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# 经济数学-向量及其线性运算.ppt_图文

M2
(vector).

a M1M2

M1

M1M2 .

M OM .

.

.

a

b

a .

a

a

a

0,

b 0,

b

a

b

a

(a,b) (b,a)

(0 )

.
0 .

M0M a

M
z

0

(

x0

,

y0

,

z0

)M
ax

( x,
x

y,

z)
x0

x

az

M

a y y y0

M 0
ax

ay

O

y

y

az z z0
z

x

z

x

az

M 0
ax

o

M
a y

ax | M0M | cos
y a y | M0 M | cos az | M0M | cos

.

ax2 a y2 az2

M 0M ax, ay, az

a {ax , a y , az }
M0M { x x0 , y y0 , z z0 }
OM { x, y, z}

(modulus)

| a | | M1M2 |

| a |

a2 x

a2 y

a2 z

(unit vector)

1 a 0 M1M20

a(direction angle)

.
z

az

M

M 0
ax

ay

o

y

0 , 0 , 0 .

x

ax2 a y2 az2 0

cos

ax

,

ax2 ay2 az2

cos

ay

,

ax2 ay2 az2

cos

az

.

ax2 ay2 az2

:
a0 {cos, cos , cos }

1 P1P2 P1P2 2 x

y

3

4

P1

(1,0,3) P2.

P1P2

, cos 1 , , cos 2 ,

3

2

4

2

cos2 cos2 cos2 1, cos 1 . 2

, 3

2 . 3

P2( x, y, z),

cos

x1 P1 P2

x1
2

1 2

x 2,

cos

y0

y0

2

y

2,

P1 P2

22

cos z 3 z 3 1 z 4, z 2,

P1 P2

22

P2 (2, 2,4), (2, 2,2).

1. a b c

b

c

a

a

b

a b

a

b

c | c || a

|

|

b

|

|

c

|

|

ac|

|

b

|

1

a

b

b

a.

(commutativity)

2

a

b

c

(a

b

)

c

a

(b

c

).

(associativity)

3

a

(a)

0.

c

a

b

a

(b)

a

b

b

a

a

b

b

b c

a

b

a {ax , ay , az }, b {bx , by , bz },

a

b

{a

x

bx ,

ay

by ,

az

bz }

a

b

{a

x

bx ,

ay

by ,

az

bz }

a a

(1) 0, aa | a | | a |

(2) 0,

a

0

(3) 0, aa | a || | | a |

a 2a 1 a 2

1(a) ( a) ()a

2 (

)a

a

a

(a

b)

a

b

a {ax , ay , az },
a {ax , ay , az }

a0

|

a a

|

a

0

b

a

b

a.

b

b

a

a

b a

,

b

a

b

a.

b .

a . a

b

a

b

a

aba a

b.

( )a 0 a 0

a 0 0 .

2 A( x1 , y1 , z1 )B( x2 , y2 , z2 ) AB M AB

AM MB

( 1) AM .
MB

M( x, y, z) z

B

AM {x x1, y y1, z z1} A M
o

y

MB {x2 x, y2 y, z2 z} x

AM MB

{ x x1, y y1, z z1} { x2 x, y2 y, z2 z},

x

x1 ( x2

x)

x

x1 x2 1

,

y

y1 ( y2

y)

y

y1 y2 , 1

z z1 (z2 z)

z z1 z2 , 1

M AB . M

x x1 x2 , y y1 y2 , z z1 z2 .

2

2

2

z

k

az

M

a

M0
x

ay

o

j

y

x

i

i , j , k x, y, z .

axi a y jazkxyz.

3

a

6i

7

j

6k

.
a

| a | 62 72 (6)2 11,

a0

|

aa

|

6

i

11

7 11

j

6

k,

11

a0

|

a a

|

6

i

7

j

6

k.

11 11 11

p 45i mj34ik5j 8k a n4m2i

3n4

j p7kx

y .

a 4m 3n p

4(3i 5 j 8k ) 3(2i 4 j 7k )

(5i j 4k ) 13i 7 j 15k,

x ax

13,

y 7 j .

1.

m

i

j

n

2 j k

m , n.

1

n

| m n |, | m n |,

m

m n {1,1,1}, m n {1,3,1}

| m n | 3, | m n | 11,

3, 11 .

2

D b
A

a

C

M
B

BC

AM

MD

1

(a

b ).

2

DC

AB

AM

MB

1 2

(a

b ).

1. _________ 2. ___________ 3. ___________ 4. _____________ 5. _____ 6. _________ ___ _________
7.___________ 8.____________
9. ____________

10.

11.ab a b_________a_,_b________________

12.____a___b____a___b___a,

b

_______

___________ .

13. M1 (0 , 1 , 2) M2 (1 ,1 , 0) M1M2

{ }-2 M1 M2={

}

14. M1 (4 , 2 , 1) M2(3 , 0 , 2),

M1M2 ________ M1M2 =_________

cos =_____cos =____cos =_____

_____ , _____ , ______

15.b 0c=____2_ai___ij__2j kc,k0=,_ba__0_2___i______3___j_____5_k____
16. xOy, yOz, zOx , , cos2 +cos2 +cos2 =____________ .
.
ABC BC D1 , D2 , D3 , D4 A
AB c, BC a D1 A , D2 A , D3 A D4 A .

B(2 ,1 , 7) X Y Z 4 ,4 7 A .
a 6 , 7 ,6 .

1.,

2.

3. 1 4. 5.

6., 7.

8.

9. 1

1110..a b212. a

b

.

13. 1,2,2, 2,4,4

14. 1, 2,1,2, 1 , 2 , 1 , 2 , 3 , ;
2 223 43

15.

1, 3

1, 3

1 3

,

2 , 38

3, 38

5 38

,

2 3

,

1 3

,

2 3

16. 2.

D1

A

(c

1 5

a)

D3

A

(c

3 5

a),

DD4 2AA(c(c5452a)a.)

(-2,3,0)

A

6 11

,

7 11

,

161

6 11

,

7 11

,

6 11

.