Аналитическая геометрия и линейная алгебра. Учебное пособие

Ɉɩɪɟɞɟɥɟɧɢɟ Ȼɚɡɢɫɨɦ ɬɪɟɯɦɟɪɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ

G G G

ɥɸɛɵɟ ɬɪɢ ɩɨɩɚɪɧɨ ɨɪɬɨɝɨɧɚɥɶɧɵɟ ɜɟɤɬɨɪɚ ɟɞɢɧɢɱɧɨɣ ɞɥɢɧɵ i j k .

G

ɍɬɜɟɪɠɞɟɧɢɟ Ʌɸɛɨɣ ɜɟɤɬɨɪ a ɬɪɟɯɦɟɪɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɦɨɠɧɨ

ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ ɥɢɧɟɣɧɨɣ ɤɨɦɛɢɧɚɰɢɢ ɛɚɡɢɫɧɵɯ ɜɟɤɬɨɪɨɜ > 11]

G

G

G

G

a xi y j zk .

G

ɑɢɫɥɚ x y z ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɤɨɨɪɞɢɧɚɬɚɦɢ ɜɟɤɬɨɪɚ a ɜ ɞɚɧɧɨɦ ɛɚɡɢɫɟ

ɉɭɫɬɶ ɜ ɬɪɟɯɦɟɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɚɞɚɧɚ ɞɟɤɚɪɬɨɜɚ ɩɪɹɦɨɭɝɨɥɶɧɚɹ ɫɢ

G G G

ɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ ɛɚɡɢɫɧɵɟ ɜɟɤɬɨɪɚ i j k

ɩɪɢɥɨɠɟɧɵ ɤ ɧɚɱɚɥɭ ɤɨɨɪɞɢɧɚɬ

ɬɨɱɤɟ O ɢ ɫɨɜɩɚɞɚɸɬ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɫ ɤɨɨɪɞɢɧɚɬɧɵɦɢ ɨɫɹɦɢ Ʉɨɨɪɞɢɧɚɬɵ

JJJG

G G G

ɜɟɤɬɨɪɚ OM

ɜ ɛɚɡɢɫɟ i j k – ɷɬɨ ɤɨɨɪɞɢɧɚɬɵ ɤɨɧɰɚ ɜɟɤɬɨɪɚ M ɜ ɡɚɞɚɧɧɨɣ ɫɢ

ɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ

G

ɉɭɫɬɶ ɜ ɬɪɟɯɦɟɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɚɞɚɧɨ ɞɜɚ ɜɟɤɬɨɪɚ

a x1 y1 z1 ɢ

G

ɥɟɝɤɨ ɜɢɞɟɬɶ ɱɬɨ

G

G

G

G

G

G

G

G

a b (x1i y1 j z1k) (x2i y2 j z2 k)

G

G

G

(x1 x2)i ( y1 y2 ) j (z1 z2)k .

Ɍɚɤɢɦɨɛɪɚɡɨɦ ɩɪɢGɫɥɨɠɟɧɢɢɞɜɭɯɜɟɤɬɨɪɨɜɢɯɤɨɨɪɞɢɧɚɬɵɫɤɥɚɞɵɜɚɸɬɫɹ

ɍɦɧɨɠɢɦ ɜɟɤɬɨɪ a x y z

ɧɚ ɱɢɫɥɨ B

G

G

G

G

G

JG

G

Ba B(xi y j zk) (Bx)i (By) j (Bz)k .

ɉɪɢ ɭɦɧɨɠɟɧɢɢ ɜɟɤɬɨɪɚ ɧɚ ɱɢɫɥɨ ɜɫɟ ɟɝɨ ɤɨɨɪɞɢɧɚɬɵ ɭɦɧɨɠɚɸɬɫɹ ɧɚ ɷɬɨ

ɱɢɫɥɨ

ɉɭɫɬɶ ɡɚɞɚɧɵ ɤɨɨɪɞɢɧɚɬɵ ɧɚɱɚɥɚ A x1 y1 z1

ɢ ɤɨɧɰɚ B x2

y2 z2 ɜɟɤ

JJG

JJG

JJG

ɂɫɩɨɥɶɡɭɹ ɩɪɢɜɟɞɟɧɧɵɟ ɫɜɨɣɫɬɜɚ ɨɩɟɪɚɰɢɣ ɧɚɞ ɜɟɤɬɨɪɚɦɢ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ

JJG

JJG JJG

JJG

x1 y2

y1 z2 z1 .

AB ( 1)OA OB. ɉɨɥɭɱɚɟɦ ɤɨɨɪɞɢɧɚɬɵ ɜɟɤɬɨɪɚ

AB x2

G

G

G G

G

G

G G

(2.2.1)

a b

a

b

FRV·a b

31

G

G

ȼ ɫɥɭɱɚɟ ɟɫɥɢ ɜɟɤɬɨɪɚ a ɢ b ɤɨɥɥɢɧɟɚɪɧɵɟ ɢ ɢɦɟɸɬ ɨɞɧɨ ɧɚɩɪɚɜɥɟɧɢɟ

G

G

ɤɨɥɥɢɧɟɚɪɧɵɟ ɢ ɢɦɟɸɬ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɟ ɧɚɩɪɚɜɥɟɧɢɟ –

ɫɱɢɬɚɟɦ ·a b

G

G

·a b Q

ɋɪɚɡɭ ɢɡ ɨɩɪɟɞɟɥɟɧɢɹ ɜɵɬɟɤɚɸɬ ɫɜɨɣɫɬɜɚ ɫɤɚɥɹɪɧɨɝɨ ɩɪɨɢɡɜɟɞɟɧɢɹ

1)

G G

G G

;

a b b a

2)

JJG G

G

G

Ba b B a b ;

G

3)

G

G

G G

a a p

a a

ɬɨɥɶɤɨ ɟɫɥɢ a – ɧɭɥɟɜɨɣ ɜɟɤɬɨɪ

G

G G

G G

G G

ɥɸɛɨɦɭ ɜɟɤɬɨɪɭ G G G

ɉɭɫɬɶ ɜɟɤɬɨɪɵ i

j k – ɛɚɡɢɫ ɬɪɟɯɦɟɪɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɬɨɝɞɚ

GG

G G

G

G

(2.2.2)

i i j j k k ;

G G

G G

G G

G G

G G

G G

(2.2.3)

i j i k j i j k k i k j .

G G G

G

ɉɭɫɬɶ ɞɜɚ ɜɟɤɬɨɪɚ ɡɚɞɚɧɵ ɫɜɨɢɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ ɜ ɛɚɡɢɫɟ

i j k

G

a x1 y1 z1 b x2 y2 z2 . ɇɚɣɞɟɦ ɫɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɷɬɢɯ ɜɟɤɬɨɪɨɜ

G G

G

G

G

G

G

G

a b x1i y1 j z1k x2i y2 j z2 k .

G G

a b x1x2 y1y2 z1z2 .

(2.2.4)

G

G G

a

a a x2 y2 z2 .

G

G

G G

a b

FRV·a b

G

G

32

a

b

JJG JJG

¸ ¸ ¸

cosC

CB CA

.

JJJG

JJJG

41 3

41

3

41

BC

AC

A Q B DUFFRV

38

C DUFFRV

3

.

41

41

a

a

¬

11

12

­

A

­.

a22

­

a21

®

a11

a12

a21

a22

ɢɥɢ det A ɢ ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ ɩɪɚɜɢɥɭ

det A a11a22 a12a21 .

ɉɪɢɦɟɪ

1

2

1¸4 2¸3 2 .

3

4

a

a

a

¬

11

12

13

­

­

a

a

­

A a

­.

21

22

23

­

­

­

a32

a33

­

a31

®

det A a

¸

a22

a23

a

¸

a21

a23

a

¸

a21

a22

11

a

a

12

a

a

13

a

a

32

33

31

33

31

32

34

det A a11 ¸a22 ¸a33 a11 ¸a23 ¸a32 a12 ¸a21 ¸a33

a12 ¸a23 ¸a31 a13 ¸a21 ¸a32 a13 ¸a22 ¸a31 .

(2.2.8)

ɉɪɢɦɟɪȼɵɱɢɫɥɢɦ ɨɩɪɟɞɟɥɢɬɟɥɶ

1 0

2 0

2 1

¸

¸

¸

1

3

2

3

2

1

2

1

3

¸ ¸ ¸

ȼɟɪɧɟɦɫɹ ɨɩɹɬɶ ɤ ɨɩɟɪɚɰɢɹɦ ɧɚɞ ɜɟɤɬɨɪɚɦɢ

G G G

Ɉɩɪɟɞɟɥɟɧɢɟ Ɍɪɨɣɤɚɜɟɤɬɨɪɨɜ a b c ɧɚɡɵɜɚɟɬɫɹɩɪɚɜɨɣ ɟɫɥɢ ɝɥɹɞɹ

G

G

G

ɢɡ ɤɨɧɰɚ ɜɟɤɬɨɪɚ c ɦɵ ɜɢɞɢɦ ɤɪɚɬɱɚɣɲɢɣ ɩɨɜɨɪɨɬ ɨɬ a ɤ b ɩɪɨɬɢɜ ɱɚɫɨɜɨɣ

ɫɬɪɟɥɤɢ

G

G

Ɉɩɪɟɞɟɥɟɧɢɟ ȿɫɥɢ ɜɟɤɬɨɪɚ a ɢ b ɧɟɤɨɥɥɢɧɟɚɪɧɵɟ ɬɨ ɜɟɤɬɨɪɧɵɦ

G

G G

ɩɪɨɢɡɜɟɞɟɧɢɟɦ ɷɬɢɯ ɜɟɤɬɨɪɨɜ ɧɚɡɵɜɚɟɬɫɹ ɬɚɤɨɣ ɜɟɤɬɨɪ c >a b@ ɱɬɨ

1)

G

G

G

G

G

c

a

b

VLQ·a b

G

G

G

ɜɟɤɬɨɪ c

ɨɪɬɨɝɨɧɚɥɟɧ ɤ ɤɚɠɞɨɦɭ ɢɡ ɜɟɤɬɨɪɨɜ a ɢ b

G G G

ɬɪɨɣɤɚ ɜɟɤɬɨɪɨɜ

a b c – ɩɪɚɜɚɹ

G G

G

G

G

Ⱦɥɹ ɤɨɥɥɢɧɟɚɪɧɵɯ ɜɟɤɬɨɪɨɜ a ɢ b ɩɨɥɚɝɚɟɦ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ

G>aGb@ .

ɍɬɜɟɪɠɞɟɧɢɟ > @ Ⱦɥɢɧɚ ɜɟɤɬɨɪɧɨɝɨ ɩɪɨɢɡɜɟɞɟɧɢɹ

>a b@

ɪɚɜɧɚ

G

G

ɩɥɨɳɚɞɢ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ ɩɨɫɬɪɨɟɧɧɨɝɨ ɧɚ ɜɟɤɬɨɪɚɯ a ɢ b.

ɋɜɨɣɫɬɜɚ ɜɟɤɬɨɪɧɨɝɨ ɩɪɨɢɡɜɟɞɟɧɢɹ

G G

G G

1) >aG bG@ >b a@;

2) >a Ga@G ; G G

3) >Ba b@ B>a b@;

G G

G

G G

G G

4) >a b c@ G>aGcG@ >b c@.

G G G

ɉɭɫɬɶ ɜɟɤɬɨɪɵ i

j k

– ɛɚɡɢɫ ɬɪɟɯɦɟɪɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɢ ɬɪɨɣɤɚ i j k –

ɩɪɚɜɚɹ ɬɨɝɞɚ

GG

G G

G

G

G

(2.2.9)

>i i@ > j j@ >k k@ ;

G G

G

G G

G

G G

G

>i j@ k >i k@ j > j i@ k

(2.2.10)

G

G

G

G G

JG G

G

G

G

ɉɭɫɬɶ ɞɜɚ ɜɟɤɬɨɪɚ ɡɚɞɚɧɵ ɫɜɨɢɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ ɜ ɛɚɡɢɫɟ i j k

G

a x1 y1 z1 b x2 y2 z2 . ɇɚɣɞɟɦ ɜɟɤɬɨɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɷɬɢɯ ɜɟɤɬɨɪɨɜ

G G

G

G

G G

G

G

>a b@ >x1i y1 j z1k x2i y2 j z2 k@.

ɂɫɩɨɥɶɡɭɹ ɫɜɨɣɫɬɜɚ ɜɟɤɬɨɪɧɨɝɨ ɩɪɨɢɡɜɟɞɟɧɢɹ ɢ ɮɨɪɦɭɥɵ

ɩɨɥɭɱɚɟɦ

G G

G

G

G

>a b@ y1z2 y2z1 i x2z1 x1z2 j x1z2 x2z1 k .

(2.2.11)

Ɏɨɪɦɭɥɭ ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɢɫɩɨɥɶɡɭɹ ɜɜɟɞɟɧɧɨɟ ɪɚɧɟɟ ɩɨɧɹɬɢɟ

ɨɩɪɟɞɟɥɢɬɟɥɹ ɬɪɟɬɶɟɝɨ ɩɨɪɹɞɤɚ

G

G

G

G G

i

j

k

x1

y1

z1

>a b@

.

x2

y2

z2

Ʌɟɝɤɨ ɩɪɨɜɟɪɢɬɶ ɱɬɨ ɪɚɫɤɪɵɜɚɹ ɷɬɨɬ ɨɩɪɟɞɟɥɢɬɟɥɶ ɩɨ ɮɨɪɦɭɥɟ

ɦɵ ɩɨɥɭɱɢɦ ɜɵɪɚɠɟɧɢɟ

ɉɪɢɦɟɪ1. Ɂɚɞɚɧɵ ɬɪɢ ɜɟɪɲɢɧɵ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ ABCD A

B C ɇɚɣɞɟɦ ɩɥɨɳɚɞɶ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ

JJG

JJJG

ȼɵɱɢɫɥɹɟɦ ɤɨɨɪɞɢɧɚɬɵ ɜɟɤɬɨɪɨɜ BA BC .

ɉɨ ɮɨɪɦɭ

ɥɟ ɧɚɯɨɞɢɦ ɤɨɨɪɞɢɧɚɬɵ ɜɟɤɬɨɪɧɨɝɨ ɩɪɨɢɡɜɟɞɟɧɢɹ

JJG

JJJG

SABC

1

416 2 26 .

2

AM

2

S

ABC

2 2

26 4

26

CB

BN

2

SABC

2 2

26 4

26

AC

CK

2

SABC

2

2

26

2

26 .

AB

6

3

G G G

Ɉɩɪɟɞɟɥɟɧɢɟ ɉɭɫɬɶ ɞɚɧɵ ɬɪɢ ɜɟɤɬɨɪɚ a b c ɋɦɟɲɚɧɧɵɦ ɩɪɨɢɡɜɟ

G

G

°¯

cG Ɂɞɟɫɶ ɜɟɤɬɨɪɧɨɟ

ɞɟɧɢɟɦ ɷɬɢɯ ɜɟɤɬɨɪɨɜ ɧɚɡɵɜɚɟɬɫɹ ɱɢɫɥɨ

aG b cG

¡aG b

G G

¢

±

G

ɬɨɝɞɚ ɤɨɝɞɚ ɜɟɤɬɨɪɵ ɤɨɦɩɥɚɧɚɪɧɵ > @.

G G G

G

ɉɭɫɬɶ ɬɪɢ ɜɟɤɬɨɪɚɡɚɞɚɧɵ ɫɜɨɢɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ ɜɛɚɡɢɫɟ i j k a x1 y1 z1

G

G

b x2 y2 z2

c x3 y3 z3 . ɋɦɟɲɚɧɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɷɬɢɯ ɜɟɤɬɨɪɨɜ ɛɭɞɟɬ

ɪɚɜɧɨ ɜɟɥɢɱɢɧɟ ɨɩɪɟɞɟɥɢɬɟɥɹ

G G

G

x1

y1

z1

(2.2.12)

a b

c

x2

y2

z2

x3

y3

z3

A

B

C

8

A

B

C

9

A

B

C

10

A

B

C

11

A

B

C

12

A

B

C

13

A

B

C

14

A

B

C

A

B

C

16

A

B

C

A

B

C

18

A

B

C

19

A

B

C

20

A

B

C

21

A

B

C

22

A

B

C

23

A

B

C

24

A

B

C

A

B

C

26

A

B

C

A

B

C

28

A

B

C

29

A

B

C

30

A

B

C

1

A

B

C

D

2

A

B

C

D

3

A

B

C

D

4

A

B

C

D

A

B

C

D

6

A

B

C

D

A

B

C

D

8

A

B

C

D

9

A

B

C

D

10

A

B

C

D

11

A

B

C

D

12

A

B

C

D

13

A

B

C

D

14

A

B

C

D

A

B

C

D

16

A

B

C

D

A

B

C

D

18

A

B

C

D

19

A

B

C

D

20

A

B

C

D

21

A

B

C

D

22

A

B

C

D

23

A

B

C

D

24

A

B

C

D

A

B

C

D

26

A

B

C

D

A

B

C

D

28

A

B

C

D

29

A

B

C

D

30

A

B

C

D

ɉɨɫɬɪɨɢɦ ɭɪɚɜɧɟɧɢɟ ɩɥɨɫɤɨɫɬɢ ɩɨ ɡɚɞɚɧɧɵɦ ɤɨɨɪɞɢɧɚɬɚɦ ɧɨɪɦɚɥɶɧɨɝɨ

JG

ɢ ɡɚɞɚɧɧɨɣ ɬɨɱɤɟ M0 x0 y0 z0 ɉɭɫɬɶ M x y z

– ɩɪɨ

ɜɟɤɬɨɪɚ N A B C

JJJJJG

A(x x0 ) B( y y0 ) C(z z0 ) 0 .

(2.3.1)