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VECTOR ALGEBRA

ANALITICAL GEOMETRY; LINEAR ALGEBRA

VECTOR ANALYSIS ; DIFFERENTIAL AND INTEGRAL CALCULUS

2

(Green, Divergence, Stokes)

20

Â Â

10

15

- n

n = 1, 2, 3

A

A

ijk ,,

nRnC

A

AA

T

),...,(1 n

aan

3

A

A

A

20

20

)

= Scalar Triple Product = Vector Triple Product

(

Dot Product

(-

Vector Product

Scalar Triple

Product

Vector Triple

Product 35

T

4

= = Sin )

Cos

5

(Projectile)

5

Hyperbola- Parabola-Ellipse-Circle

30

tvtgy

tvx

)sin()2/1(

)cos(

0

2

0

bxaxy 2

T3

5

5

5

Cylinder

5

sphere

5

Ellipsoid

The second degree surfaces in 3-space

(Quadric Surfaces) T4

12

2

2

2

2

2

c

z

b

y

a

x

dczbyax

0222 lkzhygxxyexzdyzczbyax

2222 )()()( Rczbyax

cba 2,2,2.

00 ,...., zzcz

2

2

2

02

2

2

2

02

2

)1()1(

R

c

zb

y

c

za

x

x, y, z

6

Algebraic Quadratic Equation

5

a=c a=b b=c

5

a=b

(Cone)

15

Hyperboloid of one sheet

12

2

2

2

2

2

c

z

b

y

a

x

12

2

2

2

2

2

c

z

b

y

a

x

2

2

2

2

2

2

c

z

b

y

a

x

12

2

2

2

2

2

b

y

a

x

c

z

T4

7

Hyperboloid of two sheets

Rotating Ellipsoid

20

(Elliptic Paraboloid)

(Hyperbolic Paraboloid)

15

c

z

b

y

a

x2

2

2

2

c

z

a

x

b

y2

2

2

2

zyyxx 22 22

xyz 2

'

'

cos

sin

sin

cos

y

x

y

x

T4

8

cos1r

20

60

a =

Cardioid :

Polar Coordinates

Cylindrical

Coordinates

Helix

T

),(r

r

sin

cos

bar

bar

sin2,cos2 arar

ar

4/

2

2

r

r

z

bez

arz

9

Z

40

Torus

40

Spherical Coordinates.

30

)cos1(ar

)ln( 22 yxz

12

2

2

2

c

z

a

r

sin4cos2r

1)4( 22 rz

)(zfr

)cos1(

sin

zr

zr

),,(

c

),,(),,,(),,,( zyxzr

0,0

cos2),cos1(a

T

10

AB C

OBOAOC)(

:

^

12

^

12

^

122122221111)()()(),,(),,,( kzzjyyixxPPzyxPzyxP

n , x xn )

T

11

A

B

µ

),,( zyxP

:

3

n

A

A

12

kji ,,

14

= ( ) , = ( = ( , )

t 1 t

10,1 tbtat

ABC

BCDE2

ABAC

DE

DE

AB

OBOAOC)(

:

a b AB

CAB

ABC

C

ABba AB

OC

15

))(1()1(

)(])1[(

1,

batbbtatOBOCBC

batabtatOAOCCA

tt

,CAB

16

A

B

B A

=( a1 , a2 , a3 ) AAB = ( b1 ,b2 , b3 )C = (c1 , c2, c3 )

a a A = ( a a1 , a a2 , a a3 )

CBCACBA ..).(

0

17

t

vbaab

wbaab

vb. va.v

w v

a

ab

jitajib 5432

N

sinba

18

( -1AB A

A

OCBBB

BAproj

B

B

B

BA

B

Bpproj

A

B

A

B

.

.

.

( -2

CBCACBA ..).(

ABC

A

B

c

A

BprojACorByAC

19

M N P

0).(. BABC projA

B

MP = MN + NP

)( BA

C B AC C

i=1,2,3,4,5 Pi

P4,P5 M

0)N

N.PP)(

N

N.PP( 5141

20

P1

M

P P5 P P

N

21

3i + jabai + j

ba

( i+ j ) = 0a = t( i + j ) , b .a + b = 3i +j ( i + j )

t = 2

a = 2i + 2j b = 3i + j a = i-j

b = 3i + j a = i - j

,,

x

y

z

Au

^^^

coscoscos kjiu

cAkA

bAjA

aAiA

cos.

cos.

cos.

^

^

^

^^^

kcjbiaA

222

222

222

cos,

cos,

cos

cba

c

cba

b

cba

a

22

bi

j

k

Ai...........jb

...........kc

^^^

kcjbiaA

,^

A

Au

a

1coscoscos 222

A=( a1 , a2 , a3 ) B = ( b1 ,b2 , b3 )

A B

CABACBA )(

^^^

,, ijk

33

A

B BAS

23

A B

A B c

û

24

BBSinBBA

BASinBA

B

BSinBABA

CABACBACBCB

CABACBACBCB

CABACBACBCB

CBACBACACABABACABACBA )(,,)(

A

B

C

a

b

c

c

C

b

B

a

A sinsinsin

25

bababa 2

26

4

5sin

2

5)(5

2

1)(6)(

2

123

2

1babaabbababaS

baba 2,3 ba ,

:

0])([])(([)]([ bacacbcba

dcba ,,,0)()( dcba

cba ,,

).)(.().)(.()(.)(,,, cbdadbcadcbadcba

dcbacdbadcba )](.[)](.[)()(

0)](.[)](.[)](.[)](.[ dcbacdbabdcaadcb

adacabaa 321

dcb

cbaa

dcb

bdaa

dcb

dcaa

.

.,

.

)(.,

.

)(.321

dcb ,,

o

A

B

C

ab

c

daccbba

A B C

BCABdbcabd

bbdcabbabb

)()(

0)(

27

: "

=

).( CBA

33

ABC0. CBA

ACBBCACBA ).().()(

28

).( CBA

= ( a1 , a2 , a3)

= ( a1 , a2 , a3) ,

ABCD ,,,

CABA

DCDBCBAD

..

..)).((

29

jiu sincosjiv sincos0

cossincossinsin

w vu

wvuV6

1

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