, , . , l o l 1 , l o, l o¢, l 1, l o¢ l 1.
. .
:
l o: = =, l 1: = =. (35)
, (a 1, a 2, a 3)½½ l o, (b 1, b 2, b 3)½½ l 1, A o(x o, y o, z o)Î l o, A 1(x 1, y 1, z 1)Î l 1.
x 1 x o y 1 y o z 1 z o
A = a 1 a 2 a 3,
b 1 b 2 b 3
D = det A.
8. 1. l p
cos a = =. (36)
2. l o l 1 Û D ≠ 0.
3. l o l 1 Û D = 0 .
4. l o½½ l 1 Û rank A = 2 ½½.
5. l o= l 1 Û rank A = 1.
. 1. a l o l 1 b , , .
cos a = cos b =,
cos a = cos b =½ cos b½ =.
. , l o, l o¢.
2, 3. , l o l 1 , . , Û , , Û : = 0. D.
, D ≠ 0, ,, , , l o l 1 Þ .
4, 5. l o½½ l 1 l o= l 1, ½½ . , A . , rank A = 2.
, , , ,
A . , rank A = 1.
, ||, l o l 1 ; , A . , , rank A = 2, , , Û l o|| l 1. rank A = 1, A , , ,, Û l o= l 1.
9. l o l 1 (35).
1. l o½½ l 1, l o l 1
h =, (37)
2. l o l 1 ,
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h =. (38)
. 1. l o½½ l 1. A o, . h l o l 1. : S =½ ´½, ½ ½.
h = S/ ½ ½ = (37).
2. l o l 1 . l o po½½ l 1, l 1 p1½½ l o.
l o l 1 po p1. A o , . po, p1. po p1, h l o l 1. ½ ½, ½´½ Þ
h = V/S = (38).
. A 1(x 1, y 1, z 1) l,
l: = =
(37).
.
1. A (1, 6), B (3, 0), C (6, 9) ABC. .
. , R (a, b). :
(x a)2 +(y b)2 = R 2.
, . M 1(x 1, y 1), M 3(x 3, y 3) BC AB :
x 1= = =, y 1= = =, M 1.
M3(1,3).
l 3 , AB, l 1 BC. = ( 4, 6) ^ l 3 l 3 M 3 . :
4(x +1) + 6(y +3) = 0.
= (9, 9) ^ l 3. l 1:
9(x - ) + 9(y -) = 0
x + y 6 = 0.
= l 1 I l 3. , l 1 l 3 :
x + y 6 = 0,
4 x + 6 y +14 = 0.
, 4:
x + y 6 = 0,
10 y 10 = 0.
y = 1, x = 5, O (5, 1).
. :
R =½½= =.
:
(x 5)2 + (y 1)2 = 65.
2. ABC 3 x 2 y + 5 = 0, C (5,5) O (3/2,3) AB.
A, B E, O BC. ABC.
. , , .
ax + by + c = 0.
a b (a, b). (3,-2)^ .
l = OD D. OD, .. . , . l:
x = + 3 t, (*)
y = 3 - 2 t.
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D = l I BC. , , l BC. x y l BC:
3( + 3 t) 2(3 -2 t)+5 = 0,
+ 9 t +6 +4 t +5 = 0,
13 t = , tD = .
t l D (3,2). , E : . , , . E D. tD = D, E tE = 2 tD = 1. (*), E ( 4,5;1).