Vector 3 Axis

- 1 - www.focus-physics.com

3.1

YY

X

X Z

Z

XX , YY , ZZ X Y Z O X Y Z ( Origin ) P 3 P P ( x , y , z ) P(3,4,4) XY , YZ XZ P(3,4,4) (QR,RS QS)

X

Y

Z

P 3,4,4

R

Q

S

P(3,4,4) XY Q 3,4,0 YZ R(0,4,4) XZ S(3,0,4)


1 1 1P(x , y ,z ) 2 2 2Q(x , y ,z )

2 2 22 1 2 1 2 1PQ x x y y z z

2 2 2QR 0 3 4 4 4 0

9 0 16 25 5 2 2 2RS 3 0 0 4 4 4

9 16 0 25 5 2 2 2QS 3 3 0 4 4 0

0 16 16 32 4 2

3.2

(1) (Scalar Quantity) (2) (vector quantity ) A B

A

B

A B

A

B

ABAB

AB

AB AB


AB A B A B AB AB A B A B () AB AB AB

u u

A

B

A B

A

B

u

u

u

u

u u v u v

u

v

w

u

v

w

u

v

w

u v u v u v u = v

u

v

w

u

v

w

u

v

w


u u u u - u

u

u v

v

w

w

1 10 (1) 50 009 (2) 65 040 (3) 30 150 (4) 40 300 0o 360o 100o three figure system

OA 50 009 OB 65 040 OC 30 150 OD 40 300


( 0o 360o 100o three figure system ) (1) 120 (2) 30 120o (3) 60 225o (4) 10 075o

N

120

A

B

N

A

B

120

30

N

A

B

225

60

N

A

B75

10

1 2 3 4

5 5 50

v,u,w vuw v

u w u w v

u

w v


A

B

CS

AB BC AC

AB BC

AC

P

QR

RS

QR

PQ

PS

PQ QR RS PS

(jew vector) 0 AB BC CA AA 0

A

B

C

AB BC CA AA 0

AB BC

CA

PQ QR RS SP PP 0

SP

QR

RS

QR

PQ

SP

PQ QR RS SP PP 0 v,u )v(uvu

vu u v

u

v

u

v

u v

u

2u

2u


2 u 2 u u ( -2 u ) 2u u 1 u a ( ) u a a u

1. a = 0 a u = 0 2. a > 0 a u a u

u 3. a < 0 a u a u

u

1 u v , a b

1. a u 2. a(b u ) = (ab) u 3. a( u + v ) = a u + a v 4. (a+b) u = a u + b u

2 0u 0v u v a

vau 3 0u 0v , u v Rb,a

0vbua a = 0 b = 0


PQ u,v,w

u

v

P

Q

1

u

v

w

P

Q 2 PQ u v PQ u v w

u

v

wP

Q 3 P

Q

u

w

v

4 PQ u v w PQ u v w ( ) 1. 1.1 () - A B B C AC P Q Q R R S PS

A

B

CS

AB BC AC

AB BC

AC

A B B C AC

P

QR

RS

QR

PQ

PS

PQ QR RS PS

P Q Q R R S PS

- OB OC BC


1.2 AB CB A B B C AC 1 1. A B B C AC

2. A B B C C D AD

3. A B B C C D D E AE 4. AB BC CD DE EA AA 0 2 () 1. AB CB AC 2. AB CB CD AD 3. AB BC DC AD 4. AB CB CD ED EA AA 0 3 () 1. OB OA AB 2. OC OB BC 3. AC AB BC 4. EC ED DC 4 . AB , BD , CA , DB , AF , FA , AE EA a, b, c, d, e f

A

B C

D

E

F

a

b

c

e

c

d

f

AB a BD b c a f CA b a c f


DB b c f a DB b c f a AF f e a b c e f c d FA f e e a b c f c d AE f c a b 2c f e d EA f c a b 2c f e d 5. u

u

1. 2u 2. 3u 3. 0u

1 2u 2 3u 3 0

C AB

O

m n

vu

w

nu mvwm n

2 2B x , y 1 1A x , y

m n

C x, y

2 1mx nxx

m n

2 1my nyym n


1. w u v

u v

w

2 3

1

vu

w

1 4

2

3u 2v 3u 2v

w3 2 5 5

4u 1v 4u vw

4 1 5 5

vu

2 5

3

w

u

w

v

1 3

4

5u 2v 5u 2v

w5 2 7 7

3u 1v 3u vw

3 1 4 4

2. ABCD A 1,2 AB 9 i 4 j, AD i 5 j C 1. 7,11 2. 8,11 3. 9,11 4. 8,9

A 1,2 B

C a,bD

AB 9 i 4 j

AD i 5 j AC

AC AB AD AC 1 9 i 5 4 j 8i 9 j

C a,b a 1 89b 2

a 1 8 a 7 b 2 9 b 11 7,11


3. ABCD P Q AB AQ:QB 2:3 u AB v AD PQ

1. 1 u 3v6

2. 1 7u 3v6

3. 1 u v10

4. 1 u 5v10

u AB v AD

BA

CD

u

v

32

P

Q

1 2

PQ CA AB2 5

1 2

AC u2 5

1 2u v u

2 5

1 1u v10 2

1u 5v

10

4. ABC E CE 2BA BE aCB bCA a,b b a 1. -1 2. 2 3. 3 4. 5

CE 2BA BE aCB bCA

BE BC CE CB 2BA CB 2 BC CA CB 2 CB CA CB 2CB 2CA 3CB 2CA a 3,b 2 b a 2 3 5


5. ABCD M, N BC CD u AM v AN AB

1. 3 1u v2 2

2. 3 u v2

3. 2 1u v3 2

4. 4 2u v3 3

ABCD M, N BC CD u AM v AN

A B

CD

u

v M

N

AB u MB..... 1 AB v NC CB

1AB v AB 2MB2

1 1MB AB v...... 24 2

(2) (1)

1 1

AB u AB v4 2

1 1

AB AB u v4 2

3 1AB u v4 2

4 2AB u v3 3


6. AB P AB AB C PA,PC PB

P

AC

B

1. PC 4 PA PB 2. PC 2 PA PB 3. 2PC PA PB 4. 4PC PA PB 5. 1. 4.

P

AC

B1 1

PBPA

1PB 1PBPC

1 1

2PC PA PB 3. 7 . [ . / 2551] ABCD M N BC CD

BM:MC = DN:NC =1:2 AC AM AN

A B

CD N

M

1 2

2

1


A

CD N1 2

1

A B

C

M

2

1

2

2AD 1ACAN

2 1

2AB 1ACAM

2 1

3AN 2AD 1AC .... 1 3AM 2AB 1AC .... 2 1 2

3AN 3AM 2AC 2AB 2AD 3AN 3AM 2AC 2 AB AD BC AD

3AN 3AM 2AC 2 AC

3AN 3AM 4AC

3 3AN AM AC4 4

3 3 6 1.54 4 4

8. ABCD M AD 1AM AD5

N AC 1AN AC6

MN aAB bAD a b

1. 215

2. 15

3. 13

4. 1

A

B

CD

11

4

5M

N

1AM AD

5 1AN AC

6

MN AN AM


1 1AC AD6 5

1 1AB AD AD6 5

1 1aAB bAD AB AD6 30

1 1 2a b6 30 15

9. ABC D AC F BC 1 1

AD AC , BF BC4 3

DF aAB bBC ab

A

D

CB F

1

1 2

3

DF DC CF

3 2 3 2AC CB AB BC BC4 3 4 3

3 3 2AB BC BC4 4 3

3 1AB BC aAB bBC4 12

a 3 12 9b 4 9


10. A a,b ,B 4, 6 C 1, 4 ABC P

AB A 35 A B CP i 2 j

a b

C 1, 4

A a , b

B 4 , 6

P x , y

3

2

C P i 2 j

2 2B x , y 1 1A x , y

m n

C x, y

2 1mx nxx

m n

2 1my nyym n

x 1 1

CPy 4 2

2a 12 25

2b 18 2

5

x 1 1y 4 2

2a 12 10 2b 18 10

x 2, y 2 a 1 b 4

a b 1 4 3

11. ABCD M AD 1AM AD5

N AC 1AN AC6

MN aAB bAD a b

1. 215

2. 15

3. 13

4. 1


A

B

CD

11

4

5M

N

1AM AD

5 1AN AC

6

MN AN AM

1 1AC AD6 5

1 1AB AD AD6 5

1 1aAB bAD AB AD6 30

1 1 2a b6 30 15

12. u v PR 3u 1. QR 2. PS 3. SQ

uv

P

Q

R

S 1. QR 2u 2. PS 3u v 3. SQ 2u v 13. ABC AD BA a BD b CA

A

B CD

a

b 1. a 2. a b 3. a 2b 4. a 2b ( 3)


2 u v 1. u v u kv k 1.1 k 0 1.2 k 0 1.3 k 0 2. u v au bv a 0 b 0

1. u 0 v 0 x u v

1.1 2 22x x u v xv 6x 2 x u u v

2 22x x u v xv 6x 2 x u

2 22x x u 6x 2 x u xv v

23x 7x 2 u x 1 v

2

x 1u v

3x 7x 2

u v

2

x 10

3x 7x 2

x 10

3x 1 x 2

1

1

32

1x , 1 ,23


1.2 2x 1 u 3v v 5 x u u v

2x 1 u 3v v 5 x u

2x 1 u 5 x u v 3v

2x x 6 u 2v

2

2u v

x x 6

2

20

x x 6

2x x 6 0 x 2 x 3 0 x 3,2

1.3 22x 8 u xv 3v x u u v

22x 8 u xv 3v x u

22x 8 u x u 3v xv

22x 8 x u 3 x v

2

3 xu v

2x 8 x

u v

2

3 x0

2x 8 x

2

x 30

x 2x 8

x 30

x 4 x 2

2 3 4

x , 3 3,4


3.3 . AB 1 2A(x , y ) 2 2B(x , y )

AB 2 12 1

x x

y y

2 1x x a 2 1y y b

AB ab

x

y

A 4,6

B 3,4

C 6, 6 D 6, 5

E 4, 2

O 0,0

4 0 4OA6 0 6

3 0 3OB4 0 4

6 0 6OC6 0 6

4 6 2DE

2 5 3


1. PQ P Q (1) 3,2P 6,8Q (2) 4,3P 1,1Q

(1)

3

6

36

28PQ

(2)

3

4

41

)3(1PQ

2.

() 1 (1,3) (4,8) (4 1) i 8 3 j 3i 5 j

2 (3,5) (8,2) (8 3) i 2 5 j 5i 3 j 3 ( 4,2) (3,5) ( 3 4 ) i 5 2 j i 3 j 4 (0,0) (4, 3) (4 0) i 3 0 j 4i 3 j 5 (0,0) (4,4) (4 0) i 4 0 j 4i 4 j

3. 2,6A 3,8B D ABCD C

(1) 1,1C (2) 4,3C (3) 6,5C (4) 7,1C

D y,x

(1)

1y

1xCD

23

68AB

1y

1xCD

1

2AB

ABCD

1

2

1y

1x


x 1 = 2 y + 1 = - 1 x = 3 y = - 2 D (3,-2)

(2)

4y

3xCD

1

2AB

ABCD

1

2

4y

3x

x + 3 = 2 y - 4 = - 1 x = -1 y = 3 D (-1,3)

(3)

6y

5xCD

1

2AB

ABCD

1

2

6y

5x

x 5 = 2 y - 6 = - 1 x = 7 y = 5 D (7,5)

(4)

7y

1xCD

1

2AB

ABCD

1

2

7y

1x

x + 1 = 2 y + 7 = - 1 x = 1 y = - 8 D (1,-8)


4. 3,4A 5,1B C CD ABCD D

(1) 2,1D (2) 3,4D

(1)

8

5

35

41AB

y2

x1CD

ABCD

8

5

y2

x1

-1- x = -5 -2 - y = 8 x = 4 y = -10 C (4,-10)

(2)

8

5

35

41AB

y3

x4CD

ABCD

8

5

y3

x4

4 - x = -5 3 - y = 8 x = 9 y = -5 C (9,-5)


5. A(3,2)

5

3AB B

A )y,x(11

B )y,x(22

5

3AB =

12

12

yy

xx =

2y

3x

2

2

2

x - 3 = - 3 2

x = 0

2y - 2 =5

2y =7

B (0,7)

6. B(-4,5) 2AB2

A

A )y,x(11

B )y,x(22

2

2AB =

12

12

yy

xx =

1

1

y 5

x4

- 4 - 1

x = 2 1

x = - 6 5 -

1y = - 2

1y = 7

A (-6,7)

3

x , y , z x

y

z


x

y

z

(O) P(x, y,z) 1 1 1P(x , y ,z )

2 2 2Q(x , y ,z ) 2 1

2 1

2 1

x x

PQ y y

z z

Z

X

Y

O

P x, y, z

OP

1.

() 1 (1,3,7) (4,8,10) (4 1) i 8 3 j 10 7 k 3i 5 j 3k

2 (3,5,4) (8,2,9) (8 3) i 2 5 j 9 4 k 5i 3 j 5k 3 ( 4,2,1) (3,5, 10) ( 3 4 ) i 5 2 j 10 1 k i 3 j 9k 4 (0,0,0) (4, 3,5) (4 0) i 3 0 j 5 0 k 4i 3 j 5k 5 (0,0,0) (4,4,4) (4 0) i 4 0 j 4 0 k 4i 4 j 4k


a c

b d

a c b d

a d

b e

c f

a d,b e c f

a c a cb d b d

a d a d

b e b e

c f c f

0

0

0

0

0

ab

a

b

ab

a

b

c

a

b

c

a

b

c

a c a cb d b d

a d a d

b e b e

c f c f

a a

b b

a a

b b

c c

2. 4u1

, 2v7

1. u v 2. u v 3. 3u 4. 2u 3v

1. 4 2 6u v1 7 6

2. 4 2 2u v1 7 8

3. 4 3(4) 123u 31 3( 1) 3

4. 4 72u 3v 2 31 2

= 8 212 6

= 294


3. 4

u 1

2

3u

34 3(4) 12

u 3 1 3( 1) 3

2 3(2) 6

4.

1 3u

4

2v7

3 2

4 7

2 3u

4

6v8

3 6

4 8

3 2u

2 2

1

v2

2 1

2 2 2

4. 1u

4

2v8

1 2

4 8

5 1u

2

2v4

1 2

2 4

6 8u

4

4v2

8 4

4 2

7 1u 2

1

2

v 4

2

1 2 1

2 4 2

8. 1u 2

1

4

v 5

6

1 2 1

4 5 6

9. 1u 2

4

2

v 4

8

1 2 4

2 4 8

10. 2u 3

4

4

v 6

8

2 3 4

4 6 8

1. u v u kv k 2. u v


(unit vector )

O

y

x

2 2Q x , y

1 1P x , ya

b

PQ

z

y

x

O

1 1 1P x , y , z

2 2 2Q x , y , z

PQ

1) ab

2 2a b

2) ab

2 2

a1

ba b


3) a

b

c

2 2 2a b c

4) a

b

c

2 2 2

a1

ba b c

c

5) PQ PQ

1 i (

0

1

0

0

1

) , j (

1

0

0

1

0

) k (

1

0

0

)

1

u a i b j 2 2u a b

1 u 3i 4 j 5 2 u 6i 8 j 10 3 u 5i 12 j 13 4 u 15i 8 j 17 5 u 5i 12 j 13 6 u i 2 j 4k 21 7 u 2 i 2 j 2k 2 3 8 u 4i 2 j 6k 56 9 u 3i 4 j 5k 50 10 u i j k 3


2. 1. u 6 i 8 j

2 2

u 6i 8 j 6i 8 j 6 i 8 j

u 10 10 106 8

2. u 3i 4 j

2 2

u 3i 4 j 3i 4 j 3i 4 j

u 5 5 53 4

3. u 5i 12 j

2 2

u 5i 12 j 5i 12 j 5i 12 j

u 13 13 135 12

4. u 7 i 24 j

2 2

u 24i 7 j 24 i 7 j 24 i 7 j

u 25 25 2524 7

5. 5 u 2 i 3 j

2 2

u 2i 3 j 2 i 3 j 2 i 3 j5 5 5 5

u 13 13 132 3

6.3 u 3i 4 j

2 2

u 3i 4 j 3i 4 j 3i 4 j3 3 3 3

u 5 5 53 4

7.10 u 8i 15 j

2 2

u 24i 7 j 24 i 7 j 24 i 7 j

u 25 25 2524 7

8. 7 u 7 i 24 j

2 2

u 24i 7 j 24 i 7 j 24 i 7 j7 7 7 7

u 25 25 2524 7

9. u 2i 6 j 3k

2 2 2

u 2i 6 j 3k 2i 6 j 3k 2i 6 j 3k

u 7 7 7 72 6 3

10. u i 2 j 4k

2 2 2

u i 2 j 4k i 2 j 4k i 2 j 4k

u 21 21 21 211 2 4


11. u 2i 5 j 2k

2 2 2

u 2i 5 j 2k 2i 5 j 2k 2i 5 j 2k

u 33 33 33 332 5 2

12. 18 u i 4 j 8k

2 2 2

u i 4 j 8k i 4 j 8k18 18 18 2i 8 j 16k

u 91 4 8

(direction cosines ) a

1

2

3

a

a a

a

a 0 X , Y , Z

31 2 aa a, ,a a a

z

y

x

R

S

Q

O

1 2 3P a ,a ,a

2


P(-1,2,3) Q(2,3,5)

PQ =

35

23

)1(2

=

8

1

3

PQ = 74)8(13 222

PQ 74

8,

74

1,

74

3

3.4 1 1u x i y j 2 2v x i y j u v 1 2 1 2u.v x x y y 1 1 1u x i y j z k 2 2 3v x i y j z k u v 1 2 1 2 1 2u.v x x y y z z u v u v 1. u 2 i 3 j 4 i j u v v u u v 2i 3 j . 4 i j = 2 4 3 1 = 38 = 5 v u 4i j . 2 i 3j = 4 2 1 3 = 38 = 5

2 4

u 1

2

2

v 3

4

u v

4 2

u v 1 . 3 4 2 1 3 2 4 13

2 4


1. u, v w a 1) u v v u ( ) 2) u v w u v u w ( ) 3) a u v au v u av 4) 0 u 0 5) 2u u u 6) i i j j k k 1 i j i k j k 0 2. u v 0 180 u v u v cos () 3. u v u v u v 0

v

u

v

u

v

u

u

v 180u v

u

v

u

v

u v u v cos vu


21 u.u u 2 u.v u v cos 2 2 23 u v u 2u v v 2 2 24 u v u 2u v v 2 2 2 25 u v u v 2 u 2 v 2 26 u v u v 4u v 2 27 u v . u v u v 2 2 2 28 u v w u v w 2u v 2v w 2w u 2 2 2 29 u v w u v w 2u v 2v w 2w u 2 2 2 210 u v w u v w 2u v 2v w 2w u 11 u v . w s u w u s v w v s 2 2 22 212 au bv a u 2abu v b v 3 1 3

u4

4v3

u v 0

2 3u

4

6v8

u v 0

3 2u

2

1v1

u v 0

4. 1u4

5v8

u v 0

5 1u2

2v1

u v 0

6 8u

4

4v2

u v 0

7 1u 2

1

2

v 4

2

u v 0

8. 1u 2

1

4

v 5

6

u v 0


9. 1u 2

1

0

v 4

8

u v 0

10. 2u 3

4

4

v 6

8

u v 0

4 . u v u 5 , v 8 vu

vu u v 1. 0 2. 30 ( 1 ) 0 2 2 2u v u v 2 u v cos 2 22u v 5 8 2 5 8 cos0 2u v 25 64 2 5 8 1 2u v 169 u v 169 13

2 2 2u v u v 2 u v cos 2 22u v 5 8 2 5 8 cos0 2u v 25 64 2 5 8 1 2u v 9

u v 9 3 ( 2 ) 30 2 2 2u v u v 2 u v cos 2 22u v 5 8 2 5 8 cos30

2 3u v 25 64 2 5 82

2u v 89 40 3

u v 89 40 3 2 2 2u v u v 2 u v cos 2 22u v 5 8 2 5 8 cos30

2 3u v 25 64 2 5 82

2u v 89 40 3

u v 89 40 3


5 u v 5 2 | u v | 26 u v 1. 3 2. 6 3. 8 4. 12 | u v | 5 2 | u v | 26

2 2 2 2 2 2| u v | | u v | (| u | | v | 2u v) (| u | | v | 2u v) 2 2(5 2) ( 26) 4u v

50 26 4u v 24 4u v

24u v 64

2.

6 u v 60 | u v | 37, | u v | 13 | u | | v | 1. 5 2. 7 3. 37 4. 50 2 2 2| u v | | u | 2u v | v | 37 ..(1)

2 2 2| u v | | u | 2u v | v | 13 ..(2) (1)+(2) 2 22 | u | 2 | v | 50 2 2| u | | v | 25 ..(3) (1)-(2) 4u v 24

u v 6 | u || v | cos60 6 (u v | u || v |)cos

1| u || v | 62

4 ; 2 | u || v | 24 (4) (3)+(4) 2 2| u | 2 | u | | v | | v | 49 2(| u | | v |) 49

| u | | v | 7 2. 7 u v u v

u.v u v cos

-1

u v 2 2 4cos

4 1 4 1 4 4u v

8 8cos cos

9 9

u 2 i j 2k , v i 2 j 2k


8 u i 3 j, v 2i j (u v) (u v) cos

1. 5

1 2 . 5

2 3. 5

1 4. 5

2

u i 3 j, v 2i j

u v 3i 4 j, u v i 2 j 2 2| u v | 3 4 25 5 2 2| u v | ( 1) 2 5 (u v) (u v) 3( 1) 4(2) 3 8 5 (u v) (u v) | u v || u v | cos

5 5 5 cos 5 1cos

5 5 5 1.

9. u v u v u v u 2v u 2v 2u v | u | 2 cos

1. 10

1 2. 6

1 3. 4

1 4. 2

1

u v u 2v (u v) (u 2v) 0

2 2| u | u v 2 | v | 0 (1) u 2v 2u v (u 2v) (2u v) 0

2 22 | u | 3u v 2 | v | 0 (2) (2)-(1) 2| u | 4u v 0 2( 2) 4u v 0

1u v2

.(3)

(1) 5| v |2

(3) 10

1cos 1.


10. u ai bj a 0 b 0 u (5i 2 j) 14 u

i 3cos5

a b

1. 7 2. 14 3. 18 4. 21 u (5i 2 j) (ai bj) (5i 2 j) 5a 5b 14

5a 14b2

u i | u || i | cos

2 2 29a (a b )25

3a b4

3 (5a 14) 426,4 2 23

56b 8,23

a b 6 8 14 2 11. u v 6 u v 2 u v

2 2u v u v 4u v

2 2

6 2 4u v

4 4u v u v 1 12 A,B,C O OA 3i 2 j

OB 2i 5 j 2AC AB3

2| OC |

1. 9

113 2. 9

98 3. 9

193 4. 9

159

AB OB OA (2i 5 j) (3i 2 j) i 7 j

2 2AC AC ( i 7 j)3 3

OC OA AC

2 7 8(3i 2 j) ( i 7 j) i j3 3 3

2 49 64 113| OC |9 9 9

1.


13. u 4i 3 j,| v | | u | | u v | 8 u.v 1. 6 2. 7 3. 12 4. 14

u 4i 3 j 2 2| v | | u | 4 3 5 2 2 2u v u 2u v v 2 2 28 5 2u v 5

64 25 25u v 72

2.

14. u,v,w u 5v 2w 0 u 3i 4 j u v u w | w | cos 1. 2 2. 2.5 3. 3 4. 3.5

u 3i 4 j 2 2| u | 3 4 5 u v u v 0 u 5v 2w 0 ..(1)

u 5v 2w 2 2| u 5v | | 2w | 2 225 | v | 4 | w | 25 (2) (1) 5v 2w u 2 2 225 | v | 4 | w | 4w u | u | ....(3)

(2)=(3) 50| w | cos 2.520

2.


2 3 1u , v , w3 0 4

1. 2 3 5u v3 0 3

2. 2 1 2 2 0u 2w 23 4 3 8 5

3 . u v 2 3 3 0 6

4. 1 1 6

u v w 2 3 3 0 64 4 24

5. u v w

6. 22u 2 3 13

7 . 2 2v 3 0 3

8. Cos u v u v u v cos

u v 6 2cos3 13 13u v

9 . 2 3 1 2 4

u v w . 8 12 203 0 4 3 4

10. 2 1 2 3u v u w . 2 12 6 0 203 4 3 0

11. 1 u

2

3u 2 i 3 j

13 13 13u

12. 3 v

3

0v3 3 3i

3v


(Projection Vectors)

y

x

v

u

O

A

OA (Projection Vectors) v u

2

u.vOA u

u

1. O , OA 3i 4 j,OB 5i 2 j A OB D OD

1. 7 5i 2 j29

2. 7 5 i 2 j29

3. 8 5i 2 j29

4. 8 5 i 2 j29

A

B

D

O

OA 3i 4 j, OB 5i 2 j 22OB 5 2 29

OB.OA 5 3 2 4 7

2

OB.OAOD OB

OB


2

7OD 5i 2 j

29

7OD 5i 2 j29

2. OA 3i 4 j, OB 12 i 5 j BC OA C OC

1. 5 3i 4 j16

2. 16 3i 4 j5

3. 16 3i 4 j25

4. 25 3i 4 j16

A

B

C

O

OB 12i 5 j OA 3i 4 j 22OA 3 4 5

OB.OA 12 3 5 4 16

2

OB.OAOC OA

OA

2

16OC 3i 4 j

5

16OC 3i 4 j25


3 . OA 3i 4 j, OB 12 i j AC OB C OC

1. 8 12 i j29

2. 29 12 i j8

3. 40 12 i j145

4. 140 12 i j45

y

x

A

O

C

B

OA 3i 4 j

OB 12 i j 2 2OB 12 1 145 OB.OA 12 3 1 4 40

2

OB.OAOC OB

OB

240

OC 12i j

145

OC 40 12 i j145


C AB

O

m n

vu

w

nu mvwm n

2 2B x , y 1 1A x , y

m n

C x, y

2 1mx nxx

m n

2 1my nyym n

1. ABC D AB AB AD : DB 3: 2 CA 3i 2 j, CB 2 i 3 j CD

1. 95

2. 115

3. 135

4. 145

C

AB

D

3 2

CA 3i 2 j CB 2 i 3 j

3CB 2CACD

5

3 2 i 3 j 2 3i 2 j

5

6i 9 j 6 i 4 j

5

12

i j5

2

212 13CD 15 5


2. A,B,C O OA 3i 2 j

OB 2 i 5 j 2AC AB3

2

OC

1. 1139

2. 989

3. 1939

4. 1539

CAB

O

2 1

OA 3i 2 j OB 2 i 5 j

2OB 1OAOC

2 1

2 2 i 5 j 1 3i 2 j

2 1

2 12 i 5 j 3i 2 j

3 3

7 8

i j3 3

2 2

2 7 8 113OC

3 3 9

3. A 3, 2 ,B 9,4 O 0,0 AB 3 C D OC OD

C

O 0,0

3OA

2

9OB

4

D A 3, 2 B 9,41 1 1

3OA

2

9OB4


3 9 52OA 1OB 2 1OC

2 4 02 1 3 3

3 9 71OA 2OB 1 2OD

2 4 22 1 3 3

OC OD 5 7 0 2 35

4 . A 1,1 ,B 4,10 ,C 7,9 D AB AD 2

3AB

CA DC cos

1. 25

2. 210

3. 25

4. 210

D

2 1

A 1,1

C 7,9

B 4,10

6CA

8

3CB

1

6CA

8

3CB1

1CA 2CBCD3

6 31 2

48 1CD

23

4DC

2

CA DC cos

6CA8

2 2

CA 6 8 10

4DC

2

2 2DC 4 2 20

CA.DC 6 4 8 2 40


CA.DC CA . DC cos

CA.DC CA . DC cos

40 10 20 cos 2

cos5

5 . A B 10,0 2,4

AB C AC 1

3CB

O COB

1. 210

2. 110

3. 110

4. 210

C AB

O

3 1

OA 10 i OB 2 i 4 j

1OB 3OAOC

4

1 2 i 4 j 3 10 i

4

7 i j

7 2 1 4OC OB 1

cos50 20 10OC OB


6. ABC D BC BD : BC 1:3

. 3AD 2AB BC . 21

AD BC BC6

1. . . 2. . . 3. . . 4. . .

A

BC2 1

D . 2AB 1ACAD

3

3AD 2AB 1AC .

. . 2 1AD BC AB AC BC3 3

2 1AB BC AC BC3 3

A

B

DC

2 1

120

60

aa

2 1AD BC AB BC AC BC

3 3

2 1AD BC AB BC cos AC BC cos3 3


a

22 1 1AD BC a a cos120 a a cos 60 a3 3 6

. 3. 7 . A,B, C D BC

BD : DC 2 :1 2 2 2

AD a AB b AC c AB AC a,b,c

AB AC 0 2 2 2a b c

1. 3181

2. 3281

3. 1027

4. 1127

A

B C2 1

D 1AB 2ACAD

3

1 2AD AB AC3 3

1 2

AD AB AC3 3

2 2 21 4 4

AD AB AB AC AC9 9 9

2 2 2

2 2 2 1 4 4 11a b c9 9 9 27


3.5

1

2

3

a

u a

a

1

2

3

b

v b

b

2 3 3 2

3 1 1 3

1 2 2 1

a b a b

a b a b

a b a b

2 3 1 3 1 22 3 1 3 1 2

a a a a a ai j k

b b b b b b

u v

1 2 31 2 3

i j k

u v a a a

b b b

1. u,v,w k 1) u v v u 2) u v w u w v w 3) u v w u v u w 4) u kv k u v 5) ku v k u v 6) u u 0 7) i j k , j k i , k i j 2. u,v,w u v w u v w 3. u 0 v 0 u v u v sin u v 0 180 4. u v u v u v


1. 4

u 2

4

6

v 3

0

u v

u v = i j k

4 2 4

6 3 0

= 2 4 4 4 4 2i j k3 0 6 0 6 3

= ( 2 0 3 4 ) i ( 4 0 4 6 ) j ( 4 3 2 6 k = 12 i 24 j 0k = 12 i 24 j

v

u

v sin

v

u

v sin

u v u v sin 1 1u v u v sin2 2

v r

u

h r

v u , v r h u v r u v r h u cos v r


= v r u cos = u v r cos u v r

u , v w )wv(u = 0

1 2 3 1 2

1 2 3 1 2

1 2 3 1 2

u u u u u

(u v) r v v v v v

w w w w w

1. ABCD AB 2 i j k AD i j 2k AB AD

i j k

AB AD 2 1 1

1 1 2

1 1 2 1 2 1i j k1 2 1 2 1 1

= 1 2 1 1 i 2 2 1 1 j 2 1 1 1 k = i 3 j k 2 2 2AB AD 1 3 1 11

2. u i 2 j 3k , v 2 i 3 j , r j 2k u v r

i j k

v r 2 3 0

0 1 2


= 3 0 2 0 2 3i j k1 2 0 2 0 1

= 6 i 4 j 2k u v r i 2 j 3k 6 i 4 j 2k

= 1 6 2 4 3 2 = 686 = 20 = 20 3 u i 2 j , v 3i j 5k , r 2 i 7k

1 2 0 1 2

u v r 3 1 5 3 1

2 0 7 2 0

7 20 0 0 0 42 55 5555

4. 3 A, B C A i 2j k , B 21 3k C 7j 4k

1 2 1 1 2

(A B) C 2 0 3 2 0

0 7 4 0 7

(A B) C -23 23

Vector 3 Axis

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