1

VECTOR ALGEBRA

I One Mark Question

k.3j2i ofdirection in ther unit vecto theFind1)

kjia

kji

a

aa

14

1

14

3

14

2ˆ

14

32ˆ

14194132a

k 3j2i aLet

222

equal? b & a vectors theAre .a If j.2i b & 2ji aLet 2)

b

ba

abaa

52,52 2222

But vectors are not equal since the corresponding components are distinct i.e. directions

are different.

3) Find the values of x & y so that vectors 2i+3j and xi+4j are equal.

3,2

32

32

yx

yixijibaGiven

yjxibjia

3322131221 kj-2i . 3k - 2j i

k.j-2i &3k - 2j i vectorsofproduct dot or scalar theFind4)

r.unit vecto a is a 1a

12

1

2

1

2

1

2

1 a

j2

1-i

2

1

2

j - i a

r.unit vecto a is 2

j-i that Show )5

22

2

II Two Marks Questions:

1) Find the vector parallel to the vector i - 2j and has magnitude 10 units.

5

2

aˆ

5a

21a

j2i aLet

22

jiaa

5

20

5

10b vector Reqd

5

2.10

ˆ bNow

10

units 10 magnitude having a toparallel vectors thebeLet

ji

ji

ab

b

b

2k- j ia vector theof cosinesdirection and ratiosdirection theFind2)

.

kjia

kji

a

aa

a

a

kjia

6

2

6

1

6

1ˆ

6

2ˆ

6

211

2

222

6

2,

6

1,

6

1 i.e.a of components are cosinesdirection

2- 1, 1,i.e. a of components are ratiosdirection Here

3) Show the vectors 2i-3j+4k and -4i+6j-8k are collinear.

If vectors are parallel then unit

vector along & parallel vectors are

same

3

collinear. areb &a

another of interms expressed becan vector One

24322864

432

akjikjib

kjia

13x

13x

121 x

12a x

12ax.a-x

1aGiven

12ax.a-x ,ar unit vecto afor if ,x Find)4

2

22

22

5

5

9843422

.2a...

4b. a and3,2aGiven

4b. a and 3,2asuch that are b and a vectors twoif ,b-a Find)5

2

22

222

ba

ba

bbabatkw

b

b

bba

b

ba

b

ba

b

batkw

bba

a.

1a

.

1cos

1cos1-, of valuesallfor cosa

.

a

.cos...

a that prove b and a vectorsany twoFor )6

4

bba

b

babb

bababba

bbaba

bba

a

a

b.a properties previous Froma2a

...2a

.2a

equalityin Triangle a that prove b and a vectorsany twoFor )7

2

22

22

22

2

22

22

22

35.116

35.116

35.10.216

.35.10.21.6b7a2.b5-a3

b7a2.b5-a3 Evaluate)8

bbaa

bbaa

bbabaa

bbabbaaa

kjba

kji

kji

kji

ba

baba

1919

19191414

2122127227

223

771

and find2k 2j-3ib &7k 7j-ia If )9

3and 2

27

06-2027-20,27-6

scofficient Equating

62272276

1

2762

27k6j2iGiven

27k6j2i if & Find)10

okji

o

kji

okji

okji

5

)(2)()(

)()(

)()( LHSConsider

)(2)()( that Show )11

bababa

obabao

bbabbaaa

baba

bababa

2042

113112111

111

11-1-

321

product pleScalar tri

kji andk j-i- 3k,2ji vectorsofproduct plescalar tri theFind)12

4

0123

06433

032231211

0

123

11

211

coplanar are vectorsGiven that

coplanar. arek -2j-3i&kj-i 2k,ji vectors theif Find)13

6

III Three Marks Questions:

ly.respective externally and internally 1:2 ratio in the Q and P points the

joining line divides which Rpoint a ofector position v theFind

. ba and 23 ectorsposition v with Q and P points heConsider t1)

OQbaOP

abOR

bbaa

baba

nm

OPnOQmOR

aOR

bbaa

baba

nm

OPnOQmOR

nmbaOQbaOP

4

1

2232

12

2312

externally

3

5

12

2232

12

2312

,Internally

1:2:,23given

Solution

7

PQ of cosinesdirection also and 4 2,- 1,-Q and 0) 3, P(2, points thejoining vector theFind )3

25

4,

2

1,

25

3are cosinesdirection

25

4

25

5

25

3

25

453

25

5016259453

453

3242

3242

PQ

42

32OPthen

axes along rsunit vectobekj,i,Let

4,2,10,3,2Given

^

^

222

kjiPQ

kji

PQ

PQPQ

PQ

PQ

kjiPQ

jijii

jikji

OPOQ

kjiOQ

ji

QP

4) Show that points A(1, 2, 7) , B(2, 6,3) & C(3, 10, -1) are collinear OR

Show that the points with position vectors i+2j+7k, 2i+6j+3k and 3i+10j-k are

collinear.

8

collinear. are C B, A,

satisfied. iscondition Collinear

Here

332

1323333

8,8,27,2,11,10,3

4,4,13,6,21,10,3

4,4,173,26,12

1,10,33 6, 2,7,2,1

ACBCAB

AC

ACBCAB

OAOCAC

OBOCBC

OAOBAB

OCOBOA

5

7cos

7

5 cos

14

10

1414

10

a

b . acos then b & abetween angle be Let

14123b

14321a

10343 k 2j-3i . 3k2j - i b . a

k 2j-3i b3k2j - i aLet

k. 2j-3i &3k 2j - i vectorsebetween th angle theFind )5

1

222

222

b

9

s.lar vectorperpendicumutually are c ,b,a

0a.c&0c .b that showcan We///

lar toperpendicu is0.

049

112186

49

1

263.63249

1

2637

1.632

7

1 b . aconsider

3267

1c

2637

1b

6327

1 aLet

lar.perpendicumutually are

3267

1 and 263

7

1,632

7

1 vectors that theShow )6

ly

baba

kjikji

kjikji

kji

kji

kji

kjikjikji

ba lar toperpendicu isba0 ba . ba

016824244.826 ba . baconsider

2445335 ba

8265335 ba

lar.perpendicu are b - a & ba vectors that theshow then 53,35a If )7

kjikji

kjikjikji

kjikjikji

kjibkji

80802236

03.322-2

0.ba

.c is baGiven

3222 ba

2322 ba

. find then c lar toperpendicu

is basuch that and j3i c and 2,322a If )8

r

jikji

c

kji

kjikji

kjibkji

10

23...

3...2

111...2

..

..///

..

0..a

.a.a

1,1,1

... findthen such that rsunit vecto are ,, If)9

222222

2

2

2

2

accbba

accbba

cbaaccbbaAdding

cbaac

baccb

acaba

caba

ocba

cba

accbbaocbacba

ly

kiba

ba

jiba

ba

baba

42

2k2ji2k2j3i

44

2k-2ji2k2j3i

2k-2jib &2k 2j3ia where

and vector theofeach lar toperpendicur unit vecto and vector a Find )10

33

2

3

2

24

8k-16j-16ic

c

cc then

ba and ba lar toperpendicur unit vecto thebe cLet

245766425625681616

81616

80016016

402

044

ba ba c then ba and ba lar toperpendicu vector thebe cLet

222

kji

c

kjic

kji

kji

c

11

kjiaorkjia

or

a

kjia

a

a

a

a

a

a

a

a

a

a

kajai

a

2

1

2

1

2

1

2

1

2

1

2

1

12060

21cos

41cos

cos4

31

cos2

1

4

11

cos2

1

2

1

cos2

1

2

1

cos

cos

cos

21

14cos

cos

21

13cos

a

acos then

1aGiven

a aLet

,4,3

k then j, i, with aby made angles thebe , ,Let

.a of components hence and findk then with

angle acute and and j with 4 i,with 3 angles makesr unit vecto a If )11

00

2

2

2

2

22

3

3

3

2

2

2

1

1

1

1

321

12

kjAC

kjikji

OAOCAC

kjiAB

kjikji

OAOBAB

kjiOCC

kjiOBB

kjiOAA

C

34

255

32

2532

555,5,1

5325 3, 2,

22 1, 1,

5,5,1,5 3, 2,B ,2 1, 1,A ices with vert triangleof area theFind )12

unitssq512

1

2

1ABC triangleof Area

5116936436

4360403126

340

321

222

ACAB

ACAB

kjikji

kji

ACAB

13

7

263AC alongr Unit vectoAC toparallelr Unit vecto

7

263

AC

ACAC,AC alongr Unit vecto

7494369263AC

263AC

3k-2j-i5k4j-2i

baAC Diagonal

.363ba ABCD ramparallelog of Area

363012124201122ba

112244561012

3-2-1

54-2

kji

ba

3k-2j-ib5k4j-2ia

diagonal. its toparallelr unit vecto find Also 3k.-2j-i

and5k 4j-2i are sidesadjecent whoseramparallelog of area theFind )13

^

222

22

kji

kji

kji

unitssq

jikji

14

triangle.angledright a of vertices theformly respective5k -3j-ic &

k j-2ib 4k,-4j-3ia ectorsposition v with C B, A, points that theShow)14

kjiOAGiven

CBAGiven

443

5,3,11,1,24,4,3

satisfiedislaw

OCABCAB

kjikjikjiCABCAB

kjikjikji

OCOACA

kjiBC

kjikji

OBOCBC

kjiAB

kjikji

OAOBAB

kjiOC

kjiOB

le

26253

253443

62

253

53

4432

53

2

15

triangleangledright a form c ,b ,a i.e.

satified. is theoremPythagrous

41635 that seecan We

66112

4141621

3535531

,,

222

222

2222

2222

2222

BCACAB

BCACAB

CACA

BCBC

ABAB

aformcba le

cbacbacbacba

acbcba

acbbabcbbacabaacba

,,2.2.

.0000.

....

ac bacb.ba

ac ccabcb.ba

ac cb.baLHS

cba2 ac ,cb,ba that Prove)14

coplanar are vectors

0.

5712

231161144

211

123

114

cb.aconsider

2kjic

k-2j-3ib

kj-4iaLet

coplanar are2k ji andk -2j-3i k,j-4i vectors that theShow)14

cba

16

MATHEMATICS

II PUC

VECTOR ALGEBRA

CET QUESTIONS & ANSWERS

kjdjickibjia

kjikji

33)33)33)33)

.externally 1:2 ratio in the&2 are ectorsposition v

whoseQ and P points jioning line divides which Rpoint a ofector position v theFind )1

ki

kjikji

kjikjiOPOQOR

nmkjiOQkjiOPSol

33

2222

,1

)2(2

12

2

1:2:,,2:

CAns

CACBABd

CABCABc

ACBCABb

CABCABa

:

0)

0)

0)

0)

. not true is following theof which ABC triangleIn the )2

17

CAns

magnitudesdifferentbutdirectionsamehavebavectorsthebothd

alproportionnotarebaofcomponentsrespectiveThec

bab

quantityscalarsomeforaba

:

.&)

.&)

)

.)

incorrect? are followiong theofh then whicectors,colinear v twoare b and a If )3

bAns

ba

ba

baba

babaSol

dcbabbaa

baba

:

cossin

cossin

cossin:

0)0)))

)4

22

2222

222

222

2

22

22

10)20)20)10)

then runit vectoan is&2.a ,4If)5

dcba

babba

20

416

24

:Sol

2

2222

2222

b

b

ba

bababa

CAns :

18

CAns

boradBandAoptionbothc

babothbboraa

baobaba

:

00))

0&0)00either )

? &about say you can then What & 0Given )6

2)

3)

4)

6)

is &

between aqngler then uniy vectoan is and 3

2 3,such that vectorsbe &Let )7

dcba

ba

bababa

2

1

3

23

1

sin

0 and 3

2 3,

Ginven

:

ba

baWKT

baba

solution

BAns :

45sin

45sinsin

0

0

19

DAns

collinearareCDAB

CDAB

CDAB

CDAB

CD

AB

CDAB

kjikjiCDAB

kjiCD

kjikji

OCODCD

kjiAB

kjiji

OAOBAB

kjiOD

kjiOC

jiOB

kjiOA

Solution

collineardcba

CD

kjikjijikji

:

&

coscos

1218

36

2623

36

7218

36

cos

then &between angle belet

724644282

18141

36

2322

2824

282

3236

4

52

6

32_3

52

:

)4

2)

3)

4)

is andABbetween angle then ly,respective DC,B,A,

points ofthe vectorsof sposition theare 6&323,52, If)8

222

222

20

2550cba

5025169543

000

.ba....cb.

.cbacba

0.

0.

0cb.aGiven :Sol

10)52)25b)5 )

cba then other two theof sum to

larperpendicu vector ofeach and5,4,3aGiven )9

222

222222

22

cbacba

ccccabbbaaa

cbacba

cac

acb

dca

cb

ans:b)

0cb.aGiven :Sol

0)2

21)

21

2 b)

2

21- )a

then .c.bb. and2c & 4b1,a,cba If )10

dc

acao

2

21212

4161241

c-b-

c-b-a.cc.bb.a2Adding

c-b.a.

b-a.cc.b///

.b.a

0.b.a

.cba.

222

222

222

2

2

2

2

a

a

ac

aca

caa

oaa

ly

is then lar toperpendicu is and toparallel is

where and3k -j2i j,-3iLet )11

121

21

jdbj

2)2

j3ic)5-3i)

22

3i a)

21

kji

jikji

jiji

jikji

ijikji

ji

32

3

2

22

332

22

33

2

1

01960011332

03.3132

03.332

0.0,

lar toperpendicu is Given

from3

j-3iscalar a is where

parallel are & Since

12

1

12

2

1221

1

11

ans:a)

*****