-
8/14/2019 L2 101
1/32
-
8/14/2019 L2 101
2/32
-
8/14/2019 L2 101
3/32
Oxy
l
f(x, y) = 0
l
x, y
t
x = x0 + tmy = y0 + tn
Ox k = tg
-
8/14/2019 L2 101
4/32
//x
OO y
YYAB
A(x0, y0) B(x, y)
k =y y0x x0
y = k (x x0) + y0 b = y kx0
y = kx + b
M(x1, y1) N(x2, y2)
y
y1
y2 y1 =x
x1
x2 x1
-
8/14/2019 L2 101
5/32
M1(a1, b1), M2(a2, b2)
//x
OO y
ooooooooooooooooooooooooooooooo M2
M1GGGGGGGGGGGG
GGGGGGGGGGGGGG
qq
qqqq
qqqq
qqqq
q
M(x, y)
|M1M| = |MM2|
|M1M| =
(x a1)2 + (y b1)2
|M2M
|= (x a2)2 + (y b2)2=
(x a1)2 + (y b1)2 =
(x a2)2 + (y b2)2
A = 2a2 2a1, B = 2b2 2b1, C = a2
1 + b2
1 a2
2 b2
2
-
8/14/2019 L2 101
6/32
Ax + By + C = 0
(A = B = 0 !)
A = 0, B = 0, C = 0 Ax + By = C
C
a =
C
Ab =
C
B
x
a+
y
b= 1
a, b
//
x
OO y
vvvvvvvv
vvvvvv
vvvvvv
vvvv
a
b
-
8/14/2019 L2 101
7/32
y = k1x + b1 (I)y = k2x + b2 (II)
//x
OO y
oooooooooooooooooooooooooooooooooooo
]]
^^
1 2
QQ..
l1l2
O
= 2 1
tg = tg(2 1) = tg2 tg11 + tg1tg2
=k2 k11 + k1k2
y = A1
B1x C1
B1, y = A2
B2x C2
B2
-
8/14/2019 L2 101
8/32
k1 = A1B1
, k2 = A2B2
tg =A1B2 B1A2A1A2 B1B2
= 0 k1 = k2,
= /2 k1 k2 = 1
2x + y
6 = 0, 4x + 5y + 7 = 0
y = 2x 6 = k1 = 21
, y = 45
x 75
= k2 = 45
=
tg =4
5 2
1
1 + 2 4
5
=
14
13=
-
8/14/2019 L2 101
9/32
3x + 2y 6 = 0, 6x + 4y + 5 = 0
y = 32
x + 3 = k1 = 32
, y = 32
x 54
= k2 = 32
=
tg =3
2 3
2
1 +
3
2 3
2= 0 = k1 = k2 = l1l2
x + y + 2 = 0, 3x 3y + 4 = 0
y =
x
2 =
k1 =
1, y = x +4
3=
k2 = 1 =
tg =1 (1)
1 + 1 (1) = = (k1 k2 = 1 = l1 l2)
(|OM| = p)
Ox
-
8/14/2019 L2 101
10/32
//x
OO y
tttt
tttt
tttt
tttt
tttt
tttt
ttttttt
tttt
tt
B
A
Mp
^^
xx
O
OMA, OM B
a = |OA| = |OM
|cos =p
cos
b = |OB | = |OM|cos(90 ) =
p
sin
xp
cos
+yp
sin
= 1
x cos + y sin p = 0
M0(x0, y0), O(0, 0) x cos + y sin p = 0
d,
d
M0
-
8/14/2019 L2 101
11/32
//x
OO y
qqqqq
qqqq
qqqq
qqqq
qqqq
qqqq
qqqq
qqqq
qqqqqqq
qqq
qqqqq
qqqq
qqqq
qqqq
qqqq
qqqq
q
qq.qq
O
p
pM0
= p p
M0 = 0 M0
x cos + y sin p = 0
x cos + y sin = p
M0
= | d| = x0 cos + y0 sin p = Ax1 + By1 + CA2 + B2
A(4;5) 3x 2y 4 = 0
d =
3 (4) 2 5 4
32 + 22
=26
13= 2
13
-
8/14/2019 L2 101
12/32
A1x + B1y + C1 = 0 (I)A2x + B2y + C2 = 0 (II)
A1B2 A2B1 = 0
x = B1
C2 C1B2A1B2 A2B1 ; y = C
1
A2 A1C2A1B2 A2B1
A1B2 A2B1 = 0 A1
A2=
B1B2
= t
t = C1C2
A1A2
=B1B2
=C1C2
-
8/14/2019 L2 101
13/32
f(x, y) = 0
F(x,y,z) = 0
M(x,y,z)
x,y,z
-
8/14/2019 L2 101
14/32
F(x, y) = 0
z z
//
y
OO z
yyyyyyyy
yyyyyy
yyyyyy
yyyyyy
yy
'' x
111111111
111
M0
M0
F(x, y) = 0
O
M0(x0, y0, z0) M0(x0, y0)
z
M0(x0, y0)
-
8/14/2019 L2 101
15/32
-
8/14/2019 L2 101
16/32
n =
{A,B,C
}
n = {A,B,C}
M0(x0, y0, z0)
M(x,y,z)
M0M n (M0M , n) = 0 M0M = {x x0, y y0, z z0}
=A(x
x0)+B(y
y0)+C(z
z0) = 0
M0 n
Ax0 By0 Cz0 = D
Ax + By + Cz + D = 0
-
8/14/2019 L2 101
17/32
//y
OO z
GGGGG
GGGG
GGG
xO
jjjjjjjjjjjjjj
lllllllllllllllllllllllG
GGG
GGGG
GGGG
GGGG
GGGG
GGGG
GGG
cccc
cccc
cccc
cccc
ccccccc
cccc
cc
EEM0
hhhhhhhhhhh
44 Mqqqq
qqqq
qqqq
qqq
##r0r
||||||||
==n
A, B, C, D
xD/A +
yD/B +
zD/C = 1
x
a+
y
b+
z
c= 1
zzzzzzzzzzzzzzzzzzzzzzz
yyyyyy
yyyyyy
yyyyyy
yyyyyy
yy
}}}}
}}}}
}}}}
}}}}
}}}}
}}}}
}tttttttt
tttttttttttttttttt
ppppppppppppp
77M
M1
,,M2
qqqqq
qqqq
qqqq
qq
##M3
M1(x1, y1, z1), M2(x2, y2, z2), M3(x3, y3, z3)
-
8/14/2019 L2 101
18/32
M(x,y,z)
M1M = {x x1, y y1, z z1},M2M = {x x2, y y2, z z2},M3M = {x x3, y
y3, z z3}.
(M1M,
M2M,
M3M) = 0.
x x1
y y1 z z1x2 x1 y2 y1 z2 z1x3 x1 y3 y1 z3 z1
= 0
-
8/14/2019 L2 101
19/32
// y
OOz
cccc
cccc
cccc
cccc
ccccc
x
O
iiiiiiiiiiiiiiiiii
CCCCC
CCCC
CCCC
CCCC
CCCC
CCCC
CC
jjjjjjjj
jjjjjjjj
jjj
BBBB
BBBB
BBBB
BBBB
BBBB
BBBB
BBBB
{{{{{{{{{{{{{{{{{{{
==
AA
A
mmmmmmmmmmmmmmmmmm
66 iiiiiiiiiiii
M
P
M1
M0d
{{{{ ==r
mmmmm 66 n0
p
n, n0 =
{cos , cos , cos
}
r
M(x,y,z)
n0r = p
n0r = (r, n0)
(r, n0) p = 0 x cos + y cos + z cos p = 0
= 1A2 + B2 + C2
D
-
8/14/2019 L2 101
20/32
-
8/14/2019 L2 101
21/32
a = {m,n,p}
//y
OO z
cccc
cccc
cccc
cccc
cccc
c
x
O
DD
,,GG
GGGG
GGGG
GGGG
GGGGG
GGGG
GGGG
GGGG
GGGG
M0
M
r0
r
LGG
GGGG
G
a
a = {m,n,p}
M0(x0, y0, z0) M(x,y,z)
a,M0M = {xx0, yy0, zz0}
x x0m
=y y0
n=
z z0p
-
8/14/2019 L2 101
22/32
M0M, a
M0M = t a t R
r = r0 +M0M =
r = r0 + t a, t R
x = x0 + m ty = y0 + n tz = z0 + p t
M1(x1, y1, z1), M2(x2, y2, z2)
a = M1M2 = {x2 x1, y2 y1, z2 z1}
x x1x2 x1 =
y y1y2 y1 =
z z1z2 z1
-
8/14/2019 L2 101
23/32
n1 = {A1, B1, C1}, n2 = {A2, B2, C2}
1 : A1x + B1y + C1z + D1 = 0
2 : A2x + B2y + C2z + D2 = 0
n1, n2
L
L
yyyyyy
yyyyyyyyy
y
yyyyyy
yyy
HHHHHHH
HHHHHHHHHHHHHHH
IIIIIIIIIIIIIIIIIIIIII
1
2
yyyy
A1x + B1y + C1z + D1 = 0
A2x + B2y + C2z + D2 = 0
(A1x + B1y + C1z + D1) + (A2x + B2y + C2z + D2) = 0
-
8/14/2019 L2 101
24/32
2x y + 3z + 1 = 0x + 2y
20z + 2 = 0
x = 2 y + 3z + 5 = 02y 20z + 4 = 0 = z = 1, y = 8 = M0(x0, y0,
z0) = M0(2, 8, 1).
n1 = {2, 1, 3}, n2 = {1, 2, 20}
a = n1 n2 =
i j k2 1 31 2
20
= 14i + 43j + 5k = x 214
=y 8
43=
z 15
-
8/14/2019 L2 101
25/32
L1 :x x1
m1
=y y1
n1
=z z1
p1
, a1 =
{m1, n1, p1
}L2 :
x x2m2
=y y2
n2=
z z2p2
, a2 = {m2, n2, p2}
cos =m1m2 + n1n2 + p1p2
m21
+ n21
+ p21
m22
+ n22
+ p22
L1 L2 a1 a2 m1m2 =
n1n2 =
p1p2
L1 L2 a1 a2 m1m2 + n1n2 + p1p2 = 0
d d
-
8/14/2019 L2 101
26/32
-
8/14/2019 L2 101
27/32
: A1x + B1y + C1z + D1 = 0, n1 = {A1, B1, C1} : A2x + B2y + C2z
+ D2 = 0, n2 = {A2, B2, C2}
OO n1
n2
111
11
OO
n1
//n2
-
8/14/2019 L2 101
28/32
cos = 0
=
n1
n2 =
A1A2 + B1B2 + C1C2 = 0
= n1, n2
n1 n2 = 0 = A1A2
= B1
B2= B
1
B2
A1
A2 =
B1
B2 =
C1
C2 =
D1
D2
, , L1, L2
L1, L2
-
8/14/2019 L2 101
29/32
=
(L1, L2) n1 L1, n2 L2 =
(n1, n2)
cos = cos (n1, n2) = A1A2 + B1B2 + C1C2A2
1+ B2
1+ C2
1A2
2+ B2
2+ C2
2
yyyyyy
yyyyyy
yyyyyy
yyyyyy
yyyyyy
yyyyyy
yyy
yyyyyy
yyyyyy
yyyyyy
yyyyyy
yyyyyyyyyy
yyyyy
yyyy
yyyy
yyyy
yyyy
yyyy
yyyy
yyyy
yyyyy
yyyy
yyy
yyyyyy
yyyyyyyyy
yyyyyy
yyyyyy
yyyyyy
yyyyyy
111111111111
11111111
OOn2
ooooooo77
n1
L1
L2
ooooooo
77n1OOn2
ooooooo
eeeeeeeeeeeeeeeeee DDD
DDDD
DDDD
DDDD
,,
L1
L2
-
8/14/2019 L2 101
30/32
A1x + B1y + C1z + D1 = 0
2x + B2y + C2z + D2 = 0A3x + B3y + C3z + D3 = 0
= 0
= 0
L :x x0
m=
y y0n
=z z0
pa = {m,n,p}
: Ax + By + Cz + D = 0, n = {A,B,C}
-
8/14/2019 L2 101
31/32
-
8/14/2019 L2 101
32/32
n
a
(n, a) = 0
Am + Bn + Cp = 0
//
aOO n L
n, a [n, a] = 0 A
m=
B
n=
C
p
a111
OOn
