Mavzu: vektorlarni skalyar, vektor va aralash ko‘paytmalari reja: Vektorlarning skalyar ko‘paytmasi

Vektorning vektor va aralash ko‘paytmasiga doir


misollar.

1. 
   Determinantlarni hisoblang.
   

1)


—2


2)


—3


—2


4)


—4


x + 1 —21


5)


cosx


sinx


cosx


—sinx





5


3


3


4


3
2


4.2.2. Determinantlarni hisoblang.

321			4 —3	5		^2	—1 2
1)	253	;	2)	3 —2	8	; 3)	3	5 3
	342			1 —7 —	5		1	6 x + 5
	x^2 3	2			X	24	9
4)	x —1	1	= 0;	5)	X	2	3	= 0.
	0 1	4			1	1	1

= 0;


4.2.3.


Quyidagi chiziqli tenglamalar sistemasini Kramer usulida


yeching.
1) i^3y-* =
⁷ (5% + 3y =


17
5; '


'2x — 3y = 5 4% — 5y = 7^{; 3)}


xcosa — ysina = cos2a
xsina + ycosa = sin2a

4. 
   Quyidagi chiziqli tenglamalar sistemasini Kramer usulida yeching.
   

x + y + 4 z = 1

x + 2 y - z = -6


1) < 2 x + y + 6 z = 2


2) <2x - y + z = 7


3x + 3 y + 13z = 2


3x + 5 y + 2 z — — 1


x + 2y + 3z -13 = 0
3) px + 2y + 2z -16 = 0 ;
4 x - 2 y + 5 z - 5 = 0


2x - 3 y + z = 2


4)


< 2x + y - 4z = 9
6 x - 5 y + 2 z = 17

4. 
   a va b vektorlar o‘zaro ^ = - burchak hosil qiladi. Agar |a| = 6,
   

6
|b| = 5 bo‘lsa, |[ab]| ni hisoblang.

4. 
   |a| = 10, |b| = 2 va a b = 12 berilgan bo‘lsa, |[ab]| ni
   

hisoblang.

4. 
   |a| = 3, |b| = 26 va |[ab]| = 72 bo‘lsa, a b ni toping.
   
5. 
   a va b vektorlar o‘zaro perpendikulyar. |a| = 3 va |b| = 4 ni
   

bilgan holda, quyidagilarni hisoblang:

1. 
   |[(a + b)(a-h)]|;
   
2. 
   |[(3a-h)(a-2h)]|.
   

4. 
   a va b vektorlar o‘zaro ^ = 27 burchak hosil qiladi. |a| = 1, |h| = 2 ni bilgan holda, quyidagilarni hisoblang:
   

1. 
   [a,b]²; 2) [(2a + b)(a + 2h)]2; 3) [(5 + 3 h)(3a-h)]2.
   

4. 
   Ixtiyoriy p, q, r, n vektorlar berilgan. a = [p n], b = [q n] va c = [r n] vektorlarni komplanar ekanligini isbotlang.
   
5. 
   a(3;-1;-2) va b(1;2;-1) vektorlar berilgan. Vektor
   

ko‘paytmalar koordinatalarini toping:

1. 
   [a b]; 2) [(2a + b)b]; 3) [(2a - b)(2a + b)].
   

4. 
   X(2;-1;2), B(1;2;-1) va C(3;2; 1) nuqtalar berilgan.
   

Vektor ko‘paytmalar koordinatalarini toping:

1. 
   [AB BC]; 2) [(BC - 2CA)CB].
   

4. 
   X(1; 2; 0), B(3;0;-3) va C(5; 2; 6) nuqtalar berilgan. ABC uchburchak yuzasini hisoblang.
   

4. 
   Uchburchakning 4(1;-1;2), B(5;-6;2) va C(1;3;-1)
   

uchlari berilgan. B uchidan AC yon tomonga tushirilgan balandlik
uzunligini hisoblang.

4. 
   a(2; -2; 1) va ¿(2; 3; 6) vektorlar orasidagi burchak sinusini
   hisoblang.
   
5. 
   X(3;-2;5), B(1;4;-3) va C(-6;2;4) nuqtalar berilgan
   

bo‘lsa,
1) [AB BC]AC; 2) [AB AC]BC;
aralash ko‘paytmasini toping.

4. 
   C(-2;4;3), D(1;-5;6) va E(3;7;-4) nuqtalar berilgan
   bo‘lsa,
   

1. 
   [CD DE]CE; 2) (2CD - 3DE)(DC + 3CE)(2CD - ED);
   

3) (3DE + ED)(2CD - EC)(DC + 2CE) aralash ko‘paytmasini toping.

4. 
   a = i + 6j - 4/c, b = -3i + 2j + 7/i va c = -5i- 6j + 2/c
   

vektorlar berilgan bo‘lsa,
1) [a b]C; 2) [a C]b;


3) [BC ac]ab


3) [b C]a;
5) (3a - c)(2h + a)(4c + 3b)


J\ ¹ f ¹ '-'7 ?
aralash ko‘paytmasini toping.

4. 
   a(2; -3; 1), ¿>(-3; 1; 2) va c(1; 2; 3) vektorlar berilgan bo‘lsa,
   [a, ¿]c va a[5 c] ni hisoblang.
   
5. 
   a(6; -4; 8) va ¿(-2; 4; 0) vektorlar berilgan bo‘lsa:
   

1. 
   [(a + h)(a-h)];
   
2. 
   a(a + ¿) ;
   

3)


il + i!_x
(b - -) topilsin.

4. 
   Quyidagi hollarning har birida [a b] vektor ko‘paytma topilsin:
   

1. 
   a(2; 3; 1), b(5;6;4);
   
2. 
   a(5; -2; 1), b(4; 0; 6);
   

52

3. 
   a(-2;6;-4), b(3; -9; 6).
   
4. 
   a (8; 4; 1) va b(2; -2; 1) vektorlardan yasalgan parallelogramm
   

yuzi hisoblansin.
4.2.23. a(3; 1; 2), b (2; 7; 4) va c(1; 2; 1) vektorlar berilgan:


1) abc;


2)





3) a[bc]


topilsin.

24. 
    Berilganlarga ko‘ra a, ko‘paytmasini toping.
    

1) a = k, b = i, c = j;
3) a = j, b = i, c = k;

5. 
   a = i + j, b = i-j, c = j;
   

b^b va c vektorlarning aralash
2) a = i, b = k, c = j;
4) a = i + j, b = j, c = k;

6. 
   a = i + j, b = i-j, c = k.
   

6. 
   a, b va c vektorlar o‘zaro perpendikulyar hamda |a| =4,
   

|^b| = 2 va |c| = 3 berilgan bo‘lsa, abc ni toping.

25. 
    a va b vektorlar o‘zaro ^ = “ burchak tashkil qiladi va c vektor bilan perpendikulyar. |a| = 6, |^b| = 3 va |c| = 4 berilgan bo‘lsa, abc
    

ni toping.

25. 
    a(1;-1;3), b(-2;2;1) va c(3; -2; 5) vektorlar berilgan bo‘lsa, abc ni toping.
    
26. 
    a = 2i + 3j + 4k, b = 31 + 2j + k va c = j - k vektorlar
    

berilgan bo‘lsa, quyidagilarni toping.


2) [bc]; 3) [ac]; 4) [ac]b ;
7) ([a^b]c);


5) [ab]c ;
8) (^[ac]^b); 9) (a[^bc]);
11) [(a-2c)(3^b-2a)];






1) [ab^b ] ;
6) a[bc] ;
10) [(2a- 3^b)(4a- 5c)];

12. 
    ([(2b + c)(3a - c)](^b - 2a));
    
13. 
    ([(2a - 5c)(2b + 3c)](3b - c)); 14) ((ab)c); 15) ((bc)a).
    

4.2.29. a = -1 + 3j + k, b = j + 2k va c = 2a - 3b vektorlar
berilgan bo‘lsa, quyidagilarni toping:
1) (a[c^b]); 2) (c[a^b]);


3) a[bc] ; 4) b[ac] ;

?

= c(abd) — d(abc);

5. 
   ([¿î[5

30. 
    Ayniyatni isbotlang:
    

1. 
   a[bc] + ^b[ac] + c[ab] = 0;
   
2. 
   [ab][cd] = (ac)(bd) — (ad)(bc);
   
3. 
   [a^b][cd] + [ac] [db] + [ad] [¿ici] = 0;
   
4. 
   [ab] [cd]
   
5. 
   [a^b][bc][ca] = (abc) ;
   
6. 
   a[a[a[ab]
   perpendikulyar;
   
7. 
   [a(b[cd])] = [ac](bd) — [ad](bc);
   
8. 
   a b[cd]
   

2

9. 
   [ab] [ac]² — ([ab][ac]) = a²(abc) ;
   
10. 
    [ab] [bc] [bc][ca] [ca][ab] = (abc) ;
    
11. 
    (ab)[cd] + (ac)[db] + (ad)[bc] = a(bcd)