3 Eucliden Vector

Three

Euclidean Vector Spaces

3.1 (space)

( , )x y ( , , )x y z 3 (3-space) 3

3 ( Coordinate axes) X XY XZ Y XY YZ Z YZ XZ (Origin) O

3.1.1

1 1 1( , , )P x y z

2 2 2( , , )Q x y z

2 2 2

2 1 2 1 2 1( ) ( ) ( )PQ x x y y z z

n n

| Euclidean Vector Spaces

2 3

3.1.2 , ,x y z ( , , )x y z 3 ( , , )x y z O ( , , )x y z

3.1.3

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

1 2P P

2 1 2 1 2 1

( , , )x x y y z z

3.1.4 3

1 2 3( , , )u u u u

1 2 3( , , )v v v v 3

u v 1 1 2 2 3 3, ,u v u v u v

1 1 2 2 3 3( , , )u v u v u v u v k

1 2 3( , , )ku ku ku ku

(Standard operations) 3

3 0 (0,0,0)

1 2 3( , , )u u u u u

1 2 3( , , )u u u u

3 3.1.5 u , v w 3 ,k m

1. u v 3 2. u v v u 3. ( ) ( )u v w u v w 4. 0 3 0 0u u u 5. u u 3 ( ) ( ) 0u u u u 6. ku 3 7. ( ) ( ) ( )km u k mu m ku 8. ( )k u v ku kv 9. ( )k m u ku mu 10. 1u u

| Page 77

infinite dimension n n -space 2 3-space

n (Vectors in n-Space) 2-3 3 () n 3.1.6 n n (ordered n-tuple) n

1 2( , ,..., )

na a a n

n (n-space) n

1 2

( , ,..., )n

a a a n n 3.1.7

1 2( , ,..., )

nu u u u

1 2( , ,..., )

nv v v v n

u v 1 1 2 2, ,..., n nu v u v u v

1 1 2 2( , ,..., )n nu v u v u v u v k

1 2 3( , ,..., )ku ku ku ku

(Standard operations) n

3 0 (0,0,...,0)

1 2

( , ,..., )n

u u u u u

1 2( , ,..., )

nu u u u

3 n


n Properties of vector Operations in n-Space 3.1.8 u , v w n ,k m

1. u v n 2. u v v u 3. ( ) ( )u v w u v w

4. 0 n 0 0u u u 5. u u n ( ) ( ) 0u u u u

6. ku n 7. ( ) ( ) ( )km u k mu m ku 8. ( )k u v ku kv 9. ( )k m u ku mu 10. 1u u

3.1.9 v n k 1. 0 0v 2. 0kv 3. 1v v

(Linear Combinations) 3.1.10 w

n w 1 2, ,..., nv v v n

1 1 2 2...

n nw k v k v k v

1 2, ,...,

nk k k

1 1 2 3, ,c c c

1 2 3(1, 1,0) (3,2,1) (0,1,4) ( 1,1,19)c c c

| Page 79

3.1 1. (1,3,3)u , ( 1,0,3)v (1, 2,3)w

a. 2u v

b. u w v

c. 2 3v w 2. 1 x 2 7u v x x w 3. , ,a b c ( 2,9,6) ( 3,2,1) (1,7,5) (0,5,4)a b c 4. , ,a b c (1,2,0) (2,1,1) (0,3,1) (0,0,0)a b c


3

1. (Inner Product) 2. ( Outer

Product) 3. (Cross Product)

3 () ( Outer Product) ( Dyad Product)

3.2

3.2.1

1 2( , ,..., )

nu u u u

1 2( , ,..., )

nv v v v n

(Euclidean inner product) u v u v

1 1 2 2

1

...n

i i n ni

u v u v u v u v u v

2n 3n (dot product)

- 2u u u - Tu v v u

(Properties of Euclidean inner product) 3.2.2

1 2 1 2( , ,..., ), ( , ,..., )

n nu u u u v v v v

1 2( , ,..., )

nw w w w

n ,k m 1. u v v u 2. ( )u v w u v u w 3. ( ) ( ) ( )k u v ku v u kv 4. ( ) ( ) ( )ku mv km u v 5. 0u u 0u u 0u

1 2

( , ,..., )n

u u u u , 1 2

( , ,..., )n

v v v v

1 2

( , ,..., )n

w w w w n ,k m 1. u v v u 2. ( )u v w u v u w

| Page 81

3

3. ( ) ( ) ( )k u v ku v u kv 4. ( ) ( ) ( )ku mv km u v 5. 0u u 0u u 0u

3.2.3

1 2( , ,..., )

nu u u u n u

u

2 2 2 21 2

...n

u u u u u u u

3.2.4

1 2( , ,..., )

nu u u u

1 2( , ,..., )

nv v v v n

( Euclidean distance) u v d( , )u v

d( , )u v u v

2 (2,3, 1,7), (1,0,3, 5)u v d( , )u v

| Euclidean Vector Spaces n () (norm) ( Euclidean norm)

Cauchy-Schwarz 3.2.5 Cauchy-Schwarz

1 2( , ,..., )

nu u u u

1 2( , ,..., )

nv v v v n u v u v

3.2.6 u v

n k

1. 0u

2. 0u

0u

3. ku k u

1 2

( , ,..., )n

u u u u 1 2

( , ,..., )n

v v v v n k 1. 0u

| Page 83

2. 0u 0u

3. ku k u 3.2.7 u v

n k u v u v


3.2.8 Parallelogram Equation for Vectors

u v n

2 2 2 22u v u v u v

3.2.9 u v

n 2 21 1

4 4u v u v u v

3.2.10 ,u v w

n k 1. d( , ) 0u v

2. d( , ) 0u v u v 3. d( , ) d( , )u v v u 4. d( , ) d( , ) d( , )u w u w w v

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3.2 1.

a. ( 4,3)u b. (2,2,2 2)v c. (5,6,0)w

2. (unit vector)

a. u n u

b. u n u 3. (1, 1, 4,2,3)u (2,3,4,5,6)v 4. ,u v w

n k (1.) d( , ) 0u v (2.) d( , ) 0u v u v (3.) d( , ) d( , )u v v u (4.) d( , ) d( , ) d( , )u w u w w v


3.3 (Orthogonality) (dot product) 2-3

cosu v u v

u v n

cosu v

u v

3.3.1 u v

n 0u v

3 ( 2,3, 1,4), (1,2,0, 1)u v 3.3.2 u v

n 2 2 2

u v u v

| Page 87

3.4 3 L 3

0 0 0 0( , , )P x y z

( , , )a b cv L ( , , )P x y z L 0P P

v t

0PP t v

0 0 0( , , ) ( , , )x x y y z z ta tb tc

0 0,x x ta y y tb 0z z tc

3.4.1 3 , ,a b c

0 0 0, , ( )x x ta y y tb z z tc t

0 0 0 0( , , )P x y z ( , , )a b cv

0 0 0

, , ( )x x ta y y tb z z tc t 1

0(2,2, 3)P

(4,5, 7) v 2 (2,4, 1)A

(5,0,7)B XY


( , , )x y zr ( , , )P x y z 0 0 0 0( , , )x y zr

0 0 0 0( , , )P x y z 0PP 0r r

t 0

r r v t

0r r v

0 0 0x x y y z z

a b c

3.4.2 n

0r v 0

n t t

0r r v

0r v

3

0( 1,2)P ( 2,3) v

| Page 89

4

0(1,2, 3)P (4, 5,1) v

5

4 11 393 7 26

2 8 26

x y z

x y z

x z

6 3

4


( inclination) (normal vector)

0 0 0 0( , , )P x y z ( , , )a b cn

(normal vector) ( , , )P x y z

0P P

n

00PP n

0 0 0( ) ( ) ( ) 0a x x b y y c z z

point-normal form

3.4.3 3 , ,a b c

0ax by cz d ( , , )a b cn (normal vector)

0ax by cz d , ,x y z 7 (1, 2,5)

(4,2, 3) n

3 (normal vector) 3.4.4 ( cross product vector product)

1 2 3( , , )u u u u

1 2 3( , , )v v v v u v

2 3 3 2 3 1 1 3 1 2 2 1( , , )u v u v u v u v u v u v u v

3.4.5 u v 3 u v u v

| Page 91

8 1 2(1,2, 1), (2,3,1)P P

3(3, 1,2)P

( , , )x y zr ( , , )P x y z 0 0 0 0( , , )x y zr

0 0 0 0( , , )P x y z 0PP 0r r

1 2, v 0 v 0

1 1 2 2t t

0r r v v

3.4.6 n

0r 1 2, v 0 v 0

n 1 2,v v

1 2,t t

1 1 2 2t t

0r r v v

0r 1 2,v v


9 9 5 16 0x y x

10 2 5x y z

11

9 5 1618 2 10 32

27 3 15 48

x y z

x y z

x y z

| Page 93

12 1z y z

2 3 1z y z

13 1z y z

2 3 1z y z

3.4.7

0 0 0 0( , , )P x y z 0ax by cz d D

0 0 0

2 2 2

ax by cz dD

a b c

14 (1, 4,3) 2 3 6 1 0x y z


3.4 1. P u

a. ( 4,1), (0, 8)P u

b. ( 9,3,4), ( 1,6,0)P u

2. (4,0, 5)u

3.

3 2 06 4 2 0

3 2 0

x y z

x y z

x y z

3 2 26 4 2 4

3 2 2

x y z

x y z

x y z

| Page 95

2( )f x x ( real valued function of real variable) x

2 2( , )f x y x y ( real valued function of two real variables)

4.4 n m

n

2( )f x x 2 2( , )f x y x y 2

2 2 2( , , )f x y z x y z 3 n m f n m f

(transformation map) n m : n mf 1m m n

: n nf n (operator n )

1 2 3, , ,..., mf f f f n

1 1 1 2

2 2 1 2

1 2

( , ,..., )

( , ,..., )

( , ,..., )

n

n

m m n

w f x x x

w f x x x

w f x x x

1 2( , ,..., )nx x x n 1 2( , ,..., )mw w w

m m T

: n mT

1 2 1 2( , ,..., ) ( , ,..., )n mT x x x w w w

15

1 1 2

2 1 2

3 1 2

w x x

w x x

w x x

2 3 2 3:T 1 2 1 2 1 2 1 2( , ) ( , , )T x x x x x x x x (1,2) (3,2, 1)T

| Euclidean Vector Spaces Some Notational Matters

A T

( )AT Ax x

T

( )T T x x

AT A

n m 1 2 1 2( , ,..., ) ( , ,..., )n mT x x x w w w 1 2 3, , ,..., mf f f f

1 1 1 2

2 2 1 2

1 2

( , ,..., )

( , ,..., )

( , ,..., )

n

n

m m n

w f x x x

w f x x x

w f x x x

: n mT n m

: n mT

1 11 1 12 2 1

2 21 1 22 2 2

1 1 2 2

...

...

...

n n

n n

m m m mn n

w a x a x a x

w a x a x a x

w a x a x a x

1 11 12 1 1

2 21 22 2 2

1 2

n

n

m m mn nm

w a a a xw a a a x

a a a xw

Aw x

ijA a T

16 4 3:T

1 1 2 3 4

2 1 2 3 4

3 1 2 3 4

2

3

4

w x x x x

w x x x x

w x x x x

11

22

33

4

2 1 1 13 1 1 14 1 1 1

xw xw xw x

| Page 97

1 2 (operator) y

x

y x

2 3 (operator) xy

xz

yz

2 3


3 (projection) 2 (operator) (orthogonal projection) x

(orthogonal projection) y

4 (projection) 3 (operator) (orthogonal projection) xy

(orthogonal projection) xz

(orthogonal projection) yz

| Page 99

5 2 (operator)

6 3 (operator) x

y

z


7 2 (operator)

k 2 (0 1)k

k 2 ( 1)k

7 3 (operator)

k 3 (0 1)k

k 3 ( 1)k

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4.3 1. A T

3 2:T

(1,0,0) (7,11), (0,1,0) (6,9)T T (0,0,1) ( 13,17)T

2. 2 3:T

11 2

2

1 4

2 5

3 6

xT x xx

3. 1 2, , , n

mv v v : m nT

1

2

1 1 2 2 m m

m

x

xT x v x v x v

x

(

1 2, , ,

mv v v )

4. 2 2:T 45 T

5. 2 2:T T

6.

1 1 22 1 2

7

3 20

y x x

y x x

7. A T (linear operator)

a. 3 0

0 3A

b. 1 0

0 1A

c. 1 1

1 1A

d. 1 0

0 0A

e. 0 2

2 0A

f. 0.8 0.6

0.6 0.8A


8. ( )T x Ax

a. 0 1

1 0A

b. 2 0

0 2A

c. 1 0

0 2A

d. 0 0

0 1A

e. 0.6 0.8

0.8 0.6A

1

1

| Page 77

T:R4 R3 T(a1,a2,a3,a4) =(a1,a1+a3,a3+a4) S4 , S3 R

4 ,R3 S = {(1,0,0,1),(0,0,0,1),(1,1,0,0),(0,1,1,0)} , S 1 ={(1,1,0),(0,1,0),(1,0,1)} R4 , R3 T S S3 T S S 1 T(2,1, 1,3) 1.1 1.2 T T:R2 R2 T(x,y) =(x+2y,2x3y) T {(1,0),(0,1)} {(2,1),(1,0)} {(1,1),( 2,3)} {(1,1),(0,2)} T:R2 R3 T(x,y) = (x+y , 2xy , x+2y) T 3.1 {(1,1),( 1,2)} {(1,0,0),(1.0.1),(0,1,1)} 3.2 {(1,0),(0,1)} {(1,0,1),(0,1,1)(1,1,0)} 4. A B 4.1 A = {(1,0),(0,1)} ; B = {(1,1),( 1,2)} 4.2 A = {(1, 1),(2,1)} ; B = {(1,2),(3,3)} 4.3 A = {(1,1),( 1,2)} ; B = {(1,0),(0,1)} 4.4 A = {(1,2),( 3,1)} ; B = {(1, 1),(2,1)}

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