. , :
(a,b) = x1x2 + y1y2 + z1z2.
. , a b,
.
, .
. , (a, b, c) = (a×b, c).
, .
, , . . .
1.25. 1.14 , :
1) ; 2) ; 3) ; 4) ;
5) .
1.14
(4, 2, 4) | (1, 4, 2) | (1, 1, 1) | (0, 1, 1) | |
(5, 1, 3) | (3, 1, 1) | (1, 1, 0,) | (1, 1, 0) |
1.26. = , :
1) = 2 , = 3 ; 2) = , = ;
3) = = .
, .
1.27. × , , :
1) = (1, 5, 3), = (2, 4, 3);
2) = (3, 2, 6), = (6, 3, 2);
3) = (3, 0, 4), = (1, 2, 2).
1.28. :
1) (2; 2; 2), (1; 3; 3), (3; 4; 2);
2) (3; 2; 4), (1; 4; 7), (1; 2; 2).
1.29. , , :
1) = (1, 1, 2), = (1, 2, 3), = (2, 1, 1);
2) = (5, 2, 1), = (1, 2, 1), = (1, 2, 2).
1.30. , :
1) = (1, 1, 3), = (0, 2, 1), = (1, 1, 4);
2) = (1, 2, 2), = (2, 5, 7), = (1, 1, 1).
1.31. ,
= (3, 2, 1), = (1, 0,1), = (1, 2, 1).
1.32.
1) (1; 1; 0), (2;2; 1), (3; 1; 1), (1; 0; 2).
2) (4; 4; 3), (2;1; 1), (2; 2; 1), D (1; 3; 2).
: <; S , V , .
.
1) :
= (1 + 1; 0 1; 2 0) = (2; 1; 2), = (2 + 1; 21; 1 0) = (3; 3; 1),
,
.
2) = (4; 0; 1),
, . : .
∆ :
3) :
= 0 + 4+ 6 (0+24+3)= 17.
, ,
,
4) .. ,
1.2.3. .
, . : (1, 2) φ. , 60 (. 1.2); 1', e2'. , .
|
|
e1= i =
e2 = j =
A = .
. 1.2. 60˚
. , = λ ( λ) =0. λ det (A λE) = 0.
1.33. . :
1) = 4 3 , = ; 2) = 2 + 4 ,
=
1.34. , , :
,
1.35. , :
1) = 2) =
3) = 4) =
. = (1, 2, , n). , . .
, 1.
1.10. . .
:
= .
= = (0,5 )٠(0,6 ) 2 = 0,3 0,5 0,6 + 2 0,2 = 2 1,1 + 0,1 = 0.
. , = 1, = 0,1. , = 1:
( 1) ٠ = = .
. = (0,8; 1).
0,8: 1 4: 5.
1.36. :
= .
, , 1100 . .
1.37. :
A= .
, , = 6270 . .
1.
1. a=(7;4;3) e1=(1;2;0),
e2 =(3; 1; 2), e3 = (0; 4;1).
2. , x(;1) (;1) . .
3. , = 2i j + 3k d = 2i 2j + 4k.
4. : (2; 1; 4), (1; 0; 3), (3; 1; 2).
2.
1. , (1,) .
2. = 2a 3b, |a| = 3, |b| = 2, 60.
3. e1 = (2, 2, 4), e2 = (0, 1, 0), e3 = (2, 3, −4)?
|
|
4. m a = m i 3 j + 2 k b = i + 2 j m k ?
, , , , .
1.38. , , : 1) , (2; 3); 2) , (1; 2); 3) 2 x 3 y + 1 = 0, (2; 3); 4) x + y 2 = 0, (1; 2).
1.39. , , : 1) 3 2 + 5 = 0, (2; 1); 2) 2 + 7 = 0, (0; 3).
(2; 0), (2; 4), (4; 0). : 1) ; 2) , ; 3) , ; 4) , .
1) , :
(1.1)
(2; 4), (4; 0), ,
2 8 = 4 + 8,
2 = 4 + 16,
= 2 + 8.
. 1.3.
2) :
. :
= , =
= , = .
(3; 2). () (1.1):
2 + 6 = 5 + 10, 5y = 2x+4, = 0,4 + 0,8 .
3) D.
. . AD , :
kD = = =
, , :
0 = k ( 0) (1.2)
(2; 0) k = 1/2, 0 = 0,5( (2))
= 0,5 + 1 .
4) , .
. . l // BC, k ι = k.
kι = 2. (1.2), (2; 0) k = 2,
0 = 2 ( + 2) = 2 4 .
! .
1.40. , 1) ; 2) ; 3) 4) , , , 5) .
1) (1; 1), (2; 5), (6; 2); 2) (1;1), (2; 5), (4; 2);
3) (3; 1), (2; 4), (3; 1); 4) (1;2), (6; 2), (1; 6);
5) (2; 3), (4; 5), (4; 2); 6) (1;3), (3; 4), (7; 2);
7) (1; 3), (8; 5), (3; 2); 8) (4;2), (1; 5), (3; 2);
9) (5; 1), (4; 6), (1; 0); 10) (1; 1), (2; 2), (3; 4).
1.41. , . , : 1) (4; 3); 2) (2; 3).
1.42. , OX OY : 1) = 2 b = 5; 2) = 1 b = 4.
1.43. , (4; 3) 3 . .
1.44. , 5 +10 = 0 8 + 4 + 9 = 0 + 3 = 0.
1.45. , 2 3 + 5 = 0 3 + 7 = 0, = 2.
1.46. : (3; 5), (1, 3). D, .
1.47.
+ + 5 = 0 4 = 0. , (2; 2).
|
|
1.48. , (1; 2),
Q(5;1) R(4; 3). .
1.49. (2; 1) 3 4 + 5 = 0 4 + 3 7 = 0. .