the same magnitude are
oca

Bol 4) Troe
(4) Fala. (a? md 2? ao line actes | 22 = 21 28
ip Pala

fe Ae ee
CSL Or acia
Pire
A + - lies,

Dates: IE - OL vo dolia E aim
essa cu dra ad pres

Note. Two vectors @ am B are cad te be equal if
GE) = FG w mad |? nafs meme dite) direction,

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Important Questions, Formula Sheets & much more,

4. al. JERIA = ION
Remark. In thin way, vo can construct on Infinite nomber of

posi savers
‘Write two didlorent vectors having sumo sicection,

Lat? len} si, 0)
m

Dy a

A = le
‘eras da thin way, wo can omsizur om Inaste nomber of

poso anne
4 Find o vues ex and y 00 that tho vectors 27 +3) and
sl rn) exo equal

5. Abe plate (1, 8, 9) and (4, 5,0)
Seren,

Hs pat Pen @ me 2,9) and 4,5, 6) en,
ere, pation voten al pa P= OP wi} +2 wah

ond poco verter of pot Q= OR = 47 0 5 weh

6. O sd
12 6 Bi O A

(À at vector in the distan of 2 + D is

Seb frojeh eh 1,15

a ae AA

10. Find a vector ta the dirgstion of vaster 52 = fe ai
‘whieh has magnitude 6 oni.

Sol La dm) ee

Aveta ho dales of ¡él a? wilh has magnitude à
ES

CE

11. Bhow that the vectere 3Í -8.
‘re calizo.

ol Let @ = 2) -a) oad
Hah se; ad

7. Sel. Giver: Feats AQ, 2, = 8) and MC 1,2 1)

(2-9) Eran
Y. (Patton Vestor, OA) of point As AGL, 2, =3) =i 02) 28
ma PY. paa BW -2 09-12) + À.

"Vector Ait (directed from’ to)
ERY et RA px. le: à
E - ¿A - sil
SI oe SY

e - fus

A eh oder A = AR

8. sot engl ab vectra 2 and (02) aio or?
Anm
et = À à De À aqua a 0, OF at

Pina the portion rector of a PEA whieh dividen the
Hk pl tro Bebe ad mias pion ver

ONCE D mato 211
DE (0) état.

ra - 1

ie - : AP à

(0 Torta YY. opt agit DO ifamal (ie, an
LR eme NO mer Le AU M te REGED

ch ih

(PN point R dividing PQ exter (Le. R Has bide PQ.
tol te ZA opi Dc a2 t= 2 > Loe PR

9. D Ep RQ
(hy Formal It soo
= tlsafesbesie josh deje aja
Show thet the points A, B and O wit position vantons,
2 nai-aj-4h, Bunt js dona 2a i-0)-62,
pacte Sha varier ot à re anglo lg
O um DIOR )- SEAN ah,

Home; ee bE, mes
Dr

en AN. ru ae ru
JO D :
Y a

EII «Open,

== fat) ef Join i-ajssh- ale j-h

ee en
AB PY of print O - BY. of palas A.
mi-ai-5h-@i-4j ain ima joa do 4) oad

an

10. 1

gf

E Que ey BL

y
a7

Fyes
et By Gt Bt Eu

'

a

Euer
T

lion A
ME and B aro tn collinear veetarg then whlch of tho
fellate aro incorrect: u
[GUTE! - > ¿fon cama mmm: Bl -: à

(©) the respective components of 2 and Baro
prepertinal

(D) both the vectors Zand Ÿ have same direction, but
‘Silent magaltadoe

(Option (D ts mot tre because Evo

shine vectors can have diferems
directions and also. dior

mapatutes. E

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Important Questions, Formula Sheets & much more,

12. aa ta Cr:

2 Pind the anglo Between the vectora À - 27 + hand
aj ed.
Armin # aia} eat ma Vash 2) +8,
CE = iG oido = er re
wu B= yee =
‘Hoa, 2B: = Probus of colones of 14 Produ of content of
rote tons Lh

E + P's + «Aen

ER bata O E od

"We know Bat conf

ind the projection of te vector À = jon the vector
lei it jst
CHENE

Projection et vector end ©

13. A +7 —
ere Ja Jin
5. Show thet each of the given threo vectors la a ult

Oe re ao
weten 1 i +a) + 0h), 3 st - 6) + ah),

en
Li ea} ado

Alan shew that they aro mutually perpendicular fo cack
‘tier

Sol Let Y -

Doi iy rif
ei aye ah

as

14. 8:06

es pren to co er

ace ly pra va

DN at A). A

15. (E 8 = 17 on

means. AS

IE iad the magnitude of fue voete 2° end) 0), having the
amo magaltide rusk that ta angle Between’ dom {60°

say mae pe i

Sok Givens IE = 1 | od angie © jt) berresa 2 and B in
A a :
les 28
Ap d
De, 3 (E melts « Ae ew e en
Eee Zu et
wild ei fal
Maren N > ini so
6: Lang Ja verla never ego)
II mo
1 |= ma 18

16. wales} oak
= Pata oe
A j 207. ; Jem
AS ee 0 7 eres)
cuy 6
ee « AE … A0)
= ME fa» ON

Ber -: Zus
er. £

11 Blow thot 12118 + 18 178 jis patpemiiautes te)

PUB - IB IR, for any two menzure vecters © und 8.
Bol Lat IRB IR = 18 ane

whore = nd = 1181

Let RIIIE EE TS

Now, FE =0b en). me)

17. Putting 118 I= 0 ftom @, wo havo 61 8/1 coo = 8)

le 0m Or all (ng) recen D. à can bo any vt
Note (+ Be COI
OOO

CPE

erario
sing frm of Nota leve

rer

Puta 121 =1, 18 1-1, 18 1 Atem

COST Cay
®

18. En
Boj the comemos ig not te.

OS

A A O

Therefure @ + 8 (By definition af Zero Vostar)
EOS

e ee.

Torero, rd

Bet BF 10 A + ı Aaa

A es ee ie)
15. Mo perifecs 4/08, C of «hinge ABO are (1, 8 8),
10,0) pnd (0,9) reopeetyely, nd iad ZAR
= Ay By Gla tron a AO, 2,3 BI 0,8)

Peston votos (BV) of pint A (oa GA) = (2,2)
afer} sah

19. Using) oa)
AD 20) 5.50)
fra rs fer

, 1d
Be = A,
16, Show thatthe points AA), 7, Wf, 9) und 09,20, =) are
pt

Bol Gian pull TO QE ET
eos, Gey caló 4, 3 Cle
OW - ee ri
DIA
and 00 = (8, 1, = 35 e10) À
BB = PV. of tam point B PLY. af inital point À
=a} sa) end +2} er)
DÉC EC EE EEE EE")

20. Prove tat AB AD = ©.

Show that the vectors ai = J + à, 1-33 -5% and

32-4) „ah toca too vertioes of aright angled triangle
as he given (on) erre bu EN ft pots AB,
nomena

PY. of point ins) — Ÿ + À and
BY. of paint Bia 3) ZUR ond
o O

BB BY et pcia PV. potes
Se er A
= haf sh 0)
CAT OT 8

ala) di (inf eh) moh au disais os

=F ek 00)
ad Ati = PV, of point O PY of pola À
“ala de aia ai al + 3-8

= is) ob =)
Aa (D ni wo here

21. ore sealer, en 22? 1 unit voter 1
Wiel @ie-1 Genki a=

Sal. Giron: 2° tne nen.zu woctr of aged a
Sia
‘A give: 2 0 and JR da unit vet,
= ia Tig

CON À

‘Optom Di th cu neva

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23. rr)

2 mu

B +8 amt tT wine à «33 + of + 2% ane

Expanding Sig Set mw = 106—6)- J06=0)+ 109)
= Few nt
it) Tre stor ee ee

Toerafoe, a weit vocte perpendicular to beth and Bis

24. Lo
Coy

=)

+ Ala Dot vetors 6 dad hm 07 +07 + i
The yl dea

EA
FT aa N

Putting wales of y und «fom (4), Ga) and (1) 1 60,

ee and.

25. xO Ras
E Pediondp ital ej mir ET ET ES

Bol Oled + 6) +278)» (À Sd D
Ey
a|-%

u
eo s
E

pening ang fin co.

MAA - o E - dies eee
Comparing mellicionta di J, 4 fam poto 368 we Rave
le ow =

A o 7}
md Pe

re, 2 02
From (a = 6

DRE

= 81-818 = 0-0 mh inte: 1=Bunia= =.

€ Given thet 2.8 =Oand À x B = ©, what can you

26. = Buher © = Der B= G or vectors a oad Ÿ ore
‘ulcer (oe perl) vector Ir]

‘Wo Kney fresa common sense thet vectors @ and) 3 are

perpenäirular to sac other an wei an os parallel (or linear)
1a impossible a

(1. Leb the vestors E, Bi, @ be given ws ad + asf Hank,
CO AW ay x

Tren ow thot PACE) 2 (Bad x2.

Soh, Gin: Vecinal = ail taf + ah = ed
Pal tad ret

Fx? a EE EE
= of 8
PAS ES

hee Ara hal

27. Paledies EE
nes va.
eal ++ inal sn] vad

fired we 101 = din = fixe =2 8 no

D a nomzar vor.

Bot tu Bw

Toklng 2 cameron Gr Hy, 1 Y
i

©: Ba end Ba idee)

8 Find She arca ofthe Klamgle Wit vortices AL, 1, 2) BG, & 6)
an OU, 6, 6.

Bol. Vertes of SABC are AL 1, 2) 82,6, 5) ond CU, 5 BL

Poston Vector (EV) el point Ais 0,1,2= 0 +28
LV ot plat 2, 9, 5) a

abso} ead
ep Oia G6, 8)
AS

28. (Ae x Ab = AAA
ES
10, Find the ates of the parallelogram mans näjne
a Sige don
out ld
o cn

Wee
Wo kaw that area of perallclegran = 1 x 3
A sxe
= 600 JE = 104 square wits
Nein. Asc of parsllope hoes dlagonal vectors are & eal
Falk
Folie «Pi.

29. ‘Option (Bi the erect answer.
12) Ares of n rectangle having vertices A, B, C nnd D with

outa rea fs Ed
a lo

wt mi @z we
Gives ABOD i a sotana. i

We kan tint AB = PV. oc pst

- 2100030008)

IM « À
D = Pat pt DPW of pt À

Jadiel

ADS (AD | = A en
‘Area et rectangle ABCD = (ABHAD} (m Lang x Broads)
a ts

Orton (Ci the correct nur.

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Important Questions, Formula Sheets & much m

31. LD eden rete toe OX al
igi

nd een OV ie $1

OM op ME
= oho i vc Mj
ign ign y 08 RAA

Season A 1: ÉD ON

A O? - di à « Ys a

a,
D.

marks Pr Bal. of abe cine san generals tho
Home er
mai Vector lang a Hno making ná magle 0 with positive
(mass Le (co + Gein 09,

2. Find ho scalar components and magnlinds of the vostor

Jatning tee paint FC, =) andl ity ay zn

ak. Give paint aro Mo 2) nd Ge ya 20

ri a

= PY, Porta vector) o plat Pa

O

32. ‘laplaceriont from kor Initial point of departare.
el. Lt um taba tho fl pot of doperture as rie.
Let he gel walk a diranco OA = 4 km lourd met
m

un

‘Through Vo pant A dew a line AQ pardillo a no OP (mica
da 30° st of Noth Le, In East Meta quadrant mating on ange
of 30 with Neri)

Lat the girl walk a distance AB = 3 km (given) slong thia
direction OQ (giren). 04 = 4 (- À N-Vostor OA is along
Cal

33. thea 19 6 tre thas 1 = VE +

art your answer.
Bol. Tho or nt tus (ala)

ra 2 = Te.
Biter ho CET 1 veces

2 Sema Gh sn of « ri

Gave L Vector", BATE aro ollas.

Led = i, Pl Alam D
Dan. AE ro + dl

Ale Ii AA = AB 4 BC =F Ie)
(Casa I Vectora 9, E, 7 Emma

(Bach sido a a iano Los Wan sum oth eter bm sde)

34. ANT

Raisin

Let vector 7 be the resultant of vesters @ and D
EA ajaj ds

at Gira Vectra @ mé 9 9} = à mo

mal jte.

Roquiced vector of mega 5 alta sed poralal (or
ellie) lo roman vecor = à Bio

OR to SB
we wiv) oh, 8 aaa ak and
9-10 sb, nadin wat vector parallel the
vector iR Beat. \
Bol Gran Ven te) ide

35. mob @ 04, med «sf er

AB = PY, of pect B PV ef point À

ui -af a) ol ai fun) cet
om ab wal en) «eh

A A = forte = AVES,
fod) BO = PY, of point CARN of point 3
O „A. sj Auber wat

mei say yok

= Boa) 6 A foie =ayız
BO =P plist OE oline A
PONT EE

mud cojer Doa) cadmio eo} yh
AO= (AG ja AUD PERRAS JJ = à ia
Now, Ab + mdd 4a) neh rol + 3j vo

mil +5) sohn ab
Points A, 1, O ao sion ohne or ar te varies of

36. “BRD a má ml ajena el -2) 0h
= apa jee
Comparing mefficiants of À, 7, À on both sides, wo have

CRETE ECC)
=

Rogue caia

inal ds position vec of » pot wich divides En ine
Join Wie twe pointe Pam Q loas perllan voctors are

GF «By ed (3-88) external in tae sudo 11 2 Ale,
‘Siew ak is ha middle pia of lina og BQ.

We kw that pon acer
mod era fn a ma?

‘Agia pan vector of the mido poic ofthe Hn sagen BO.

PV of point E PV.of point Saeed AO

37. Rd
Bei nai aj eek dj

mal ej +28
mod dB al aj aj ado
wal jo odds da ss
a re
y si poa ge
aa e Tiere

Tots fia ea of ferpllegrn
Wi à

Bagel: Ali}. nein

: 16
“u oh
¡Wo ka lah aren of parallelogram = Va? x iI
= fo sate = VITE = 4008
= TEL SDE = 1148 04, unit.
Tu en a nn ee yn
METEO 2
D D Er mer a gl D >

38. = teen
E
A

2 0
on Un Fy CDA, 0 a eae and has cu 0 a pe)

DRDS.

a ı
routed econ ar m = “Be. wel ps

DR 43 ar? Gir 27 al

BE. A » deleselrinsn

A an,

Gleunz Vector ara = ind) «3h

maß mal aj ont

By atten seta prod of ie

vectors, @ = D is « vector

porgondicula o ato @ aod E,

aoe, vecor dich a alas parent to bth
2 = E) mirad or wa oar ale

39. 12. Tin ear poder of an eta 1 = fe 8 wide nik
verte aig tin mut tha rocken 2) 0 4) = Eh aa
24/22) 23% te ual o one Pind the vales st

eg ry

=.) + EY > D

Bee Ge ye

aria! aa

opie ned.
a

Paz a as

40. A > E en

41. ale ac br

HEB la Bo ance between re vector mi 7, ten. Bao,
ES

mese mocos id ococ meca

B20
LRU Les 62 0 > 00020

12, A Lon Big Long al cm re amp
ot ne ey o open one Be go

42. SS

= ec HN 0) Ser Oe Le
DS

aiii
oi

‘Chosen tha correct answer
ea} Ae À

weile
Bee. ; (ini:
CR -- dat
Opa) he correct mme:
I be Qs angle Botweon any two vectors 2 and, en
MEE 1 = 18 x Pi, when 0 de equal te
or
fencer

PCI 0