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..
.. , ..
2009
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2 1
. ( -). -, (). . . , , . , . . -
1.1.
. , . , , , ( ) . ., , , , . .
1.1. ( ) , (. 1.1).
,
A
B
. 1.1
- , , .. , , , .
: - , -. - .
: A , B , AB
AB.
AB , .
|AB| |AB| , A B. - , . , , , . - - 0. , -. - , -
. , .
1
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2 1. 2
1.2.
1.2. , 1 .
, - . :
( ) -, ;
, .
, - . 1.2 .
1.3. , , .
A B
C D
A B
CD
. 1.2
, , , . .
1.4. , : ; .
1.4 , . , .. , , , . , . . : a a . a, . -
. , , - , , . . , - , , . , , .
1, , , .
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2 1. 3
1.1. , . , , -
. , , - .
- , .. . , .. .
1.1. , , . , ( ) , , () . ( -, ) (-), . (-) .
1.3.
. .
1.5. a + b a b c, . b a. c, a, b (. 1.3).
1.2. -. a b , . , , (. 1.4). , a b , - , .
a
bc
. 1.3
a
b
c
. 1.4
1.3. 1.5 -, , , . #
1. : a + b = b + a.
J , - , . , - , . I
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2
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2
-1
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-1
2
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2
-1
2 1. 4
2. : (a + b) + c = a + (b + c).
J . a A (. 1.5), a =
AB. b
B, b =BC. , c C b,
c =CD.
a
b
c
a + b b + c
(a + b) + c
a + (b + c)
A
B
C
D
. 1.5
AD =
AB +
BD =
AB + (
BC +
CD) = a + (b + c),
AD =
AC +
CD = (
AB +
BC) +
CD = (a + b) + c,
.. AD , -
. I
3. 0, a a+0 = a.
J , , - . . I
4. a b, a + b = 0.
J , (a), a, .. , a, , a, . a, a. , a (a) , .. . I
5. a b x, a + x = b. x .
J (a) + b, 24
a + x = a +((a) + b
)=(a + (a)
)+ b = 0 + b = b.
x a+x = b, , - (a), 1, 2, x = (a)+b. ,
(a) +(a + x
)=((a) + a
)+ x = 0 + x = x = (a) + b.
, x . I
5 .
1.6. b a a b x, a + x = b.
a+x = b x = ba ( - 1.6) , .
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2 1. 5
a
b
b
a
. 1.6
-. a b, a , b, b a (. 1.6). , -
.
1.7. a a, a, || |a|, a > 0 < 0.
1.4. = 0, , , 0a 0 |a| = 0, .. . , (, ), 0 : 0a .
1.2. a 1 , a, ..(1)a = (a). # ,
.
6. : ()a = (a).
J , , a. , . . , a. , a. , ,.. . I
7. : (a+b) = a+b.
J = 0 , ( 0), . 6= 0, . . 1.7 : ) > 0; ) < 0. I
8. : (+ )a = a + a.
J . , , . , , - a. , : (+ ) |a| = |a|+ |a|. , , . . , > 0,
< 0. - . > 0, < 0, a a , .
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2
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2 1. 6
ab
+ab a
b
a+b
a+b
ab
ba
a+b
. 1.7
a || > || || < ||. , , | + ||a|. , , ( + )a, .. ., , .
, = 0, ( + ) |a| = |a| + |a| ( + 0)a == a + 0a, 3 1.7. I
1.4.
L A. A L - (. 1.8, ). ( O) A L. L A , A L O L , A (. 1.8, ). A L, L.
L
A
O
90
90
O
A
L
. 1.8
AB ( ) -
L (. 1.9). OAOB,
A
OA
L
B
OB
. 1.9
OA OB L,
AB L.
, -, . .
OAOB
AB l
OAOB, ,
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2 1. 7
. OAOB -
, , , .
OAOB , ,
AB l l a.
, , . , . l : -
. - l. -
. 0 . 0 , -. , . , , , ., , 0 (). , - , .
1.1. a l |a|, a l, ..
l a = |a| cos(a, l),
(a, l) a l.
J l L, A. a A, B (. 1.10).
C B L. AC
a =AB L.
LA C
B
l
a
LAC
B
l
a
. 1.10
a l ( . 1.10, ), - l C A. a l |AC| = |AB| cos AC ABC. (. . 1.10, ), l C
A. , AC l ,
a |AC|. ABC , AC, , |AC| = |AB| cos( ) = |AB| cos. =
2 a = 0, C A
AC
. cos
2= 0, , -
. I
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2 1. 8
1.2. - ,
:l(a + b) = l a + l b, l(a) = l a.
J . 1.11. , . 1.11, , l a == |AB|, l b = |BC|, l(a+b) = |AC| = |AB||BC|. , . 1.11, ,l a = |AB| , > 0, l(a) = |AE| = |AB|. ( C AB , 6 0 ) . I
A C B
a
b
Ll
LA B l
a
a
E
. 1.11
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.
. . - , . - V1, V2, V3 . . . , . . , . , . - , .
. .
2.1.
- . a1, . . . , an,
ni=1
iai = 1a1 + . . .+ nan, (2.1)
1, . . . , n . a1, . . . , an. i, i = 1, n, - . .
, - a1, . . . , an. , , , .
2.1. a1, . . . , an , 1, . . . , n,
1a1 + . . .+ nan = 0 (2.2)
. - , .
1 = . . . = n = 0, , , 1a1 + . . . + nan = 0. , : a1, . . . , an , (2.2) , 1, . . . , n . ,
(
), .
2.1. a1, . . . , an, n > 1, , - , .
J . , a1, . . . , an . 2.1 , (2.2)
9
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2 2. . 10
, 1. , , , , . 1,
a1 = 21
a2 . . .n1
an,
.. a1 a2, . . . , an. . , , a1 -
: a1 = 2a2 + . . .+nan. , a12a2 . . .nan = 0, .. a1, . . . ,an 1 = 1, 2 = 2, . . . , n = n, . . 2.1, a1, . . . , an . I
, . . ,
. ,
, ( , ). . -
.
2.2. , .
J a b , , a, ,.. a = b . 1.7 , a b . a b . , ,
, . 0, b. : = |a|/|b|. - . . , 1.7, , a = b. 2.1, a b . I
2.1. , , - : , . .
2.3. , .
J a, b, c , , 2.1, , a, : a = b + c. b c A. b, c A , .. a, A , , -. , , .. . a, b, c . , -
, . . , . O. - A, B, C (. 2.1). C , , O, A O, B. A B, -
OACB, ,OC =
OA+
OB.
OA
a =OA ,
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2 2. . 11
:OA =
OA.
OB =
OB, R. ,
OC =
OA+
OB, .. c a b.
2.1, a, b, c . I
O A'
B
B'
A
C
. 2.1
2.4. .
J , 2.3. a, b, c d. , , , . , a b , a+b = 0 , , . 0 , . , , ,
. O. a, b, c, d A, B, C, D(. 2.2). D , OBC, OCA, OAB, A, B, C OA, OB, OC -. OAC BC BDA, a, b, c , O. OC DC ,
OD =
OC +
OC . , OC
OAC B, OC =
OA +
OB,
OD =
OA +
OB +
OC .
AO
B
C
A'
B'
C'
C"
DA"
B"
. 2.2
, OA 6= 0 OA, OB 6= 0 OB, OC 6= 0 OC
, , , , , , OA =
OA,
OB = OB
OC =
OC.
OD =
OA+
OB +
OC. ,
OD , ,
2.1, . I
2.2.
( , -) , , , : V1 , ..
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2 2. . 12
, V2 , .. - , V3 . V1. V1 -
V1. , , ,.. . V1 , .. e 6= 0. x V1 x = e. x e, x . , . -, = |x|/|e|, , x e , . V2.
V2 V2. , .. - e1, e2. 2.3, x, , . - . , x e1 e2.,
x + 1e1 + 2e2 = 0, (2.3)
. , 6= 0, (2.3) , , e1, e2 . , (. 2.2). 6= 0, (2.3) . , ,
x = 1e1 + 2e2, (2.4)
x e1, e2, 1, 2 x e1, e2., (2.4) 1 2 .
, 2.3 ( , ). , e1 e2. , x = 1e1 + 2e2 = 1e1 + 2e2, ,
8 (. . 5) , (1 1)e1 + (2 2)e2 = 0. - , e1, e2 ( , . 2.2). , 1 = 1, 2 = 2, x . V3. -
V3. V3 , .. e1, e2, e3. x (. 2.4). , V2, x e1, e2, e3:
x = 1e1 + 2e2 + 3e3. (2.5)
(2.5) , e1, e2,e3 . x (2.5) e1, e2, e3, 1, 2 3 x e1, e2, e3.
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2 2. . 13
V2 V3, , -. , . , , - V3 , V3 , . , , .. V3, - . , , . ,
. , x V3 e1, e2, e3 x == 2e1+3e24e3, , : {2; 3;4}. , x = {2; 3;4}, ., V1, V2 V3 ,
, . , ,
, .
2.3.
, .
2.5. -. .
J , , V3. V3 e1, e2,e3. x y :x = x1e1+x2e2+x3e3, y = y1e1+y2e2+y3e3. , :
x + y = (x1e1 + x2e2 + x3e3) + (y1e1 + y2e2 + y3e3) = (x1 + y1)e1 + (x2 + y2)e2 + (x3 + y3)e3.
. , xi yi , ei (i = 1, 2, 3), . x = (x1e1 + x2e2 + x3e3) =
= (x1)e1 + (x2)e2 + (x3)e3. x . , x . I
. , -, V3. d , a, b, c, . 2.2. d
da = |OA||OA|
, db = |OB||OB|
, dc = |OC ||OC|
,
, (,
OA
OA da) .
2.6. , , .
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2
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2 2. . 14
J V3. V3 e1, e2, e3. x y:
x = x1e1 + x2e2 + x3e3, y = y1e1 + y2e2 + y3e3. (2.6)
, .. R,
x1 = y1, x2 = y2, x3 = y3, (2.7)
x = x1e1+x2e2+x3e3 = y1e1+y2e2+y3e3 = (y1e1+y2e2+y3e3) = y 2.1, 2.2, x y . . - , x y . 2.2 2.1 , x,
, .. x = y. (2.6),
x1e1 + x2e2 + x3e3 = (y1e1 + y2e2 + y3e3), (x1 y1)e1 + (x2 y2)e2 + (x3 y3)e3 = 0. 0 , (2.7). I
2.1. , , .
J (2.7) , ,
x1y1
=x2y2
=x3y3. (2.8)
, (2.8) , -, . V2 (2.8) . I
2.1. e1, e2 V2. a = 2e1 3e2, b =e1 + 3e2 , 2/(1) 6= 3/3. V2. e1 c = 3e1 6e2. e1, a b: a + b = e1. -
, e1 = a + b. c, . c = 1a + 2b.
c, a, b e1, e2, . ,
3e1 6e2 = 1(2e1 3e2) + 2(e1 + 3e2) = (21 2)e1 + (31 + 32)e2.
, 1, 2 {
21 2 = 3,31 32 = 6.
, , 1 = 1, 2 = 1. , c = a b.
2.2. , , - . , .
, . 2.2 i, j, k V3, , A, B, C D - . d =
OD
, .
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2
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2 2. . 15
V3 , , - i, j, k, V2 i, j V1 i. V3 O
X. |OX|2 = |OXi|2 + |OXj|2 + |OXk|2 (. 2.3), Xi, Xj, Xk X .
OXi, OXj, OXk x =
i, j, k. x {x1;x2;x3} i, j, k V3:
|x| =x21 + x
22 + x
23, x V3. (2.9)
O
X
Xii
j
k
Xj
Xk
. 2.3
V2 x1, x2 :
|x| =x21 + x
22, x V2, (2.10)
V1 x1 :
|x| =x21 = |x1|, x V1. (2.11)
x V3 i, j, k , . cos, cos , cos - x (. 2.4). , .
O i
j
k x
. 2.4
. x V3 i, j, k {x1;x2;x3} cos, cos , cos ,
x1 = |x| cos, x2 = |x| cos , x3 = |x| cos . (2.12)
(2.9) , |x|2 = |x|2 cos2 ++ |x|2 cos2 + |x|2 cos2 , |x| 6= 0 :
cos2 + cos2 + cos2 = 1. (2.13)
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2 2. . 16
, , [0, ], (2.13), , cos, cos , cos . - {cos; cos ; cos }. (2.9) (2.13) , cos, cos , cos . V2 -
, (. 2.5). , x - {x1;x2} : x1 = |x| cos, x2 = |x| sin.
O i
j
x
'
. 2.5
2.4.
. , - , .. .
2.3. a b , |a| |b| cos |a| |b| . ,
, . a b ab, -
(a, b). 1.1, -
. a , ab a b a - b a: ab = |a| a b. b 6= 0 ab = |b| b a. (.. 90),
. .
2.7. , , .
J 2.3, a b |a| |b| cos. a b:
: ab > 0; : ab < 0; : ab = 0. , ,
. , . , , , , . I
.
1. : ab = ba.
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2 2. . 17
J 2.3, , , . I
2. :
(a)b = (ab).
J b = 0 , . b 6= 0, , b 1.2,
(a)b = b(a) = |b| b(a) = |b| b a = (ab). I
3. :
(a + b)c = ac + bc.
J . c = 0 . c 6= 0, c. 1.2,
(a + b)c = |c| c(a + b) = |c| (c a + c b) = |c| c a + |c| c b = ac + bc. I
aa a a2.
4. : a2 > 0, a2 = 0 , a = 0.
J , a2 = aa = |a| |a| cos 0 = |a|2 . - , a2 > 0 . a2 = 0 |a| = 0, .. , a . I 2.2. 2 3
. ( 1) . -, a(b) = (b)a = (ba) = (ab), a(b + c) = (b + c)a = ba + ca = ab + ac. #
.
2.2. a = 3c 2d , |c| = 5, |d| = 4, c d 60. |a| =
a2, , a, ,
a2 = (3c 2d)(3c 2d) = 9c2 12cd + 4d2 = 9 |c|2 12 |c| |d| cos+ 4 |d|2 == 9 25 12 5 4 0,5 + 4 16 = 225 120 + 64 = 169.
, |a| =a2 = 13.
2.3. ABC A 120, AC A B. BC AM . BC AM (. 2.6)
BC
AM . 2.3 ,
cos =
AM BC|AM ||BC|
.
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2
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2
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2
-1
2
-1
2 2. . 18
A
B
C
120M '
. 2.6
|AB| = s. |AC| = 3s, BC = AC AB, AM =
AB +
BM =
AB + 0,5
BC =
AB + 0,5
(AC AB
)= 0,5
(AC +
AB)
AM BC = 0,5(AC +AB)(AC AB) = 0,5(|AC|2 |AB|2) = 0,5(9s2 s2) = 4s2.
AM
BC:
|AM | =AM AM = 0,5
(AC +
AB)(
AC +
AB) =
= 0,5
(AC2 + 2
AB AC +AB2 = 0,5
9s2 + 6s2 cos 120 + s2 = 0,5s
7,
|BC| =BC BC =
(AC AB)(AC AB) =
=
(AC2 2AB AC +AB2 =
9s2 6s2 cos 120 + s2 = s
13,
,
cos =4s2
0,5s
7 s
13=
891.
, BC AM = arccos(8/
91). #
a b V3 i, j,k: a = {xa; ya; za}, b = {xb; yb; zb}. , a = xai + yaj + zak,b = xbi + ybj + zbk. 1
4 ,
ab = (xai + yaj + zak)(xbi + ybj + zbk) =
= xaxbii + xaybij + xazbik + yaxbji + yaybjj + yazbjk + zaxbki + zaybkj + zazbkk =
= xaxbi2 + yaybj
2 + zazbk2 = xaxb + yayb + zazb.
, i, j, k ij = ik = jk = 0, i2 = j2 = k2 = 1. ,
ab = xaxb + yayb + zazb, (2.14)
.. - . 2.7 (2.14)
a b:xaxb + yayb + zazb = 0. (2.15)
, , 2.3 , ab = |a| |b| cos, = (a, b) a b. , , . ,
cos =ab
|a| |b|,
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2
-1
2
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2
-1
2
-1
2
-1
2
-1
2 2. . 19
cos =xaxb + yayb + zazb
x2a + y2a + z
2a
x2b
+ y2b
+ z2b
. (2.16)
, a, b V2 - i, j: a = xai + yaj, b = xbi + ybj, , (2.14)(2.16):
ab = xaxb + yayb; (2.17)
xaxb + yayb = 0;
a b
cos(a, b) =xaxb + yayb
x2a + y2a
x2b
+ y2b
.
2.4. t, a = {t; 1 t; 7} b == {t+ 1; 2;2}, , . (2.15) ,
t(t+ 1) + 2(1 t) 14 = 0
t. , , t = 3 t = 4 .
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2
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2
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2
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2
-1
2
-1
2
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2
-1
2 3
, . , . . . . . - . . .
3.1.
V3. .
3.1. a, b, c , a b a , c (. 3.1). ( ) .
a
b
c
. 3.1
V3, . , , .. V3 : . , , .
3.2. a b - c, :
1) c a b;2) c |c| = |a| |b| sin, a b;3) a, b, c (. 3.2).
a
b
c
. 3.2
20
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2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 3. 21
a b , |a| |b| sin = 0. 3.2,, 2, -. a b ab,
[a, b]. , , . , F , -
M , O OMF (. 3.3).
.
MOF
OM F
. 3.3
1. , , -.
J . , - . . ab = 0, |ab| = 0, ..|a| |b| sin = 0, a b. , -, : |a| = 0, |b| = 0 sin = 0. a b. I
.
2. a b , |ab| , (. 3.4).
a
b
. 3.4
J ,
( ) . I
3. : ab = ba; (a)b = (ab); (a + b)c = ac + bc.
J , , a b , ab = ba 1 . a b , , . 3.2 ab ba - , , . , ab ba , ( 2). , ab ba . , a, b,ab , b,a,ab . , - , b,a,ab, ab ba .
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2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 3. 22
3.2, , ab b a, .. ab = ba. . a
b, = 0 (a)b (ab) -, , , . , (a)b = (ab). , a b , 6= 0. ,
, ., , a, b a , a, b a,b , ab (a)b. (ab) (a)b . , , ,
|(ab)| = || |ab| = || |a| |b| sin,
a b,
|(a)b| = |a| |b| sin = || |a| |b| sin = || |a| |b| sin,
a b sin = sin, 6= 0. , , , -
. , , (a)b (ab) . > 0, a a . , (a)b ab
. ab, (ab) , (a)b (ab). < 0, a a . ,
(a)b ab . ab . (a)b (ab) . (. 3.2).
3.1. , , - . - :
a(b) = (b)a = (ba) = (ab),a(b + c) = (b + c)a = (ba + ca) = ab + ac.
3.1. S , a = 3c 2d b = c + d , |c| = 1, |d| = 4, c d 30.
S = 0,5 |ab| .
, ,
ab = (3c 2d)(c + d) = 3cc + 3cd 2dc 2dd = 3cd + 2cd = 5cd.
S = 0,5 |ab| = 0,5 |5cd| = 2,5 |c| |d| sin = 5. #
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 3. 23
, . - . i, j, k. -
( 9 ) :
ij = k, ji = k,jk = i, kj = i,ki = j, ik = j.
(3.1)
, -. (3.1) : -
, , ( , , ) i, j, k . . 3.5 . . 3.5 (3.1) .
i
j
k
. 3.5
a b, i, j, k: a = {xa; ya; za}, b = {xb; yb; zb}.
a = xai + yaj + zak, b = xbi + ybj + zbk.
,
ab = (xai + yaj + zak)(xbi + ybj + zbk) == xaxbii + xaybij + xazbik ++ yaxbji + yaybjj + yazbjk ++ zaxbki + zaybkj + zazbkk =
= (yazb ybza)i + (zaxb zbxa)j + (xayb xbya)k =
=
ya zayb zb i xa zaxb zb
j + xa yaxb ybk.
, , 1- , - . , . , . , , . 1. , i, j, k -
ab =
i j kxa ya zaxb yb zb
. (3.2)
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 3. 24
3.2. ,
n1 = {3; 1;2} n2 = {1;1; 1}.
, n1 n2 , ,:
3
16= 11
.
. , . , , , , ,
n1n2 =
i j k3 1 21 1 1
= i 5j 4k.: (i 5j 4k), R.
3.2.
3.3. a, b, c , (ab)c .
a, b, c : abc. .
3.1. abc , , , , a, b, c , , .
J ab , a b, 2 , S (. 3.6). e , a b - ab,
ab = Se.
a b
ce
a
hb
S
'
. 3.6
abc Se c S |c| cos, ab c. , |c| cos c e, , .. | |c| cos|, h. c e. < 90, c e a, b. , a, b, e
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 3. 25
a, b, c . . > 90, , .. a, b, c , . , = 90 , , c a b, . I
3.2. a, b, c , , , ( ). . - , a, b, c ab c , .. (ab)c = 0. , , -. #
.
1. :
abc = bca = cab = bac = cba = acb.
J , , -. , . I
3.3. ,
abc = (ab)c = a(bc),
.. , , . , -.
2. a, b, c , .
J 3.1 3.2. I
3. : (a)bc = (abc).
J bc = e 2 - ,
(a)bc = (a)(bc) = (a)e = (ae) = (a(bc)
)= (abc). I
4. (a1 + a2)bc == a1bc + a2bc.
J bc = e 3 ,
(a1 + a2)bc = (a1 + a2)(bc) = (a1 + a2)e = a1e + a2e == a1(bc) + a2(bc) = a1bc + a2bc. I
3.4. 3 4 . , ..
a(b)c = (abc), ab(c) = (abc),
a(b1 + b2)c = ab1c + ab2c, ab(c1 + c2) = abc1 + abc2,
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 3. 26
-.
3.5. , . , - . , . . x1 x2 ,
y x1y = x2y, (3.3)
x1 = x2. , (3.3) , (x1 x2)y = 0. y, y = x1x2. (x1x2)2 = 0, x1 x2 = 0, .. x1 = x2. ,
(a1 + a2)b = a1b + a2b
, c((a1 + a2)b
)c = (a1b)c + (a2b)c,
(a1 + a2)bc = a1bc + a2bc.
, 4 . #
a, b, c :a = {xa; ya; za}, b = {xb; yb; zb}, c = {xc; yc; zc}. , :
abc = a(bc) = a( yb zbyc zc
i xb zbxc zc j + xb ybxc yc
k) == xa
yb zbyc zc ya xb zbxc zc
+ za xb ybxc yc =
xa ya zaxb yb zbxc yc zc
=xa xb xcya yb ycza zb zc
. , 2 -
: , , , .
3.3. ,
a = {2; 1;2}, b = {1; 0;1}, c = {1; 1; 1}
. ,
, 6 , - abc . , V = |abc|/6 = 1,
abc =
2 1 2
1 0 11 1 1
= 6.
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 3. 27
3.3.
.
3.4. A F M1 M2 A = |F | |M1M2| cos (. 3.7, ).
A = Fs, s =M1M2.
F
M1
M2'
vP
L
O
!
. 3.7
n f i, i = 1, n, A Ai
F =ni=1
f i,
A =ni=1
Ai =ni=1
f is =( ni=1
f i
)s = Fs.
, , s. , A = Fs = 0 , , .
3.5. - L, O (. 3.7, ). v P . v = OP, . 3.6. v q -
B f = qvB. v B , vB = 0 . v B , f 6= 0, , , : W = fv = q(vB)v = qvBv = 0, - . , m mv2/2. , v B , .. vB = 0.
f B, , B, R, ,
mv2/R = q|vB| = q|v||B| sin 90.
R =m|v|q|B|
.
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 4
.
. -- , ; . : , . f(x, y) = 0 F (x, y, z) = 0 . : , - , , ,
. .
. . .
, . -
, . , -, , , - , , , (, ). , , . - .
4.1.
. - , . , - . , . , . () , -
O . V1, V2, V3 : , . - () M
OM .
: () O ; , -
; ( ) , , -
. ( ): , . : -
28
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 4. . 29
, . , .
, . - ;
- M OM , O
. -
. , -
, , . , .. -
, , ., V2 (.. ) (), - ( ) .,
, . - , , - Oijk, O , i, j, k (), Oxyz, .
4.2.
, , -, .. ( ) . . , .. . Oijk ,
, ij k , (. 4.1). , (b1; b2; b3) i = {11;21;31}, j = {12;22;32}, k = {13;23;33} . (x; y; z) (x; y; z) . ,
= xi + yj + zk
= xi + yj + zk. (4.1)
i'
j'k'
i
j
kM
O
O'
. 4.1
OM
OM
OM =
OO+
OM ,
OO O -
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 4. . 30
, ..OO = b1i + b2j + b3k.
OM =
OO +
OM = b1i + b2j + b3k + x
i + yj + zk =
= b1i+b2j+b3k+x(11i+21j+31k)+y
(12i+22j+32k)+z(13i+23j+33k) =
= (11x+12y
+13z+b1)i+(21x
+22y+23z
+b2)j+(31x+32y
+33z+b3)k, (4.2)
.. OM . -
(4.1) . (4.1) (4.2),
x = 11x + 12y
+ 13z + b1,
y = 21x + 22y
+ 23z + b2,
z = 31x + 32y
+ 33z + b3.
(4.3)
(4.3), , x, y, z. x, y, z , -. (4.3) x, y, z , . ,
11 12 1321 22 2331 32 33
=11 21 3112 22 3213 23 33
= ij k = 1, i, j , k 1.
ij (4.3) -, b1, b2, b3 . , , - :
x = x + b1,
y = y + b2,
z = z + b3.
(4.4)
(4.4) -
OO.
. - , , .
x = 11x + 12y
+ b1,
y = 21x + 22y
+ b2,(4.5)
{1i;2i}, i = 1, 2, i, j (i, j), (b1; b2) O
.
- :
x = x + b1,
y = y + b2.
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 4. . 31
, , :
x = 11x + 12y
,
y = 21x + 22y
.(4.6)
. , , > 0 ( < 0) ( ) . (4.6) . , i j : i = {cos; sin}, j = { sin; cos} (. 4.2).
x
x'
yy'
O i
i'
jj' '
'
. 4.2
, - :
x = x cos y sin,y = x sin+ y cos.
(4.7)
-, :
x = x cos y sin+ b1,y = x sin+ y cos+ b2.
(4.8)
(4.8) x, y, , - , :
x = x cos+ y sin+ b1,
y = x sin+ y cos+ b2,
b1 = b1 cos + b2 sin, b2 = b1 sin + b2 cos. ,
, (
), ( OO). -
i i, j j
. , -
. .
, :x = 11x
+ 12y + 13z
,
y = 21x + 22y
+ 23z,
z = 31x + 32y
+ 33z.
(4.9)
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 4. . 32
(4.9) , - , .. . , , . , i i ii, i j j . k k - . , , ( , i j ).
4.3.
, - .
. , . -
O AB, A(xa; ya; za)
B(xb; yb; zb). {l;m;n} AB. A B
-OA
OB. ,
OB OA =
= {xb xa; yb ya; zb za} AB =
OB OA , {l;m;n} =
= {xb xa; yb ya; zb za}, ..
l = xb xa, m = yb ya, n = zb za. (4.10)
AB = {l;m}
A(xa; ya) B(xb; yb) -
l = xb xa, m = yb ya. (4.11)
(4.10) (4.11) : ;
; -
.
. , M1M2 , : |M1M | : |MM2| = p : q. M1M2
M1M
MM2
(. 4.3). , -
. , ,MM2 =
M1M .
MM2 M1M , .. = q/p. M1M2 =
M1M +
MM2 =
M1M +
qp
M1M =
p+qp
M1M,
M1M =
pp+q
M1M2.
M1
MM2
. 4.3
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 4. . 33
M1 M2 M1M2 - Oijk : M1(x1; y1; z1), M2(x2; y2; z2). M .
OM =
OM1 +
M1M =
OM1 +
p
p+ q
M1M2 =
= {x1; y1; z1}+p
p+ q{x2 x1; y2 y1; z2 z1} =
{px2 + qx1p+ q
;py2 + qy1p+ q
;pz2 + qz1p+ q
}.
, M (x; y; z),
x =px2 + qx1p+ q
, y =py2 + qy1p+ q
, z =pz2 + qz1p+ q
. (4.12)
M M1M2, p = q = 1, (4.12) , M , ..
x =x1 + x2
2, y =
y1 + y22
, z =z1 + z2
2. (4.13)
M(x; y), M1M2 |M1M | : |MM2| = p : q, M1(x1; y1) M2(x2; y2) x =
px2 + qx1p+ q
, y =py2 + qy1p+ q
,
x =x1 + x2
2, y =
y1 + y22
.
4.1. A(4; 4; 4), B(2; 6; 4), C(4; 4; 2) ABC . . M -
ABC. N BC. (x; y; z) B C, , x = 3, y = 5, z = 3. AN M |AM | : |MN | = 2 : 1, (x0; y0; z0) (4.12)
x0 =2 (3) + 1 4
2 + 1= 2
3, y0 =
2 5 + 1 42 + 1
=14
3, z0 =
2 3 + 1 42 + 1
=10
3.
. ( -) . . , ,
, . M1 M2 Oijk: M1(x1; y1; z1), M2(x2; y2; z2). M1M2 = {x2 x1; y2 y1; z2 z1}.
M1M2, -
i, j, k, (2.14) :
M1M2
2 = (x2 x1)2 + (y2 y1)2 + (z2 z1)2. , M1M2 |M1M2| =
(x2 x1)2 + (y2 y1)2 + (z2 z1)2.
4.4.
, , , .
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 4. . 34
, , , . - , , .
4.2. : A(4; 4; 4),B(1; 2; 3), C(3;1; 2). 4ABC (4.10) AB AC:AB = {1 4; 2 4; 3 4} = {3;2;1}, AC = {3 4;1 4; 2 4} = {1;5;2}.
(3.2) :
ABAC =
i j k3 2 11 5 2
= i 5j + 13k. |ABAC| =
(1)2 + (5)2 + 132 =
195, ,
, S4ABC =|ABAB|
2=
195
2. #
, , , , - . SABC (. 3.2), 1/6
, AB,
AC
AS. ,
VSABC = |ABACAS|/6.
4.3. V SABC, :A(2;1; 1), B(5; 5; 4), C(3; 2;1), S(4; 1; 3). (4.10), , :
AB = {5 2; 5 (1); 4 1} = {3; 6; 3}, AC = {3 2; 2 (1);1 1} = {1; 3;2}, AS == {4 2; 1 (1); 3 1} = {2; 2; 2}, :
ABACAS =
3 6 31 3 22 2 2
= 18, V = 16 |ABAC AS| = 3.
4.5.
() , . , .
4.1. F (x, y, z) = 0 - x, y, z, M(x; y; z) S , F (x, y, z) = 0 S, S - .
, F (x, y) x y, .
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 4. . 35
4.2. F (x, y) = 0 - x y, M(x; y) , F (x, y) = 0 , .
, - F (x, y, z) = 0, F (x, y, z) , x, y, z . F (x, y, z) -, , . , , , , . . -, S . F (x, y, z) = 0, (x; y; z) S, F (x, y, z) = 1 . F (x, y, z) = 0 S. F (x, y, z) = 0 (F (x, y) = 0
), F .
4.3. n x1, . . . , xn
F (x1, . . . , xn) =m
i1+...+in=0
ai1,...,inxi11 . . . x
inn ,
i1, . . . , in ; ai1,...,in , ai1,...,in , i1 + . . .+ in = m, . m n .
. - F = a0,...,0 -. . , F == 2x4y+5z1 , F = xy+3 . n < k m n k , .. . F (x1, . . . , xn) = 0, n ,
.
4.4. , F (x, y, z) = 0, F - x, y, z.
F F = 0 , -.
4.5. ( ) , F (x, y) = 0, F - x, y.
() . x, y, z , x, y, z , (4.3), F (x, y, z) = 0. , , x, y, z x, y, z. ( )
, .
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 4. . 36
, , , -. , () . , (-
), . , , F (x, y, z) = 0, F , (F (x, y, z)
)2= 0, . ,
( ), -.
4.6. , - ( ) , - ().
, ( ) (, , ), , 1 2. .
-, F1(x, y, z) = 0, F2(x, y, z) = 0, {
F1(x, y, z) = 0,F2(x, y, z) = 0.
. , , - .
x = (t), y = (t), z = (t). (4.14)
. - , t . () , - . (4.14), ,
. , , (4.14).
4.4. ,
x2 + y2 z2 = 0, x z + 1 = 0,
. z x2 + y2 (1 + x)2 = 0 y2 = 2x + 1. x, , :
x =y2 1
2, z =
y2 + 1
2.
, t = y, - :
x =t2 1
2, y = t, z =
t2 + 1
2. #
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 4. . 37
, .
4.5. R -
x = R cos cos, y = R cos sin, z = R sin,
, . || 6 /2, < 6 .
4.6.
, .. , - ax + by + c = 0, a b 1. .
4.1. .
J L . M0(x0; y0) L, n = {a; b} . M(x; y) L ,
M0M
n (. 4.4).
90
n
M0
ML
. 4.4
M0M = {x x0; y y0} n,
: a(x x0) + b(y y0) = 0 ax + by + c = 0, c = ax0 by0. n 6= 0, a 6= 0, b 6= 0. . -
ax + by + c = 0, a2 + b2 6= 0. ., a 6= 0, x = c/a, y = 0. , - . M0(x0; y0) , .. ax0 + by0 + c = 0. - . , .
a(x x0) + b(y y0) = 0. (4.15)
, - n = {a; b} M0M , M (x; y). ,
1, a b , : a2 ++ b2 6= 0.
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 4. . 38
M(x; y) ax+ by+ c = 0, n -
M0M , .. M , M0
n. I
4.7.
ax+ by + c = 0, a2 + b2 6= 0, (4.16)
.
4.1 , a b . , . . , -, () . L (4.16). M0(x0; y0) L,
(4.16), .. ax0 + by0 + c = 0. M1(x1; y1), L, (4.16)
ax1 + by1 + c = ax1 + by1 ax0 by0 = a(x1 x0) + b(y1 y0) = nM0M1 6= 0.
nM0M1
M0M1 -
n. M1 M2 L(. 4.5, ), , (4.16), - . M1 M2 , L (. 4.5, ).
M2
M1
M0
n
L
M2M0
M1
n
L
. 4.5
4.6. , 3x 4y + 5 = 0 A(4; 4) B(6; 6). A , -
1, B 1., A B . #
(4.15) . L L .
4.7.
. , , , . , , . Oxy.
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 4. . 39
. L , M0(x0; y0) , Ox (. 4.6). , 6= /2.
x
y
O
M0
M
b
L
'
. 4.6
M(x; y) L , M0
Ox , tg.
yy0xx0 = tg. y,
y = kx+ b, (4.17)
k = tg; b = y0 x0 tg. y = kx + b -
. k ( ) . b Oy.
. L M0(x0; y0) s = {l;m}, (. 4.7). s L.
M0s
L
M
. 4.7
M(x; y) L, , M0M
s, .. V1. s , V1. , t
M0M = ts. ,
M0M =
= {x x0; y y0}, s = {l;m}, :
x x0 = lt, y y0 = mt,
x = x0 + lt, y = y0 +mt. (4.18)
(4.18) . M(x0; y0), , t = 0.
M0M = ts - r0 r M0 M -
,
r r0 = ts, r = r0 + ts. (4.19)
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 4. . 40
. .
M0M s, 2.1,
:
x x0l
=y y0m
. (4.20)
(4.20) . , (4.18) t.
, . L M1(x1, y1) M2(x2, y2) .
M1M2 L (4.20) ,
M1(x1, y1), s =M1M2,
x x1x2 x1
=y y1y2 y1
. (4.21)
(4.21) , .
. L A(a, 0) B(0, b) - , , , .. a 6= 0 b 6= 0 (. 4.8).
x
y
O a
b
L
. 4.8
L (4.21) A B, x a0 a
=y 0b 0
,
x/a+ 1 = y/b, x
a+y
b= 1. (4.22)
(4.22) .
. L n p > 0 . - , L. , O
L (. 4.9).
n
x
y
O
p
M
L
'
. 4.9
n Ox, - . n : n = {cos; sin}.
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 4. . 41
, M(x; y) L, , - M - p O : n
OM = p (. . 4.9). n
OM
OM n, n -
, OMn = p.
OMn
, L
x cos+ y sin p = 0. (4.23)
(4.23) . Ox . ax+by+c = 0 -
a2 + b2,
c. - {a; b} , . c = 0, (p = 0). .
4.7. 3x 4y + 10 = 0
a2 + b2,
32 + 42 = 5.
35x+
4
5y 2 = 0.
p = 2, cos = 3/5, sin = 4/5, = arccos(3/5).
4.8.
. . -. L1 L2,
L1: a1x+ b1y + c1 = 0, L2: a2x+ b2y + c2 = 0.
, L1 L2 , - n1 = {a1; b1} n2 = {a2; b2}, .
L1 L2 a1a2
=b1b2. (4.24)
a1b2a2b1 = 0, :
L1 L2 a1 a2b1 b2
= 0. (4.25) L1 L2 ,
. n1 = {a1; b1} n2 = {a2; b2} - n1n2 = 0, .., (2.17),
a1a2 + b1b2 = 0. (4.26)
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 4. . 42
, . : k1 = a1/b1, k2 = a2/b2. (4.24) (4.26) :
: k1 = k2; : k1k2 = 1. L1 L2
2 . . . , , cos( ) = cos. - . ( L1 L2)
cos =|n1n2||n1| |n2|
=|a1a2 + b1b2|a21 + b
21
a22 + b
22
.
. . k1 = tg1 k2 = tg2 L1 L2,
tg = tg(1 2) =k1 k21 + k1k2
.
, , L2 L1. L2 , . ( ) 180.
= arctgk1 k21 + k1k2
.
4.9.
M L . , L M0,
M0M .
. -
. L (4.23). M(x; y) L, n
-
M n L -
n, .. x cos + y sin. p
(. 4.10). . > 0, O - , < 0, . . M(x; y) L n
OM
p (. . 4.10), .. = x cos+ y sin p. (M,L) M(x; y) L,
: (M,L) = || = |x cos+ y sin p|.2 180.
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 4. . 43
n
x
y
O
M
L
p
'
. 4.10
- , M(x; y) L, :
(M,L) =|ax+ by + c|
a2 + b2. (4.27)
4.8. AH, AM AD ABC, A. A(1;3), B(7; 3), C(1; 7). : -
LAH , LAM LAD, AH, AM AD (. 4.11).
HD
M
A
B
C
. 4.11
LAM , , - . (x1; y1) BCx1 = (7 + 1)/2 = 4, y1 = (3 + 7)/2 = 5, LAM
, ,x+ 1
4 + 1=
y + 3
5 + 3.
8x 5y 7 = 0. LAH , , -
. , BC AH
LAH . (4.15), A LAH :
(6)(x+ 1) + 4(y + 3) = 0.
3x 2y 3 = 0. LAD, , AD -
N(x; y), LAB LAC .
(N,LAB) = (N,LAC), (4.28)
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 4. . 44
, A LAB LAC. , , LAB LAC :
LAB:x+ 1
7 + 1=y + 3
3 + 3, LAC :
x+ 1
1 + 1=y + 3
7 + 3.
LAB: 3x 4y 9 = 0, LAC : 5x y + 2 = 0. (4.28) (4.27)
|3x 4y 9|32 + (4)2
=|5x y + 2|
52 + (1)2.
, :
3x 4y 9 = 55x y + 226
.
(3 25/
26)x+ (4 5/
26)y + (9 10/
26) = 0.
, , B C -
LAD . , , ..
(3 25/
26)x+ (4 + 5/
26)y + (9 10/
26) = 0.
B ,
(3 25/
26)7 + (4 + 5/
26)3 + (9 10/
26) =
= 21 12 9 + (175 + 15 10)/
26 = 170/
26,
C,
(3 25/
26)1 + (4 + 5/
26)7 + (9 10/
26) =
= 3 28 9 + (25 + 35 10)/
26 = 34 < 0.
, B C ,
(3 + 25/
26)x+ (4 5/
26)y + (9 + 10/
26) = 0.
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 5
: ; , ;
. * -
. . . .
5.1.
Ax + By + Cz + D = 0, - A, B, C . Oxyz .
-.
5.1. .
J , 4.1. , M0 n, -. . , , , . M , n
M0M . M -
(. 5.1, ), n M0M , ,
2.7, :
nM0M = 0. (5.1)
M0 M
n
M1
M2
M0
n
. 5.1
M , n M0M
, nM0M > 0 n
M0M < 0 (.
2.7), , (. 5.1, ).
45
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 5. 46
M0, M n (x0; y0; z0), (x; y; z) {A;B;C}.
M0M = {x x0; y y0; z z0}, ,
(5.1) (2.14)
n M0M , M
A(x x0) +B(y y0) + C(z z0) = 0. (5.2)
Ax+By + Cz +D = 0, (5.3)
D = Ax0By0Cz0 A, B, C , n = {A;B;C} . , (5.3), .. . ,
, Ax + By + Cz + D = 0, A2 + B2 + C2 6= 0, . (x = x0, y = y0, z = z0), . . , A 6= 0 y0 = 0, z0 = 0 x0 = D/A. M0(x0; y0; z0), . Ax0 +By0 + Cz0 +D = 0 , D = Ax0By0Cz0. , Ax + By + Cz Ax0 By0 Cz0 = 0, (5.2). (5.2) - n = {A;B;C} M0M , M (x; y; z). , M0 n = {A;B;C}, ., (5.2) . I
Ax + By + Cz + D = 0 . - A, B, C : n = {A;B;C} . . , , () . , , -
, , (5.2) -.
5.1. , - A(2; 5; 7) M0(3;4; 1).
OA = {2; 5; 7} , -
(5.2) 2(x3)+5(y+4)+7(z1) = 0. , 2x+ 5y + 7z + 7 = 0.
5.2.
. r0 r - M0 M .
M0M = r r0, (5.1)
M , M0 n (. 5.2, ),
n(r r0) = 0, (5.4)
.
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 5. 47
M0 M
n
r0 r
O
M0 e1
e2 M
. 5.2
-, .. V2. e1, e2, .. , , M0 . - M , ,
M0M
(. 5.2, ), .. V2. ,
M0M e1, e2, .. t1 t2,
M0M = t1e1 + t2e2. - r0 r M0 M ,
r = r0 + t1e1 + t2e2, t1, t2 R. (5.5)
(5.5) , (x0; y0; z0), (x; y; z) M0, M {e1x; e1y; e1z}, {e2x; e2y; e2z} e1, e2. r r0 + t1e1 + t2e2,
x = x0 + t1e1x + t2e2x,
y = y0 + t1e1y + t2e2y,
z = z0 + t1e1z + t2e2z.
(5.6)
, . , M1, M2 M3 . , . , M . - . M ,
M1M2,
M1M3
M1M .
(. 3.2). , . , (xi; yi; zi) - Mi, i = 1, 2, 3, (x; y; z) M ,
M1M = {xx1; yy1; zz1},
M1M2 = {x2x1; y2y1; z2z1},M1M3 = {x3x1; y3y1; z3z1} -
x x1 y y1 z z1x2 x1 y2 y1 z2 z1x3 x1 y3 y1 z3 z1
= 0. (5.7) , x, y, z , -
. , 1- , y2 y1 z2 z1y3 y1 z3 z1
(x x1) x2 x1 z2 z1x3 x1 z3 z1 (y y1) + x2 x1 y2 y1x3 x1 y3 y1
(z z1) = 0.
.
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 5. 48
, -
M1M2
M1M3. , -
, , , , .. . . , .
M1(a; 0; 0), M2(0; b; 0), M3(0; 0; c), abc 6= 0, , (. 5.3).
- Mi, i = 1, 2, 3.
x
y
z
a
b
c
O
M1
M2
M3
. 5.3
M1M2 = {a; b; 0},
M1M3 = {a; 0; c},
M1M = (xa; y; z), (5.7)
x a y za b 0a 0 c
= 0. , bc(x a) + acy + abz = 0, abc ,
x
a+y
b+z
c= 1.
.
5.2. , - (1; 1; 2) . ,
, a 6= 0, xa
+ ya
+ za
= 1. (1; 1; 2) , .. 4/a = 1. a = 4 x+ y + z 4 = 0. . .
n,
, p O - (. 5.4). , p = 0, n . M , ,
OM n p, .. n
OM =
= nOM = p, n .
M (x; y; z) n = {cos; cos ; cos } (, n cos, cos , cos ). n
OM = p
,
x cos + y cos + z cos p = 0.
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 5. 49
x
y
z
O
Mp
n
. 5.4
, . Ax+By+Cz+D = 0
A2 +B2 + C2, D. -
{A;B;C} -, . , .. D = 0, .
5.3.
M0. n M1 , (M0, ) M0 . (. 5.5)
(M0, ) = | nM1M0| = |n
M1M0|, (5.8)
|n| = 1.
M1
M0
n
. 5.5
Ax+By + Cz +D = 0, {A;B;C}
n ={A;B;C}A2 +B2 + C2
.
(x0; y0; z0) (x1; y1; z1) M0 M1. Ax1+By1+Cz1+D = 0, M1 ,
M1M0:
M1M0 = {x0x1; y0y1; z0z1}. n
M1M0
(5.8),
(M,) =|A(x0 x1) +B(y0 y1) + C(z0 z1)|
A2 +B2 + C2=
=|Ax0 +By0 + Cz0 (Ax1 +By1 + Cz1)|
A2 +B2 + C2=|Ax0 +By0 + Cz0 +D|
A2 +B2 + C2,
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 5. 50
Ax1 + By1 + Cz1 = D. , , - , .
5.4.
, , 1: A1x + B1y + C1z + D1 = 0, 2: A2x + B2y + C2z + D2 = 0. ( ) n1 = {A1;B1;C1} n2 = {A2;B2;C2} (. 5.6), .
n1
n2 2
1
'
'
. 5.6
, 2.3 ,
cos =(n1,n2)
|n1| |n2|=
A1A2 +B1B2 + C1C2A21 +B
21 + C
21
A22 +B
22 + C
22
.
, , . (. 2.7). -, , , 1 2
A1A2 +B1B2 + C1C2 = 0.
, . - (. 2.1). -
A1A2
=B1B2
=C1C2.
5.1. , . , . #
. - , :
A1A2
=B1B2
=C1C2
=D1D2
.
A1A2
=B1B2
=C1C26= D1D2
, , .
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 6
. . ; - ; . , . , . , . . .
6.1.
. - . 1: A1x + B1y + C1z + D1 = 0,2: A2x+B2y+C2z+D2 = 0 , . M(x; y; z) - , , .. {
A1x+B1y + C1z +D1 = 0,
A2x+B2y + C2z +D2 = 0,(6.1)
.
. . L M0 s. , ,
. M L, ,
M0M -
s (. 6.1). s 6= 0, s V1 . t
M0M =
= ts. M0M =
OM OM0 = r r0, r r0 - M M0
, M L
r = r0 + ts, (6.2)
.
sM0
ML
. 6.1
. , {l;m;n} s L M0(x0; y0; z0) L - . (x; y; z) M . M L -
M0M = {x x0; y y0; z z0} s (. . 6.1),
51
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-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 6. 52
(. 2.6). t , x x0 = tl, y y0 = tm, z z0 = tn.
x = x0 + lt,
y = y0 +mt,
z = z0 + nt,
(6.3)
(6.3) . - (6.3) : - , t = 0, , , - t = 0 t = 1., (6.3), -
l, m, n , , x0, y0, z0 , l, m, n .
. -, (6.3) t
x x0l
=y y0m
=z z0n
. (6.4)
(6.4) . , ,
M0M s, (. -
2.1). .
l, m, n, (6.3), . , l = 0 (6.3) , x = x0. , ( , ) , .
, . . M1(x1; y1; z1) M2(x2; y2; z2),
M1M2 = {x2 x1; y2 y1; z2 z1}.
M1 , (6.4).
x x1x2 x1
=y y1y2 y1
=z z1z2 z1
, .
6.1. M1(1; 2; 3) M2(3; 2; 1) x 13 1
=y 22 2
=z 31 3
. ,
y = 2. y 2 = 0, xOz 2.
.
t. , , .
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 6. 53
6.2. B, A(2; 3;1) L:
x 11
=y + 2
1=
z 12
.
B: ) A , L; ) M L ; ) AM AB , M AB (. 6.2).
L
A
BM
. 6.2
L, n L: n = {1;1; 2}. A - (5.2): 1(x 2) + (1)(y 3) + 2(z + 1) = 0. M ,
L: x = 1 + t, y = 2 t, z = 1 + 2t. - , t (1 + t 2) (2 t 3) + 2(1 + 2t + 1) = 0, M . t = 4/3 - , x = 1 4/3 = 1/3,y = 2 + 4/3 = 2/3, z = 1 8/3 = 5/3. AB, ee , (4.13), A B. , (xB; yB; zB) B, 2 + xB
2= 1/3, 3 + yB
2= 2/3, 1 + zB
2= 5/3. xB = 8/3, yB = 13/3, zB = 7/3. #
. , . , (6.4)
x x0l y y0
m= 0,
x x0l z z0
n= 0, (6.5)
. -
. 1 2,
(6.1) , ,
A1 B1A2 B2,A1 C1A2 C2
, B1 C1B2 C2,
, . ,
: =
A1 B1A2 B2 6= 0,
. , (6.1) z
x y. (6.1) M1
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 6. 54
M2 . , , ., s , -
(6.1), n1n2 - (. 6.3).
n2 n1
s
12
. 6.3
, , - , - , , .. . -
.
6.3. , 1: x y + z 2 = 0, 2: x+ y z = 0. ,
z = 0 x y{x y = 2,x+ y = 0.
x = 1 y = 1 . , (1;1; 0) . n1n2 -
n1 = {1;1; 1} n2 = {1; 1;1} 1 2. (3.2)
n1n2 =
i j k1 1 11 1 1
= 2j + 2k,.. s = {0; 2; 2}. s {0; 1; 1}.
x 10
=y + 1
1=z
1. #
- . (6.1) - x y, z :
x =
C1z +D1 B1C2z +D2 B2A1 B1A2 B2
=B1
(C2z+D2)B2
(C1z+D1) = 1z+1,
y =
A1 C1z +D1A2 C2z +D2A1 B1A2 B2
=A2
(C1z+D1)A1
(C2z+D2) = 2z + 2.
z t z = t, :x = 1t+ 1,
y = 2t+ 2,
z = t.
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 6. 55
6.2.
. . . , . ., : Ax+By+Cz+D = 0,
L x x0l
=y y0m
=z z0n
.
M0(x0; y0; z0) s = {l;m;n} , n = {A;B;C}. L , s -
. , n s, .. . Al +Bm+ Cn 6= 0. , -
s n, Al + Bm + Cn = 0.
, ,
. ,
:
L {Ax0 +By0 + Cz0 +D = 0,Al +Bm+ Cn = 0;
L {Ax0 +By0 + Cz0 +D 6= 0,Al +Bm+ Cn = 0;
L c Al +Bm+ Cn 6= 0.
L :
L:
{A1x+B1y + C1z +D1 = 0,
A2x+B2y + C2z +D2 = 0,
.
A1x + B1y + C1z + D1 = 0,A2x + B2y + C2z + D2 = 0,Ax + By + Cz + D = 0.
(6.6)
, . - , . , (6.6)
A1 B1 C1A2 B2 C2A B C
. , (6.6) , .
. L:x x0l
=y y0m
=z z0n
: Ax+By+Cz+D = 0 0 ( )
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 6. 56
90 ( ). |cos|, s n (. 6.4). (. (2.16)),
sin = | cos| = |Al +Bm+ Cn|A2 +B2 + C2
l2 +m2 + n2
.
= arcsin|Al +Bm+ Cn|
A2 +B2 + C2l2 +m2 + n2
. (6.7)
s
nL
'
. 6.4
, - .
A
l=B
m=C
n. (6.8)
6.3.
: ; ( ); ; , .. . : -
. L1 L2 :
L1:x x1l1
=y y1m1
=z z1n1
, L2:x x2l2
=y y2m2
=z z2n2
. (6.9)
M1(x1; y1; z1) L1, M2(x2; y2; z2) L2 s1 = {l1;m1;n1} L1, s2 == {l2;m2;n2} L2. , s1 s2 -
, :
l1l2
=m1m2
=n1n2. (6.10)
, M1M2:
x2 x1l1
=y2 y1m1
=z2 z1n1
. (6.11)
, M2 L1. , (6.10) (6.11) .
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 6. 57
, -, .. (6.10) . ,
, s1, s2 M1M2 .
- , (. 3.2):
=
x2 x1 y2 y1 z2 z1l1 m1 n1l2 m2 n2
= 0. (6.12) (6.12) , 6= 0
. :
L1 L2 (6.10) (6.11);L1 L2 , L1 6= L2 (6.10), (6.11);L1 L2 (6.12), (6.10);L1 L2 (6.12).
L1:
{A1x+B1y + C1z +D1 = 0,
A2x+B2y + C2z +D2 = 0,L2:
{A3x+B3y + C3z +D3 = 0,
A4x+B4y + C4z +D4 = 0,
:A1x+B1y + C1z +D1 = 0,
A2x+B2y + C2z +D2 = 0,
A3x+B3y + C3z +D3 = 0,
A4x+B4y + C4z +D4 = 0.
(6.13)
(6.13). , . -, . - . , . n1n2 n3n4, ni = {Ai;Bi;Ci}, i = 1, 2, 3, 4. , - . .
6.4.
L1:x 1
1=y 2
3=z + 1
2, L2:
{x y z + 1 = 0,x+ y + 2z 2 = 0.
s1 L1 : s1 == {1; 3;2}. s2 L2 - , :
s2 = n1n2 =
i j k1 1 11 1 2
= i 3j + 2k. s1 = s2, . , . M0(1; 2;1) L1
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 6. 58
L2. 1 = 0. , M0 L2 .
. , - .
(. 6.5) ,
s1
s2
L1
L2
'
. 6.5
. , L1 L2 s1 s2, :
cos =|s1s2||s1| |s2|
.
, si = {li;mi;ni}, i = 1, 2. - (2.9) (2.14) ,
cos =|l1l2 +m1m2 + n1n2|
l21 +m21 + n
21
l22 +m
22 + n
22
.
6.4.
. M1(x1; y1; z1) L,
L:x x0l
=y y0m
=z z0n
,
. , - M0(x0; y0; z0) s = {l;m;n} . - s
M0M1.
M1 L h (. 6.6).s
L
M0
M1
h
. 6.6
,
(M1, L) =|M0M1s||s|
,
. - ,
(M1, L) =
y1 y0 z1 z0m n2 + x1 x0 z1 z0l n
2 + x1 x0 y1 y0l m2
l2 +m2 + n2
. (6.14)
. , , . . , -. . , -
, , , .. - (6.14). ,
L1:x x1l1
=y y1m1
=z z1n1
, L2:x x2l2
=y y2m2
=z z2n2
,
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 6. 59
(L1, L2) =
y2y1 z2z1m1 n12 + x2x1 z2z1l1 n1
2 + x2x1 y2y1l1 m12
l21 +m21 + n
21
.
, -. , , L1 L2 .
, s1, s2 M1M2, , . - (. 6.7).
h
M1
M2
L2
s2
s1L1
. 6.7
h -. , .
, - -
. (L1, L2) :
(L1, L2) =|s1s2
M1M2|
|s1s2|.
,
(L1, L2) =
l1 m1 n1l2 m2 n2
x2 x1 y2 y1 z2 z1
m1 n1m2 n22 + l1 n1l2 n2
2 + l1 m1l2 m22.
-12 -12 -12-12
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2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 7
I
: , , . . . (, , ). . . .
-
Ax2 +Bxy + Cy2 +Dx+ Ey + F = 0, (7.1)
A, B, C . ,
, .. . . , , .
7.1.
7.1. , F1 F2 , .
. F1 F2, 2a. F1 F2 2c. , 2a F1 F2, , . , a > c. , , (. 7.1).
F1F2
M
. 7.1
, , a > c. a = c F1 F2, c = 0, .. - , a. , , , a > c > 0.
60
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-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 7. I 61
F1 F2 7.1 (. . 7.1) , , 2c, , F1M F2M , M , . |F1F2| = 2c a,
F1 F2. , ,
F1 F2, , F1F2 (. 7.2, ). . O , , ( A, B, C D . 7.2, ) .
F2 F1A B
C
D
O
F2 F1
M
O x
y
. 7.2
a , b =a2 c2 .
, c > 0 a , ( A B . 7.2, ), b ( C D . 7.2, ).
. F1 F2, 2a. 2c , 2c = |F1F2| < 2a. 7.1 , M , |F1M |+ |F2M | = 2a. Oxy ,
, (. 7.2, ). , . F1(c; 0), F2(c; 0).
, |F1M |+ |F2M | = 2a :(x c)2 + y2 +
(x+ c)2 + y2 = 2a. (7.2)
, . . (7.2) :
(x c)2 + y2 = 4a2 4a
(x+ c)2 + y2 + (x+ c)2 + y2.
(x+ c)2 + y2 = a+ x, (7.3)
= c/a. , :
(x+ c)2 + y2 =a2+2ax+2x2, , , (a2c2)x2
a2+y2 =
= a2 c2. a2 c2 = b2 > 0, x2
a2+y2
b2= 1, a > b > 0. (7.4)
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 7. I 62
(7.4) , . -
(7.2) , . , , . , .
F1 F2, |F1F2| = 2c, . , F1F2, - - . , . , , F1F2. , (7.4) a. ? a. , a.
x2
a2+y2
b2= 1, b2 = a2 c2, (7.5)
.. (7.4) (7.5) . ,
x2
a2+
y2
a2 c2= 1,
x2
a2+
y2
a2 c2= 1
a 6= a . , , x -:
y2(
1
a2 c2 a
2
(a2 c2)a2
)= 1 a
2
a2,
y2c2(a2 a2)(a2 c2)(a2 c2)
= (a2 a2),
a 6= a, (a2 c2)(a2 c2) = b2b2 > 0. , (7.4) a > 0 b =
a2 c2 > 0.
.
. - . (7.4). , , y > 0, y x: y = b
1 x2/a2, , , .
. a - (7.4) x2+y2 = a2. a/b > 1 , , x2 + (ya/b)2 = a2,.. .
7.1. a/b < 1 ,.. , , (7.4), a < b. , Oxy (. 7.3) . Oxy 90, x = y, y = x.
. . ,
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2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 7. I 63
F1
F2
O x
y
. 7.3
(7.4), = 2c/2a = c/a. (7.4) a b a < b, , c ==b2 a2, = 2c/2b = c/b. c = 0, , = 0.
0 < < 1. , - , , , a = c b = 0. , , a - , a. a b (7.4).
(7.3) (7.4), (7.4) (7.2). (7.3). , (7.3) - , , , |F2M | M(x; y) : |F2M | = a+ x. -
, (7.2) , . , M(x; y) (. . 7.2)
|F1M | = a x, |F2M | = a+ x, (7.6)
.
7.1. 5 - 0,8 .
a = 5 = 0,8, b. b =
a2 c2, c = a = 4, b =
52 42 = 3.
x2
52+
y2
32= 1.
, (. 7.4).
A B
C
D
F1F2 x
y
. 7.4
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-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 7. I 64
A(5; 0), B(5; 0), C(0;3), D(0; 3), . . 7.4 F1,2(4; 0) . . (7.6) |F1M | =
= (a/ x). , a/ x a > c , F1 . d: x = a/ M(x; y), .
|F1M |a/ x
= .
, M(x; y) , F1M d , (. 7.5).
F1F2
M
O
dd'
x
y
. 7.5
d d, d
, x = a/. d , d. d d . , , a/ = a2/c (. . 7.5). p -
.
p =a
c = a
2
c c = a
2 c2
c=b2
c.
: F1M F2M M (. 7.6).
F2 F1
M
O x
y n
NA B
. 7.6
. F1 , , ,
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2
-1
2
-1
2
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2
-1
2
-1
2
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2
-1
2 7. I 65
, , . , , F1, F2, . .
7.2.
7.2. , - , .
7.2. , , - . , . #
. , , - . , , . ( F1 F2)
. ( 2c) -, F1M F2M , M , . |F1F2| = 2c
2a, , F1 F2. , , , -
, , , F1F2 (. 7.7). - , . a, , -.
F2 F1
M
. 7.7
F1F2, , , - . 2a ,
2c, F1MF2 (. . 7.7) |F1M | |F2M | 6
6 |F1F2|. a = c M , F1F2. , , a < c. , a = 0 - , F1 F2. , , F1F2 . .
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-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 7. I 66
. F1 F2 2a. 2c , 2c = |F1F2| > 2a. 7.2, M(x; y),
|F1M | |F2M |
= 2a. Oxy , , (. 7.8). - , .
F2 F1 x
y
O
. 7.8
F1(c; 0) F2(c; 0). ,
|F1M ||F2M | = 2a
(x c)2 + y2 (x+ c)2 + y2 = 2a, (x; y) M . , :
(x c)2 + y2
(x+ c)2 + y2 = 2a,
: (x c)2 + y2 = (x+ c)2 + y2 4a
(x+ c)2 + y2 + 4a2. x a =
(x+ c)2 + y2,
(x+ c)2 + y2 = |x+ a| , (7.7)
= c/a. : (2 1)x2 y2 == c2 a2 , = c/a b2 = c2 a2,
x2
a2 y
2
b2= 1. (7.8)
b > 0 ., , F1(c; 0) F2(c; 0)
a (7.8). , - . F1 F2. - . , ( , F1F2, , ) , , F1 F2 . M(x; y) (7.8), a . , -
, x2
a2 y
2
b2= 1, b2 = c2 a2. ,
x2
a2 y
2
c2 a2= 1,
x2
a2 y
2
c2 a2= 1
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2
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2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 7. I 67
. , a 6= a . , , , x :
y2(
a2
(c2 a2)a2 1c2 a2
)= 1 a
2
a2
y2c2(a2 a2)(c2 a2)(c2 a2)
= (a2 a2),
a 6= a , (c2 a2)(c2 a2) = b2b2 > 0. , (7.8) a > 0 b =
c2 a2 > 0.
.
. (7.8) . - , , . , .. x > 0, y > 0, y:
y =b
a
x2 a2. (7.9)
y(x) . {x: x > a}
, x = a . x = a.
y(x): y(x) =bx
ax2 a2
. , x > a
. , limxa+0
y(x) = +, , x = a . y(x) y = ab(x2 a2)3/2 x > a, . , . , -
:
k = limx+
y(x)
x=b
alim
x+
x2 a2x
=b
a,
b = limx+
(y(x) kx
)=b
alim
x+
(x2 a2 x
)=b
alim
x+
a2x2 a2 + x
= 0.
y = (b/a)x. (7.9) (. 7.9),
(7.8), .
a x
y
O
. 7.9
, , - . 7.10. ,
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2
-1
2
-1
2
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2
-1
2
-1
2
-1
2
-1
2 7. I 68
F2 F1AB x
y
O
. 7.10
. , y = (b/a)x . , ,
, , , ( ). ( A(a; 0) B(a; 0) . 7.10). (2a) (2b)
2a 2b, , -, (. 7.11). - , . ,
. b/a , . - . . , (7.8), = c/a., [0, 1) ( 0 ), - (1,+).
F2 F1a-a
-b
b
x
y
O
. 7.11
Oxy 2a,2b, . y = (b/a)x y = (b/a)x, . , (. 7.12). - (7.8),
x2
a2 y
2
b2= 1. (7.10)
, (7.10) - . , .
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2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 7. I 69
F2 F1a-a
-b
b
x
y
O
. 7.12
7.2. a = 4 2c = 10. , . a , , -
, b. c = 5, b =2 2,
b =
52 42 = 3. , x2
42 y
2
32= 1. ,
(. 7.13). - . y = 3x/4, (4; 0), (5; 0).
44
3
3
55 x
y
0
. 7.13
. . (7.7). F2M M(x; y):
|F2M | =
(x+ c)2 + y2 = (x+ a), (7.11)
, . ,
, . (7.7) x a =
(x c)2 + y2,
|F1M | =
(x c)2 + y2 = (x a), (7.12)
, (7.11), , . (7.11), (7.12) .
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2
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2
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2
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2
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2
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2
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2
-1
2 7. I 70
( 0 < a < c). M , .. |F1M | 6= 0 |F2M | 6= 0. (7.11) (7.12) , :
|F2M ||x+ a/|
= ,|F1M ||x a/|
= . (7.13)
d: x = a/ (. 7.14). |x+ a/| - M(x; y) d. (x a/) |x a/| M d: x = a/. (7.13) , , F2 (F1) d
( d) , . d d .
F2 F1a"
a"-
d' d
y
O x
M
. 7.14
, , - . p , -
, . ,
p = c a
= c a2
c=c2 a2
c=b2
c.
, . , , , - , (. 7.15).
F2 F1
. 7.15
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2
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2
-1
2
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2
-1
2
-1
2
-1
2
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2 8
II
8.1. ,
, .. a = b, - 2 arctg(b/a) = 2 arctg 1 = /2, .. . . , -
, , . - , - Oxy. i, j , i, j Oxy (. 8.1).
x
y
i
'i'j
j
x'y'
O
. 8.1
Oxy /4. -
(. 4.2) i =
2
2i +
2
2j, j =
2
2i +
2
2j.
, x, y - x,
y : x =
2
2x +
2
2y,
y = 2
2x+
2
2y.
(x)2 (y)2 = a2, a ( ) . x, y,
1
2(x+ y)2 1
2(x y)2 = a2,
xy =a2
2. (8.1)
(8.1) .
8.1. xy = a2
2
(8.1).
8.1. , xy == 8 . -
. , .. x = 0, y = 0, . a2/2 = 8, a2 = 16 a = b = 4. c =
a2 + b2 =
42 + 42 = 4
2,
, , : A(2
2; 2
2), B(2
2;2
2), F1(4; 4),
F2(4;4) (. 8.2).
8.2.
, . , - , -
71
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2
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2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 8. II 72
A
B
F1
F2
x
y
. 8.2
. 0 < < 1 , > 1 . , . , = 1, . - , .
8.1. , - , .
, -. , . , ,
. . . . - , (. 8.3).
FA
d
x
y
M
. 8.3
. - , , (. . 8.3). , . p. -
. F (p/2; 0), d x = p/2.
M(x; y), F d, (
x p2
)2+ y2 =
x+ p2
. (8.2)
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2
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2
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2
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2
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2
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2
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2
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2 8. II 73
(8.2) .
y2 = 2px, (8.3)
., -
(8.2), , .
. y2 = x, , - 1/(2) , , (8.3).
8.2. , , (25; 10).
y2 = 2px. (25; 10) , 100 = 50p p = 2. , y2 = 4x , x = 1 , (1; 0).
. . , (. 8.4). , M MF .
F
. 8.4
8.3.
Ax2 +Bxy + Cy2 +Dx+ Ey + F = 0, A2 +B2 + C2 6= 0, (8.4)
B = 0 ( -), A = C = 0 ( ), . -
, , /2,/2 , , , . , , ( ).
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2
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2
-1
2
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2
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2
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2
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2
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2 8. II 74
8.2. xOy x, y. /2 x = y, y = x; /2 x = y, y = x; x = x, y = y. .
. (B = 0), , . , ax2 + bx+ c, a 6= 0, x :
ax2 + bx+ c = a(x2 +
b
ax)
+ c = a
(x2 +
b
ax+
b2
4a2
)+ c b
2
4a= a(x x0)2 + d,
x0 = b
2a; d = c b
2
4a.
B = 0 .
1. A 6= 0 C 6= 0 (8.4) x ( D 6= 0) y ( E 6= 0)
A(x x0)2 + C(y y0)2 = F , (8.5)
x0 = D
2A, y0 =
E
2C, F = F + D
2
4A+
E2
4C.
(8.5) F 6= 0, , a2 = |F | / |A|, b2 = |F | / |C| (8.5), :
(x x0)2
a2 (y y0)
2
b2= 1,
(x x0)2
a2 (y y0)
2
b2= 1,
(x x0)2
a2+
(y y0)2
b2= 1,
(x x0)2
a2+
(y y0)2
b2= 1.
, . (, ) - ( , ), x = x x0, y = y y0
(x)2
a2 (y
)2
b2= 1,
(x)2
a2 (y
)2
b2= 1, (x
)2
a2+
(y)2
b2= 1.
, , a > b. a b - a < b, x = y, y = x. (8.5) F = 0, A(xx0)2 +C(yy0)2 = 0.
AC > 0 (x0; y0), AC < 0
|A|(x x0)
|C|(y y0) = 0.
2. A 6= 0, C = 0. E 6= 0, .. - y , x (D 6= 0). , A(xx0)2 = Ey+F ,
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2
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2
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2
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2
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2
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2
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2
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2 8. II 75
x0 = D/(2A); F = D2/(4A) F . A(x x0)2 == E(y y0), y0 = F /E. p = |E| /(2 |A|), (x x0)2 = 2p(y y0) ( AE < 0) (x x0)2 = 2p(y y0) ( AE > 0). x = xx0, y = yy0 - (x)2 = 2py (x)2 = 2py . , y2 = 2px x = y, y = x x = y, y = x . y (E = 0),
Ax2 + Dx + F = 0, A 6= 0, .. x. , (8.4) . :
(x x0)2 = a2, (x x0)2 = 0, (x x0)2 = a2,
x0 = D/(2A), a =|F +D2/(4A)| 6= 0. -
x = x0 a, x = x0. ( ).
3. A = 0, C 6= 0 ( x = y, y = x). D 6= 0, .. - x , y ( E 6= 0) , C(y y0)2 = Dx + F , y0 = E/(2C);F = E2/(4C) F . C(y y0)2 = D(x x0), x0 = F
/D. p = |D| /(2 |C|), - (y y0)2 = 2p(x x0) ( CD < 0) (y y0)2 = 2p(x x0) ( CD > 0). x = x x0, y = y y0 (y)2 = 2px, (y)2 = 2px. y2 = 2px x = x, y = y. x (D = 0),
Cy2 + Ey + F = 0, C 6= 0, .. y. , y :
(y y0)2 = a2, (y y0)2 = 0, (y y0)2 = a2,
y0 = E/(2C); a =|F + E2/(4C)| 6= 0. -
(, , ) , Ox.
, (8.4) , ..
Bxy +Dx+ Ey + F = 0, B 6= 0. (8.6)
(x x0)(y y0) + F = 0, x0 = E/B; y0 = D/B;F = F/B x0y0. |F | = a2/2, F
(x x0)(y y0) =a2
2, (x x0)(y y0) =
a2
2. (8.7)
a 6= 0, . - , x = x x0, y = y y0 xy = a2/2 - xy = a2/2.
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2
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2
-1
2
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2
-1
2
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2
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2
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2 8. II 76
a = 0, (8.7) , x = x0 y = y0.
- , AC > 0, B = 0 ( A = C), . AC > 0,B = 0 .
AC < 0, B = 0 , . A = C = 0, B 6= 0. .
, , A C, , ( ). .
, - , Oxy : , -; , , , , .
8.3. , 9x2 + 4y2 18x+ 16y 11 = 0. -
. . , 9(x2 2x + 1 1) + 4(y2 + 4y + 4 4) 11 = 0,9(x 1)2 + 4(y + 2)2 = 36, (x 1)
2
22+
(y + 2)2
32= 1. ,
x = x 1, y = y + 2
(x)2
22+
(y)2
32= 1,
a = 2, b = 3. a < b, - . c =
b2 a2 =
5, = c/b =
5/3.
O (1;2) (. 8.5). , , Oxy 90.
O'
O
2
1
x'
x
y'y
F1
F2
A
C
D
B
. 8.5
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2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 8. II 77
:
Oxy Oxy
A(2; 0) A(1 2;2)B(2; 0) B(1 + 2;2)C(0;3) C(1;2 3)D(0; 3) D(1;2 + 3)
F1(0;5)
F1(1;2
5)
F2(0;5)
F2(1;2 +
5)
y = 9/5 y + 2 = 9/
5
8.4. 4x2 +16y2 +8x64y+4 = 0 . ,
4(x2 + 2x+ 1 1) + 16(y2 4y + 4 4) + 4 = 0, 4(x+ 1)2 + 16(y 2)2 = 64, (x+ 1)2
42+
(y 2)2
22= 1.
x = x+1, y = y2,
(x)2
42+
(y)2
22= 1
a = 4, b = 2. a > b, . c =
a2 b2 = 2
3, = c/a =
3/2.
O (1; 2) (. 8.6). :
O1
2O'
x
x'
yy'
F2F1A
C
D
B
. 8.6
Oxy Oxy
A(4; 0) A(1 4; 2)B(4; 0) B(1 + 4; 2)C(0;2) C(1; 2 2)D(0; 2) D(1; 2 + 2)
F1(23; 0)
F1(1 2
3; 2)
F2(23; 0)
F2(1 + 2
3; 2)
x = 8/3 x+ 1 = 8/
3
8.5. 4x2 9y2 24x+ 18y 9 = 0 . . , -
4(x26x+99)9(y22y+11)9 = 0, 4(x3)29(y1)2 = 36, (x 3)2
32 (y 1)
2
22= 1.
x = x 3, y = y 1
(x)2
32 (y
)2
22= 1.
O(3; 1), a = 3, b = 2. c =
a2 + b2 =
13, = c/a =
13/3 (. 8.7).
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 8. II 78
x
y'
x'
y
F2
3
1F1
A B
. 8.7
:
Oxy Oxy
A(3; 0) A(3 3; 1)B(3; 0) B(3 + 3; 1)
F1(13; 0
)F1(313; 1
)F2(
13; 0)
F2(3 +13; 1
) x = 9/
13 x 3 = 9/
13
y = 2/3x y 1 = 2(x 3)/3
8.6. x2 + 2x 6y+ 7 = 0 , . x, (x2 + 2x + 1 1) 6y + 7 = 0, (x + 1)2 = 6(y 1). x = x + 1, y = y 1 (x)2 = 2 3y. , p = 3, p/2 = 1,5. O(1; 1) (. 8.8). :
x
x'
yy'
O'
O1
1
d1,5
F
. 8.8
Oxy Oxy
F (0; 1,5) F (1; 1 + 1,5) x = 1,5 x+ 1 = 1,5
8.7. xy x 2y + 6 = 0 , , . . x = x 2, y = y 1 xy = (2
2)2/2, ..
a = b = 2
2. c =a2 + b2 =
= 4, = c/a = 4/(2
2) =
2. O(2; 1) (. 8.9).
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 8. II 79
xx'
y y'
F1
F2
O'O 12
A
B
. 8.9
:
Oxy Oxy
A(2; 2) A(2 2; 1 + 2)B(2;2) B(2 + 2; 1 2)
F1(22; 22)
F1(2 2
2; 1 + 2
2)
F2(22;2
2)
F2(2 + 2
2; 1 2
2)
y = x 22 y 1 = 2
2(x 2)
y = 0 y 1 = 0x = 0 x 2 = 0
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 9
. . . -. . . . . .
9.1.
. . . .
9.1. , - L ( ), , L (. 9.1).
L
. 9.1
, O - , Oz . xOz S (. 9.2), - .
L
d
M
M1M2
x
y
z
O
S
. 9.2
80
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 9. 81
, S xOz (x, z) = 0. - M(x; y; z). Oz d =
x2 + y2.
M , M1(x1; 0; z), M2(x2; 0; z) z, M , x1 = d, x2 = d S.
0 = (x1, z) = (d, z) = (x2 + y2, z), 0 = (x2, z) = (d, z) = (
x2 + y2, z)
M , M
(x2 + y2, z) = 0. (9.1)
(9.1) , - S = {(x; z): (x, z) = 0}, xOz. S (9.1) x x2 + y2.
. - xOz , M(x; y; z) M (x; y/k; z), k > 0. k . k > 1 , , xOz, - , .. . 0 < k < 1 . Oxyz Q
F (x, y, z) = 0. - xOz k Q F (x, ky, z) = 0. , (x; y; z) Q, (x; ky; z) Q.
9.2.
, -, (. 9.3).
. 9.3
, - Oz , xOz (. 9.4). x2
a2+z2
b2= 1. x
x2 + y2 (. 9.1),
x2 + y2
a2+ z
2
b2= 1 . ,
Oz
x2
a2+y2
a2+z2
b2= 1. (9.2)
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 9. 82
x
yz
O
. 9.4
xOz, . k ,
x2
a2+k2y2
a2+z2
b2= 1, , ,
x2
a2+y2
b2+z2
c2= 1. (9.3)
(9.3) . - . a, b c, - (. 9.5). , .
a
b
c
Ox
y
z
. 9.5
- (, , (9.2)) ( ). (a == b = c = r), r, x2 + y2 + z2 = r2.
9.3.
, - . . , (). (. 9.6). ee (. 9.7). - . -
, , , Oz, xOz .
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 9. 83
. 9.6 . 9.7
x2
a2 z
2
b2= 1. x
x2 + y2 (. 9.1),
x2
a2+y2
a2 z
2
b2= 1. (9.4)
x2
a2 z
2
b2= 1. x
x2 + y2,
x2
a2+y2
a2 z
2
b2= 1 (9.5)
. xOz -
. - k
x2
a2+k2y2
a2 z
2
b2= 1 x
2
a2+k2y2
a2 z
2
b2= 1.
(. 9.8)
x2
a2+y2
b2 z
2
c2= 1 (9.6)
(. 9.9)
x2
a2+y2
b2 z
2
c2= 1. (9.7)
(9.6), (9.7), -.
9.4.
(. 9.10). - , , Oz xOz . x2 = 2pz, p > 0. - x
x2 + y2 (. 9.1): 2pz = x2 + y2.
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 9. 84
O x
y
z
c
. 9.8
x
y
z
Oa
b
. 9.9
x
y
z
O
. 9.10
xOz - k , 2pz = x2 + k2y2.
x2
a2+y2
b2= 2z. (9.8)
, . a = b .
9.5.
L, , - (. 9.11). , . , , Oz -
, . Ox , L xOz z = k1x. x
x2 + y2 (. 9.1).
z = k1x2 + y2. , z2 = k21(x
2 + y2), - 2 = k21a
2, x2
a2+y2
a2=
z2
c2.
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 9. 85
x
y
z
O
L
. 9.11
xOz -
k . x2
a2+
k2y2
a2=
z2
c2, ,
,x2
a2+y2
b2=z2
c2. (9.9)
(9.9) . - a = b , .
9.6.
, , -, (. 9.12). , , (. 9.13). , , - (. . 9.13). , , - . .
. 9.12 . 9.13
, - , . , - Oz. , xOy (. 9.14).
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 9. 86
xy
z
O
. 9.14
xOy (x, y) = 0 -. M(x; y; z) , ( N(x; y; 0) xOy) - . (x, y) = 0 , x, y z. : -, , ., - .
, - , , .
Ax2 + 2Bxy + Cy2 + 2Dx+ 2Ey + F = 0, A2 +B2 + C2 6= 0. .
(. 8.3). :
(. 9.15, ) x2
a2+y2
b2= 1;
(. 9.15, ) x2
a2 y
2
b2= 1;
y2 = 2px (. 9.15, ).
, (, -) , (, ) .
x
y
z
O
x
y
z
O
x
y
z
O
. 9.15
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 9. 87
9.7.
(x, y, z) = 0 (9.10)
. - , , z = c, c . c {
(x, y, z) = 0,
z = c(9.11)
. M(x; y; z) - : ) z = c; ) x y xOy, .. N(x; y; 0),
(x, y, c) = 0. (9.12)
, .. (9.12), . ,
,
.
(9.12). , xOy (9.11). -, O Oz , Ox Oy, Ox Oy Oxyz. (9.12) (9.11) .
9.1. (9.8)x2
a2+ y
2
b2= 2z .
z = c x2
a2+
y2
b2= 2c.
c < 0 , c = 0 Oxyz,
c > 0 x2
(a2)2
+y2
(b2)2
= 1.
c , (. 9.16, ). ,
x
y
z
O
x
y
z
O
x
y
z
O
. 9.16
-12 -12 -12-12
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2
-1
2 9. 88
,
. x = c (. 9.16, ),
y = c (. 9.16, ) 2
a2+
y2
b2= 2z
x2
a2+
2
b2= 2z .
( c). . P1, y = 0, - P2 x = 0 (. 9.17, ). P2 , :
P2 P1; P2 P1; P2 P1.
P1
P2
x
y
z
O
x
y
z
P2
P1
O
. 9.17
. P1 P2 , .. P1, P2 . #
x2
a2 y
2
b2= 2z (9.13)
(9.8) . , (9.13) . . -
y = c c :
x2
a2 c
2
b2= 2z.
x = c c :
c2
a2 y
2
b2= 2z.
P1 , y = 0, P2
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