Matematika 1 (determinanti, vektori, analiticka geometrija).pdf
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Transcript of Matematika 1 (determinanti, vektori, analiticka geometrija).pdf
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1
-
( )
-
1. 1.1. 1.2.
2.
2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7.
3.
3.1. 3.2.
3.2.1. 3.2.2. . 3.2.3. .
3.3. 3.4. 3.5.
4.
4.1. 4.2. 4.3.
5.
5.1. . 5.2.
6.
6.1. 6.2.
6.2.1. 6.2.2. ( )
6.3. 6.4. 6.5. , 6.6. 6.7.
-
7.
7.1. 7.1.1. 7.1.2. 7.1.3. 7.1.4.
7.2. 7.3.
8.
8.1. 8.2.
8.2.1. 8.2.2.
8.3. 8.4. 8.5.
8.5.1. ( ) 8.5.2. 8.5.3. 8.5.4.
8.6. 8.7. 8.8. 8.9. 8.10.
9. 9.1.
9.1.1. 9.1.2.
9.2. - 9.3. 9.4. 9.5.
9.5.1 9.5.2. 9.5.3.
9.6. 9.6.1. 9.6.2.
. 1 . .
-
1.
1.1.
1.1.1. ( : , ...) :
.a b
ad bcc d
= 1.1.1:
( )4 3 4 1 2 3 4 6 102 1
= = + = , 2 22ctg 1
ctg tg 1 cos 1 cos sincos tg
x 2x x x xx x
= = = x .
:
1. , ..,
a b a cad bc
c d b d= = .
1.1.2: 2 3
4 3 11 2
= = 2 1 4 3 13 2
= = .
2. () , ..,
( )a b c dad bc bc adc d a b
= = = . 1.1.3:
2 34 3 1
1 2= = 1 2 3 4 1
2 3= = .
3. () ,
..,
0a b
ab aba b
= = . 1.1.4:
2 36 6 0
2 3= = .
4. ()
, ..,
( )a b ka kbk k ad cb kad kcbc d c d
= = = . 1.1.5:
4 2 4 3 8 1216 12 4
1 2 1 2 = = = ( )2 34 4 4 3 4 1
1 24 = = = .
1-1
-
5. () () , ..,
0a b
akb kabka kb
= = . 1.1.6:
2 318 18 0
6 9= = .
6. ()
() , ..,
( ) ( )a b a kc b kdad bc a kc d c b kdc d c d
+ += = + + = . 1.1.7:
2 34 3 1
1 2= = 2 1 3 3 2 3 5 9 10 9 1
1 2 1 2+ + = = = .
1.1.1.
1 1 12 2
a x b y ca x b y c
+ =2+ =
. (1.1)
1 12 2
a bD
a b= , ;
1 12 2
x
c bD
c b= , D
;
1 12 2
x
a cD
a c= , D
. (1.1) :
1. :
0D
xDxD
= yDyD
= . : . 1.1.8:
2 5 83 1
x yx y+ = + = ,
2 513
3 1D = = , 8 5 13
1 1xD = = ,
2 826
3 1yD = = .
, e: 1x = 2y = .
1-2
-
2. . 0x yD D D= = = : . 1.1.9:
2 3 46 9 12
x yx y+ = + = , .
0x yD D D= = =
4 2,3
xx .
3. , 0D = 0xD 0yD , . : . 1.1.10:
52 2 8x y
x y+ = + = , , .
0D = 2 0xD = .
1.2. 1.2.1. ( : , ...):
11 12 13
21 22 23
31 32 33
a a aa a aa a a
.
:
1. . 1.2.1:
( ) ( ) ( ) ( ) ( ) ( )2 3 4 2 31 1 3 1 1 2 1 0 3 3 1 4 1 2 1 1 4 2 3 2 0 1 3 17
1 2 0 1 2 = + + + + =
2. ().
1.2.2:
( )2 3 4
1 3 1 3 1 11 1 3 2 3 4 2 6 3 3 4 1 17
2 0 1 0 1 21 2 0
= + = + =
: :
+ + + + +
.
, :
1-3
-
1. . 1.2.3:
2 3 41 1 3 17
1 2 0 =
,
2 1 13 1 2 174 3 0
=
.
2. () . 1.2.4:
2 3 41 1 3 17
1 2 0 =
, ( )
1 1 32 3 4 17 11 2 0
7= =
3. () . 1.2.5:
1 1 31 1 3 0
1 2 0
=
.
4. ()
. 1.2.6:
3 2 3 3 3 4 6 9 121 1 3 1 1 3 5
1 2 0 1 2 0
= =
1 ( )
2 3 43 1 1 3 3 17 51
1 2 0 = =
.
5. () () . 1.2.7:
2 3 44 6 8 01 2 0
=
.
6. ()
() . 1.2.8:
2 3 41 1 3 17
1 2 0 =
( ) ( )2 1 3 3 1 3 4 3 3 1 6 51 1 3 1 1 3
1 2 0 1 2 0
+ + + 17 = =
.
1.2.1.
11 12 13 1
21 22 23 2
31 32 33 3
a x a y a z ba x a y a z ba x a y a z b
+ + = + + = + + =. (1.2)
1-4
-
11 12 13
21 22 23
31 32 33
a a aD a a a
a a a= , ;
1 12 13
2 22 23
3 32 33
x
b a aD b a a
b a a= , D -
;
11 1 13
21 2 23
31 3 33
y
a b aD a b a
a b a= , D -
;
11 12 1
21 22 2
31 32 3
z
a a bD a a b
a a b= , D -
. (1.2) :
1. :
0D
xDxD
= , yDyD
= zDzD
= . 1.2.3:
3 2 12 3 5 8
0
x y zx y zx y z
+ = + = + =,
3 2 12 3 5 3= 1 1 1
D
=
, 1 2 18 3 5 60 1 1
xD
= =
,
3 1 12 8 5 91 0 1
yD = =
, 3 2 12 3 8 31 1 0
zD
= =
, e: 2x = , 3y = 1z = .
2. . 0x y zD D D D= = = = 1.2.4:
2 4 12 5 1
2
x y zx y z
x y z
+ = + = = 0x y zD D D D= = = =, .
( )1 2 ,1 ,z z z + + .
3. , 0D = 0xD 0yD 0zD , . 1.2.5:
2 2 52 24 2 2 3
x y zx y zx y z
+ = = = , , 0D = 21 0xD = .
.
.
1-5
-
1.2.6:
1
21
ax y zx ay zx y z
0+ + = + + = + + =
. (1.3)
: :
21 1
1 2 3 21 1 1
aD a a a= = + ,
1 1 10 21 1 1
xD a 0= = ,
( )1 1
1 0 2 2 2 2 11 1 1
y
aD a= = = a , ( )2
1 11 0 11 1 1
z
aa a a aD a = = . =
, 0D = , .., 2 3 2 0a a + =
. , 1a = 2a = 1a = 0D = , 0xD = , 0yD = 0zD = ;
2a =0D = , 0xD = , 2yD = 2zD = .
: 1a 2a 0D , (1.3)
2
0 03 2
xDxD a a
= = = + , ( )
( )( )2 1 2
1 2 2yD ay
D a a= = =
a
( )( ) ( )
11 2
z a aD azD a a a
= = =2 .
, (1.3) .
1a = 0x y zD D D D= = = =
2a = 0D = 2 0yD = , (1.3) . 1.2.2. , ..,
11 12 13
21 22 23
31 32 33
000
a x a y a za x a y a za x a y a z
+ + = + + = + + =.
, : 0x y zD D D= = =1. 0D 0x y z= = =
. 2. ,
. 0D =
1.2.7: 3 0
3 20
x y zx y zx y z
+ + = 0+ + = + + =
( 0x y z= = = ) 3 1 11 3 2 2 01 1 1
D = = ,
1-6
-
2 4
2 50
x y zx y z
x y z
00
+ = + = =
1 2 42 1 5 01 1 1
D
= =
.
1.1. 3 2
22 1
2x y z a
x y azax y z
+ + = + + = + + = 1.2.
22 4
3 4
x y z ax y az
ax y z
+ + = 1+ + = + + = 1.3. ( )
( )1
2 2 2
3 2 1 7 5
x y zx ay a z a
x a y z
+ + = + + = + + =
1-7
-
2.
: . . : , , , , ... . AB
JJJG
B ( 2.1), : : B ; : (
); (, ): AB AB
JJJG.
: , , ...
A
B
aGbG
2.1 2.2 ( ) . . 1 . 0 0
G.
( 2.2). , . aG
, . bG
a b= GG
2.1.
: - 2.1.1; - 2.1.2.
aG bG
baGG +
aG
bG
baGG +
2.1.1 2.1.2
2-1
-
: : ; a b b a+ = +G GG G : ; a b b a+ = +G GG G : ; 0 0a a+ = + =G GG G aG : ( ) ( ) 0a a a a+ = + = GG G G G .
: ( )a b a b = + G GG G 2.1.3:
aG
bG
( )baba GGGG +=aG
bG
baGG
aG
bG
2.1.3
2.2.
baG
G : G
b aG ; b
G a G , .., b a= G G ;
bG
aG 0 > aG 0 < .
2.2.1: , aG 2aG 12
aG 32
a G . : 2.2.1.
aG2
aG aG2a
GaG
23 aG
2.2.1
: G
b b aaG = G G ; : ( ) ( )a a = G G ; : ( ) a a a + = + G G G ; : ( )a b a b + = + G GG G .
:
0 0a = =G 0a = GG ; 0a GG a a = =G G ; 0 . a b a = =G GG G b
aG aG 1 .
2-2
-
2.3.
2.3.1. 1 2, , , na a aJJG JJG JJG 1 2, , , n
a
.
1 1 2 2 n na a + + + JJG JJG JG" J
1 2, , , na a aJJG JJG JJG .
2.3.2. 1 2, , , na a aJJG JJG JJG b
G
1 2, , , n a
1 1 2 2 n nb a a = + + + G JJG JJG JG" J
. bG
1 2, , , na a aJJG JJG JJG
2.3.3.
1 2, , , na a aJJG JJG JJG
1 2, , , n 1 1 2 2 0n na a a + + + =
JJG JJG JJG G" .
2.3.4. 1 2, , , na a aJJG JJG JJG
1 1 2 2 0n na a a + + + =JJG JJG JJG G"
1 2 0n = = = =" .
: aG b
G aG bG . . aG b
G
0 0a b + = = =G GG . , aG b
GcG aG bG .
, , ..,
aG bG
cG dG
1 2 3, , 1 2 3d a b c= + +
G GG G .
2.4.
, k () ( 2.4.1): iG
jG G
xyz ;
x, y z , ; .
, iG
jG
kG
- , ..,
dG
, ,x y z x y zd i j = + +
G Gk
G G.
( , , )x y zd =G .
dG
( , , )x y zA ( 2.4.1).
x
y
z
kG
iG
jG x
y
z
dG
2-3
-
2.4.1 ( ), ,x y za a a a=G , ( ), ,x y zb b b b=G ( ), ,x y zc c c c=G . :
, ,x x y y z za b a b a b a b= = = =GG
( ), ,x x y y z za b a b a b a b = GG 2.4.1: , ( )2,1,3a =G ( )3,1,0b =G , ( ) ( )2 3,1 1,3 0 5, 2,3a b+ = + + + =GG .
( ), ,x y za a a a = G 2.4.2: , ( )2,1,3a =G 3 = , ( ) ( ) ( )3 2,1,3 3 2,3 1,3 3 6,3,9a = = =G .
( )aG bG yx z
x y z
aa ab b b
= =
2.4.3: , ( )2,1,3a =G ( )4,2,6b =G ( ) 2 1 34 2 6= = .
, ( )aG bG
cG 0x y z
x y z
x y z
a a ab b bc c c
= .
2.4.4: ( )1,2, 4a = G , ( )2,1, 5b = G ( )4, 2, 6c = G ( )
1 2 42 1 5 04 2 6
=
.
( ), ,x y zA a a a , ( ),x y zB b b b
( ), ,x x y y z zAB b a b a b a= JJJG . 2.4.5: ( )2,1,3A ( )4,2,5B .
( ) ( )4 2,2 1,5 3 2,1,2AB = =JJJG .
2.5.
2.5.1. aG bG
. a b GG ( )cos ,a b a b a b = G GG G G G .
aG
bG
( )ba GG,
aG b
G , ..,
( ), ,x y za a a a=G ( ), ,x y zb b b b=G
x x y y za b a b a b a bz = + + GG .
2.5.1: ( )2,1,3a =G ( )3,1,0b =G
2-4
-
2 3 1 1 3 0 7a b = + + =GG . :
a b b a = G GG G
2.5.2: ( )2,1,3a =G , ( )3,1,0b =G , 2 3 1 1 3 0 7a b = + + =GG , 7b a =G G .
( ) ( ) ( )a b a b a b = = G GG G G G 2.5.3: ( )2,1,3a =G , ( )3,1,0b =G , 2 = , ( ) 2 7 14a b = =GG , ( ) ( )4, 2,6 14a b b = =G GG , ( ) ( )6, 2,0 14a b a = =GG G
( )a b c a b a c + = + G GG G G G G 2.5.4: ( )1,2, 4a = G , ( )2,1, 5b = G , ( )4, 2, 2c = G , ( ) ( )6, 1, 7 32a b c a + = =GG G G , 24 8 32a b a c + = + =GG G G 0a b a b = G GG G
2.5.5: , ( )2,1,3a =G ( )1,4, 2b = G , ( )2 1 1 4 3 2 0a b = + + =GG . aG b
G
2a a a =G G G , 2a a a a= =G G G G 2 2 2x y za a a a= + +G 2.5.6: ( )2,1,3a =G , 2 2 22 1 3 14a = + + =G .
( )cos , a ba ba b =
GGGG GG
( ) 2 2 2 2 2 2cos , x x y y z zx y z x y z
a b a b a ba b
a a a b b b
+ + =+ + + +
GG
2.5.7: , ( )2,1,3a =G ( )3,1,0b =G , ( ) 2 2 2 2 2 22 3 1 1 3 0 7 7 2 35cos , 1014 10 1402 1 3 3 1 0a b + + = = = =+ + + +
GG .
2.6.
2.6.1. a . - G bG
:
a b GG
( )sin ,a b a b a b = G GG G G G ; a b GG
aG bG
; a b GG -
: - , - aG b
Ga b GG -
.
aGbG
( )ba GG,
baGG
, .., aG bG ( ), ,x y za a a a=G
( , , )x y zb b b b=G
2-5
-
x y z
x y z
i j ka b a a a
b b b =
GG GGG .
2.6.1: ( )2,1,3a =G ( )3,1,0b =G
( )2 1 3 3 9 3,9, 13 1 0
i j ka b i j k = = + =
GG GG GG GG .
:
a b b a = G GG G ( ) ( ) ( )a b a b a b = = G GG G G G
a b c a c b c ( )( )a b c a b a c + = + G GG G G G G G+ = + GG G G G G a b a b a b = G GG G G G 0a a = GG G a b GG aG bG
( 6) ( )sin , aa b a b a b a h P = = =G G GG G G bG
ah
aG
2.6.1
2.6.2: . ( )2,1,3a =G ( )3,1,0b =G
: ( )2,1,3a =G ( )3,1,0b =G
( )2 1 3 3 9 3,9, 13 1 0
i j ka b i j k = = + =
GG GG GG GG ,
( ) ( )2 223 9 1 9P a b= = + + = 1GG .
2.7.
2.7.1. , aG b
GcG .
( )a b cGG G ( ) : ( ) ( )a b c a b c= G GG G G G .
, , .., aG b
GcG ( ), ,x y za a a a=G ,
( ), ,x y zb b b b=G ( , , )x y zc c c c=G
2-6
-
( ) x y zx y zx y z
a a aa b c b b b
c c c=GG G
2.7.1: ( )3,2, 4a = G , ( )2,1, 5b = G
( )4, 2, 2c = G
( ) 3 2 4, , 2 1 5 364 2 2
a b c
= =
GG G .
:
( ) ( ) ( )a b c b a c b c a= =G G GG G G G G G
- ( )a b cGG G aG , b
G
. ( 2.7.1) cG
( )a b cVH
B a b= =
GG GGG
aG , b
G
cG
aG bG . 2.7.1
2.7.2: ( )3,2, 4a = G , ( )2,1, 5b = G ( )4, 2, 2c = G aG b
G.
: ( )3,2, 4a = G , ( )2,1, 5b = G
( )4, 2, 2c = G
( ) 3 2 4, , 2 1 5 364 2 2
a b c
= =
GG G ,
36 36V = = .
a b GG
( )3 2 4 6 7 6,7, 12 1 5
i j ka b i j k = = + =
GG GG GG GG
( ) ( )2 226 7 1 8B a b= = + + = 6GG .
36 18 864386
VHB
= = = .
2-7
-
2.1. ( )1,0,2a =G , ( )3,1,0b =G , ( )1, 1,4c = G , c .
GaG bG
2.2. ( )1,0,2a =G , ( )3,1,0b =G , ( )1, 1,4c = G , b
G
aG c . G
2.3. , ( )1, ,3a =G ( ),15,9b =G . , aG
bG
.
2.4. , ( )1,0,a =G , ( )3,1,0b =G , ( )1, 1,4c = G . , aG
.
2.5. ( )2,1,0a =G , ( )1,0,1b =G , ,
(0,3,2c =G )( )0,10,4d =G
aG , bG
cG .
2.6. a b GG a b GG : 2a p q= G G G , 3b p q= +G G G , 1, 3p q= =G G pG qG
3 = .
2.7. aG , 2a p q= G G G , 1, 3p qG G= = pG qG3 = .
2.8. 2a p q= G G G 3b p q= +G G G ,
1, 3p q= =G G pG qG 3 = .
2.9. 2a p= G G qG 3b p q= +G G G , 1, 3p q= =G G pG qG
3 = .
.
2.10.
( )2,1,0a =G( )0,3, 2b =G .
2.11. ABC: , , .
( )2,1,2A( )4,2,2B ( )2,4,4C
2-8
-
2.12. ABJJJG
ACJJJG
, , , ( )2,1,2A ( )4,2,2B ( )2,4,4C .
2.13. , ( )2,1,2A ( )4,2,2B , ( )2,4,C x . x . 5AC =JJJG , AB
JJJG AC
JJJG.
2.14. , ( )2,1,0A ( )4,2,0B , ( )3,1,1C ( )2,4,D x . x , ABCD .
2.15. SE ABCS ( )1, 1, 1A , , ( )2,1,0B ( )1,0,1C ( )4, 1, 1S .
2.16. , .
( )1,2,1a =G( )0,1, 2b =G ( )3,0,2c =G
2.17. , ( )1,2,0a =G ( )2,0,1b =G jG . b
G. aG
2-9
-
3.
.
3.1.
AB , .., ( )1 1 1, ,A x y z ( )2 2 2, ,B x y z . ( ), ,M x y z AB AM MB :AM MB = :
1 2
1x xx += + ,
1 2
1y yy += +
1 2
1z zz += + .
A
BM
3.1.1: ( )2,1,3A ( )3, 1,0B . AB 1:3.
: : 1:AM MB = 3 , .., 13
= .
1 1 12 , , 24 2 4
M :
1 212 3 9 13 211 413
x xx + += = = =+ + 4
, ( )
1 2
11 1 2 1311 413
y yy + +
2= = =+ +
=
1 213 0 9 13 211 413
z zz + += = = =+ + 4
.
3.2.
3.2.1.
. :
: 0Ax By Cz D + + + = . - .
( , ,n A B C =G ) nG
3.2.1: 2 4 4 0x y z + = (3.1) . , ( )2,0,0A ( )0, 1,0B , ( )0,0,4C (?). ,
( , , 4 2 4 )M x y x y + (3.1) , (
3-1
-
ABJJJG
, ACJJJG
AMJJJJG
). ( )2, 4,1n = G (3.1) 0AB n =JJJG G . . :
: x y zm n p
+ + =1. x, y z m, n p, , 3.2.1.
m
x
y
z
np
m
3.2.1
3.2.2:
12 3 1x y z+ + =
x, y z 2, 3 1, , , . ( )2,0,0A ( )0,3,0B (0,0,1C )
3.2.3:
2 3 6 0x y z + + =
13 2 6x y z+ + =
2 32 3 6 0 2 3 6 1 1
6 6 6 3 2 6x y z x y zx y z x y z + + = + = + = + + = .
.
: 2 2 2
0Ax By Cz D
A B C
+ + + = + +
,
D. 3.2.4:
2 3 6 0x y z + + =
( )22 12 3 6 0
2 3 1
x y z + + = + +
, .., 2 3 1 6 014 14 14 14
x y z + = .
. ( 1 1 1, , )M x y z ( ), ,n A B C=G nG
( ) ( ) ( )1 1 1 0A x x B y y C z z + + = .
3.2.5: ( )2,1,3M ( )2,3,1n = G
( ) ( ) ( )2 2 3 1 1 3x y z + + = 0
3-2
-
2 3 2 0x y z + + = .
.
( )1 1 1 1, ,M x y z , ( )2 2 2 2, ,M x y z ( )3 3 3 3, ,M x y z .
1 1
2 1 2 1 2 1
3 1 3 1 3 1
1x x y y z zx x y y z zx x y y z z
=0.
3.2.6: ( )1 2,1,0M , ( )2 3,1,2M ( )3 3,0,1M
2 5x y z 0+ =
2 1 03 2 1 1 2 0 03 2 0 1 1 0
x y z =
, ., 2 1 0
1 0 21 1 1
x y z 0=
.
3.2.2. . 1 1 1 1: 0A x B y C z D + + + = 2 2 2 2: 0A x B y C z D + + + = . ( )1 1 1, ,n A B C =G ( )2 2 2, ,n A B C =G . :
1. 1 1 12 2 2
1
2
A B C DA B C D
= = = .
3.2.9: : 2 3 4 0x y z + = : 4 2 6 8 0x y z + =
2 1 34 2 6 8
4 = = = . 2. .
1 1 1
2 2 2
|| || 12
A B C Dn nA B C D
= = G G . 3.2.7:
: 2 3 4 0x y z + = : 4 2 6 1 0x y z + + =
2 1 34 2 6 1
4 = = . nG
nG
3-3
-
3. .
0n n n n = G G G G . 1 2 1 2 1 2 0A A B B C C + + =
3.2.8:
: 2 3 4 0x y z + = : 8 2 1 0x y z + + + =
( )2 1 1 8 3 2 0 + + = .
nG
nG
4.
( )1 2 1 2 1 2
2 2 2 2 2 21 1 1 2 2 2
cos n nn n
A A B B C C
A B C A B C
, = =+ +=
+ + + +
G GG G
.
nG
nG
3.2.10: : 4 7 4 4 0x y z + + = : 2 2 1 0x y z + + + =
026arccos 15.6427
=
( )2 2 2 2 2 2
4 2 7 2 4 1 26cos .274 7 4 2 2 1
+ + , = =+ + + +
3.2.3. .
1 1 1 1: 0A x B y C z D + + + = , 2 2 2 2: 0A x B y C z D + + + = 3 3 3 3: 0A x B y C z D + + + = .
( )1 1 1, ,n A B C =G , ( )2 2 2, ,n A B C =G ( )3 3 3, ,n A B C =G , a
1 1 1 1
2 2 2
3 3 3
2
3
A x B y C z DA x B y C z DA x B y C z D
+ + = + + = + + = . (3.2)
.
3-4
-
1. (3.2) - , ..,
1 1 1
2 2 2
3 3 3
0A B C
D A B CA B C
=
.
M
3.2.11: 3 2 1 02 3 5 8 0
0
x y zx y zx y z
+ =+ =
+ =
( )2,3,1M
3 2 12 3 5 31 1 1
D
= = 1 2 18 3 5 60 1 1
xD
, = =
,
3 1 12 8 5 91 0 1
yD = =
, 3 2 12 3 8 31 1 0
zD
. = =
2.
||n n
G G , ||n n G G
.., 1 1
2 2
1
2
A B CA B C
= = 1 13 3
1
3
A B CA B C
= = .
3.2.12: 3 2 1 06 4 2 8 09 6 3 2 0
x y zx y zx y z
+ = + = + + =
3 2 16 4 2
= = 18
3 2 19 6 3 2
1 = = .
3.
||n n G G , , ; ||n n
G G ||n n G G
..,
1 1
2 2
1
2
A B CA B C
= = , 1 13 3
1
3
A B CA B C
= = ,
2 2
3 3
2
3
A B CA B C
= = .
3.2.13:
3-5
-
: 3 2 1 0: 6 4 2 8 0: 9 6 5 2 0
x y zx y zx y z
+ = + = + =
3 2 16 4 2
= = 18
, 3 29 6
15
= 6 49 6
25
= .
4.
1 2 1 2 1 2 1 2
3 3 3 3
A A B B C C D DA B C D = = =
- .
3.2.14:
2 4 12 5 1
2 0
x y zx y z
x y z
00
+ = + + = + =
( )4 51 2 2 1 1 1
1 1 1 2 = = = .
5.
0D =
1 1 1
22 2
A B CA B C
= = , 3 3 32 2 2
A B CB C
= = , A
3 3
1 1
3
1
A B CA B C
= = . (3.3)
: (3.2) - . (3.3) , .
0D =
: : 2 1 0: 3 8 0: 2 2 0
x yx z
x y z
+ = + = + =
2 1 03 0 1 01 2 1
D = =
, 2 1 03 0 1 , 1 2 1
3 0 1 , 1 2 1
2 1 0 .
3.3.
3-6
-
: 0 0
1 2
: 03
x x y y z zp
m m m = = .
p ( )0 0 0, ,M x y z p.
( 1 2 3, ,pn m m m=G )pnG
M
p
: 0 1
0 2
0 3
:x x m t
p y y m tz z m t
= + = + = +.
3.3.1:
) 1 2:
2 1x y zp 3
1 + = = .
( )2,1, 1pn = G ( )1, 2,3M .
2 1: 2
3
x tp y t
z t
= + = = +.
) 3 2
: 24 1
x tp y t
z t3
= + = + = .
2 3:
3 21
4x y zp += = .
:
1 1 1 1
2 2 2 2
0:
0A x B y C z D
pA x B y C z D
+ + + = + + + =.
p 1 1 1 1: 0A x B y C z D + + + = 2 2 2 2: 0A x B y C z D + + + = .
p , .., -
1 1
2 2
1
2
A B CA B C
= = .
p
3.3.2: : 2 4 1
:2 1 0
x y zp
x y z0+ + = + + =
.
.
( )1 1 1, ,M x y z p ( 1 2 3, ,n m m m=G
)
3-7
-
nG
nG
pM
1 1
1 2
: 13
x x y y z zpm m m = = .
3.3.3: ( )1, 2,3M ( )2,1, 1n = G
1 2:2 1
x y zp 31
+ = = . 3.3.4:
2 4 1:
2 5 0x y z
px y z
0+ + = + + =.
:
0x =
4 1 02 5 0
y zy z+ = + =
.
3y = 1z = , .., ( )0, 3,1M .
-
( ) ( ) ( )2,1,4 1,1, 2 6,0,3pn = = G .
3 1:6 0 3x y zp + = = .
.
( )1 1 1 1, ,M x y z ( )2 2 2 2, ,M x y z p
1 1
2 1 2 1 2 1
: 1x x y y z zpx x y y z z = = .
p
1M2M
3.3.5: ( )1 1, 2, 1M ( )2 2,1,3M
( )( )
( )( )
21:2 1 1 2 3 1
y zxp1 = = , ..,
1 2:1 3 4
1x y zp + += = . .
1 11 2
: 13
x x y y z zpm m m = = 2 2
1 2
: 23
x x y y z zqn n n = = . p q
( )1 1 1 1, ,M x y z ( )2 2 2 2, ,M x y z , , ( )1 2 3, ,pn m m m=G ( )1 2 3, ,qn n n n=G . :
3-8
-
.
31 2
1 2 3
|| ||p qmm mp q n n
n n n = =G G
pnG
pqnG
q
3.3.6: 1 2:
1 3 41x y zp + += =
1 4:3 9 12
x y zq 7+ + += =
1 3 43 9 12 = = .
.
||p qp q n n G G 1M q 31 2
1 2 3
mm mn n n
= = 1 2 1 2 1 21 2 3
x x y y z zn n n = =
pnG
pqnG
q
3.3.7: 1 2:
1 3 41x y zp + += =
2 7:3 9 12
x y zq 11+ = =
1 3 43 9 12 = =
1 2 2 7 1 11 13 9 12+ = = = .
.
p q 1 2M M
JJJJJJG, pnG ,
(qnG
)1 2 p qM M n nJJJJJJG G G ( ) 2 1 2 1 2 11 2 1 2 3
1 2 3
0p q
x x y y z zM M n n m m m
n n n
= =JJJJJJG G G .
p
1Mq
2MpnG
qnG
3.3.8: 1 2:
1 3 41x y zp + += =
1 3:3 1 2
x y zq = =
1 3 43 1 2
( ) ( )0 1 1 2 3 11 3 43 1 2
=
0 .
.
p q .
q
pnG
qnG
( )qp,
ppnG
3-9
-
( )1 1 2 2 3 3
2 2 2 2 2 21 2 3 1 2 3
cos p q
p q
n np q
n n
m n m n m n
m m m n n n
, = =+ +=
+ + + +
G GG G
.
3.3.9: 1 2:
4 7 41x y zp + += = 2 1:
2 2 1x y zq 3 = =
( ) 0arccos 26 27 15.64=
( )2 2 2 2 2 2
4 2 7 2 4 1 26cos , .274 7 4 2 2 1
p q + + = =+ + + +
.
1 1 2 2 3 30 0p q p qp q n n n n m n m n m n = + + =G G G G
3.3.9: 1 2:
2 11
1x y zp + += =
2 1:1 3 1
x y zq 3 = = ( )2 1 1 3 1 1 0 + + = .
3.4.
1 11 2
: 13
x x y y z zpm m m = = : 0Ax By Cz D + + + = .
, ( 1 2 3, ,pn m m m=G ) )( , ,n A B C =G . : .
|| 0p pp n n n n =G G G G . 1 2 3 0m A m B m C + + =
nG
pnG p
3.4.1:
1 2:1 3
13
x y zp + += = : 3 2 1 0x y z + + = ( ) ( ) ( )1 3 3 1 3 2 + + = 0 .
.
p , .., 1 2 3 0m A m B m C+ + .
3.4.2: 1 2:
4 7 41x y zp + += = : 2 2 1 0x y z + + =
4 2 7 2 4 1 26 0 + + = .
3-10
-
.
p p .
( ) ( )0, 90 ,pp n n = G G ,
( ) ( ) ( )1 2 3
2 2 2 2 2 21 2 3
sin , sin , cos ,2 p p
p
p
p n n n n
n n m A m B m Cn n m m m A B C
= = = + += = + + + +
G G G G
G GG G
nG
pnG
p
( ) ,p
3.4.3: 3.4.2 026arcsin 74.36
27= ( )
2 2 2 2 2 2
4 2 7 2 4 1 26sin , .274 7 4 2 2 1
p + + = =+ + + +
.
31 2||pmm mp n n
A B C = =G G
3.4.4: 1 2:
1 3 41x y zp + += = : 3 9 12 1 0x y z + + =
1 3 43 9 12 = = .
nG
pnG
p
3.5.
( )1 1 1 1, ,M x y z ( )2 2 2 2, ,M x y z ; 1 1 1
1 2 3
:x x y y z zp
m m m = = 2 2 2
1 2 3
:x x y y z zq
n n n = = ;
1 1 1 1: 0A x B y C z D + + + = 2 2 2 2: 0A x B y C z D + + + = . : .
1M 2M ( )1 2 2 1 2 1 2 1, ,M M x x y y z z= JJJJJJG , ..,
( ) ( ) ( ) ( )2 21 2 2 1 2 1 2 1,d M M x x y y z z= + + 2 . 2M1M
3.5.1: ( )1 2,1,3M ( )2 3,0,1M ( ) ( ) ( ) ( )2 2 21 2, 3 2 0 1 1 3d M M = + + = 6 .
.
3-11
-
1M
( ) 1 1 1 1 1 1 11 2 2 21 1 1
,A x B y C z D
d MA B C
+ + + =+ +
.
.
1M
3.5.2: ( )1 2,1,5M : 2 3 4 0x y z + = ( ) ( )1 22 2
2 2 3 1 5 4 2 1,7142 3 1
d M + = = =
+ +4 .
.
, .., 1 1
2 2
1
2
A B CA B C
= = , ( ) ( 3,d d M = ), 3M
.
3.5.3:
: 4 6 2 12 0x y z + = : 2 3 4 0x y z + =
3M
( ) ( ) ( )3 22 22 2 3 1 5 4 2 1, ,
7142 3 1d d M
+ = = = =+ +
4 , ( )3 2,1,5M . .
1M p
( 1 2 3, ,pn m m m=G )
pnG 1 4M M
JJJJJJG,
pnG (4 1 1 1, , )M x y z
p, ..,
( ) 1 41, pp
n M Md M p
n
=JJJJJJGGG . p
nG
p
1M
4M
3.5.4: ( )1 2,1,3M 1 2: 1 3 41x y zp + += =
( ) 1 41 10 5 26, 1326p
p
n M Md M p
n
= = =JJJJJJGGG ,
, , ( )1,3,4pn = G ( )4 1, 2, 1M 26pn =G , ( )1 4 1, 3, 4M M = JJJJJJG , ( )1 4 0, 8,6pn M M = JJJJJJGG 1 4 10pn M M =JJJJJJGG .
.
3-12
-
p q ( )1 2 3, ,pn m m m=G qnG (5 5 5 5, , )M x y z q , ..,
( ) ( ) 5 45, , pp
n M Md p q d M p
n
= =JJJJJJJGGG .
4M p.
pnG
p
5M
4M
q
3.5.5: 1 2 1:
1 3 4x y zp + += =
2 1 3:2 6 8
x y zq = =
( ) ( ) 1 45 10 5 26, , 1326p
p
n M Md p q M p
n
= = = =JJJJJJGGG ,
q. .
(5 2,1,3M ) . p q ( )1 2 3, ,pn m m m=G
( 1 2 3, ,qn n n n=G )pnG , qn
G5 4M MJJJJJJJG
, pn
G qnG ,
( )4 4 4 4, ,M x y z ( )5 5 5 5, ,M x y z p q, , ..,
( ) ( )5 4, p qp q
n n M Md p q
n n=
JJJJJJJGG GG G .
p
q
pnG
qnG
qnG
4M
5M
3.5.6:
1 2 1:1 3 4
x y zp + += = 2 1 3:
2 6 2x y zq = =
( ) ( )5 4 60, 110 10
p q
p q
n n M Md p q
n n+= = .897
JJJJJJJGG GG G ,
, , ( )1,3,4pn = G ( )2,6, 2qn = G ( )30, 10,0p qn n = G G , 10 10p qn n =G G , , ,
( )4 1, 2, 1M ( )5 2,1,3M ( )5 4 1, 3, 4M M = JJJJJJJG , ( )5 4 60p qn n M M =JJJJJJJGG G .
3.1. :
a. 1: ;
(2,1,0M )3: 2
b. , ( )2,1,0A ( )1,2,0B ( )1,3,2C .
3-13
-
3.2.
( )2,1,0M2 3
2 1 11x y z = = .
3.3.
,
( )2,1,0A(1,3,2B ) 2 2
1 2 11x y z + = = .
3 2 5 6 0
:4 3 4 0
x y zp
x y z+ + + = + + + =
3.4. ( )2, 1, 0A ; 3.5. 1:
3 2 3x y zq = = ;
3.6. 1 : 3 2 3 8 0x y z + + = ; 3.7. 2 : 8x y z 0 + = .
3.8. 2 1:2 1 3
x y zp 3+ = = 2:
2 1 1x y zq = = .
.
3.9. : 4 7:1 8 3x y zp = = 1:
1 2x y zq
2+ = = . .
3.10.
: 2 3 38 0x y z + = 2 1:2 1 3
x y zp 3+ = = .
3 1:4 4
x y zpa + = = 2 . a
3.11. p ( )7, 9, 6A ; 3.12. p 2 4 5x y z 0 + = .
3.13. ( )3,5,4A 2 1:
1 2 2x y zp = = .
3.14. 2 1:4 8 8
x yp z= = 1 1:
2 4 41x y zq = = .
3-14
-
3-15
1. 1 2. 3. Vektori.pdf 2.2.1: , .
Nizi.pdf
Izvodi.pdf
Integral.pdf 9.1.3. . . 9.1.6. , c , ,
9.2.2. 9.2.1. ,
9.2.3. C 9.2.3. - : 9.3.1. : ) ; ) . 9.5.1. : 9.5.2. .
9.5.5. : 9.6.1. x- x- 9.6.2. x- 9.6.3. , , y- x- 9.6.4. y-
Dodatok 2 - Mat formuli.pdf 2
-
Naslovna-sodrzina.pdf . 1. .
Dodatok - Mat formuli.pdf
-
Dodatok - Ispitni prasanja IM.pdf
Dodatok - Ispitni prasanja M1.pdf
