Agma Notes
Transcript of Agma Notes
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u ,v , . . .
u A B
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AB A
AA
u =AB u AB u |u |
|u | = |AB|=
A
B
u
u
u
u ,v u //v
AB, BC, CD
DA M AC
BD ABCD
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ABCD
BCDA
CDAB
DABC
ADCB
DCBA
CBAD
BADC AC BD
ABCD ABDC
A= B C=D
ABCD
CDEF
ABFE
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AB AB
u = AB A
AB
AB
A
B ABB A
A
AB ABBA
AB
AB
CD AB
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AB
CD
AB CD
u =AB v =CD
CD
AD
D
A
B
u v
D A B
u v
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O
O
u = OA v = OB O u + v O C OACB
u = OA v = OB u +v
OC C |AC| = |OB| |BC| =
|OA|.
u + u
u
u =OA a aR a u
O |a| |u |
u |a|
a
au
u a >0 a
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u + OO = u
1 u = u 0 u = OO
(ab) u =a (b u )
(a + b) u =a u + b u
a (u + v) =a u + a v
u = OA
v = OB
w = OC
D E Z1 Z2
OA+
OB =
OD
OD+
OC=
OZ1
OB+OC=
OE
OA+
OE=
OZ2
(u + v) + w = OZ1, u + (v + w ) = OZ2.
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Z2 OAZ2E
OZ1=
OZ2 OAZ1E
OZ1
AE ODZ1C Z1
OZ1 CD
CD AE ACED
OCEB OBDA E D
ACED
(1)u u + (1)u =OO
u
u v u = v + (u )
AB AB AC AC
AB+
AC AB+ AC
a R,a AB a AB.
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ABB A
ACCA ABDC ABDC ADDA
ABDC ACCA BDCA
BDCA
ABDC
BDDB
BDDB ABB A ADDA
u = OA v = OB a R
v =a u
a v u : a= |OB||OA| . u v v =a u u v v = a u .
u v v =a u
u +v = (1+ a) u
u v
u v
u =OA v =OB
w O,w = OC
a, b R
w =au + bv .
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u v
OC
C
w
OB
u ,v OA A
OA u OA =a u
a R C OA OB B
OB = b v b R
OACB
w = OC = OA+ OB = au + bv .
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u 1,u 2, . . . ,u n O
u 1,u 2, . . . ,u n
a1u 1+ a2u 2+ . . .+ anu n
a1, a2, . . . , an
u 1,u 2, . . . ,u n
u 1,u 2, . . . ,u n
u 1,u 2, . . . ,u n O
i, 1 i n a1, . . . , ai1, ai+1, . . . , an
u i = aiu 1+ . . .+ ai1u i1+ ai+1u i+1+ . . .+ anu n
u 1,u 2, . . . ,u n
O
O
v = OA v
u 1
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u =OB v
u (, v)
a R
u = a|v |v
|a| =|u |
u
a > 0
u
v a < 0 u v
OB (
OB)
(u ) u
Chasles A, B, C
(,v) (AB) + (
BC) = (
AC) .
u =OA v =OB v A A u v
prv (u )
OA
prv(u + w ) = prvu + prvw
prv(a u ) =a prvu prvu =OA prvw =
OB prv(u +w ) =
OC
B
C
D
OAA
BC D (OA) = (BD) = (BC) Chasles
(OC) = (OB )+(BC (OC) = (OB )+(OA).
OC =OA+
OB .
OA
(, 1
|v | v)
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pr
vu =OA u v pr
u
v =OB v u
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u ,v O
|u |(pru v) = |v |(prvu ) .
OAA OBB
|prvu | = 0
|u ||v | =
|OA||OB|
=|OA||OB |
=|prvu ||pru v |
(prvu ) prvu v
u
v
O v u v O u pru v u (pru v)
|v | (prvu ) u
v u v = |v | (prvu ).
u ,v u v = 0.
u ,v ,w
O
a R
u v = v u
(a u ) v =a (u v)
u (v + w ) = u v + u w
u u 0 |u | = u u
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pru u = u
u
u u u = |u | (pruu ) = |u | |u | = |u |2 0.
u = OA v = OB
AOB = 0 (u ,v)
(u ,v)
<
u v (u ,v) = (v , u ) = (u ,v)
(prvu ) = |u | cos(u ,v)
u v = |u ||v | cos(u ,v) .
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E2 O
O
u =OA
v =OB
u
v (,u ) (,v) O (,u ), (,v) (O,u ,v) O
w O
w
w =a u + b v .
(a, b)
w (O,u ,v) a u , b v
w (O,u ,v) C C
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OC
OC
C
(O,u ,v)
E2
R2 = {(x, y)|x, y R}
u ,v i =
1
|u |u , j = 1|v |
v .
(O,
i ,j )
(a, b) w =a i + b j
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w
a= w
i , b= w
j
i i =j j = 1 i j = 0
w i = (a i + b j ) i= a
i i + b j i
= a
w j =b .
(O,
i ,
j) u ,v O (u1, u2), (v1, v2) u =u1 i +u2 j v =v1 i +v2 j u +v (u1+ v1, u2+ v2)
u + v = (u1+ v1) i + (u2+ v2) j ,
a R a u (au1, au2)
a u =au1 i + au2 j .
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u v u
v = (u1i + u2j )
(v1
i + v2
j)
= u1v1i i + u1v2i j + u2v1j i + u2v2j j= u1v1+ u2v2.
u
|u | =u u
=
(u1
i + u2
j ) (u1i + u2j)
= u12i i + u1u2i j + u2u1j i + u22j j
=
u12 + u22.
O
AB A
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B B C
AB
AC
AB+ BC
A B
v A A
B
u
A
B v u
F1 A F2 B
F1+
F2.
u = AB
u u
u [u ]
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[u ] = {v v u }. u
v u v
[u ] = [v] u v .
[u ] = [v] [v] = [w ] [u ] = [w ]
12
, 24
, 510
, 341682
12
13
1
2
+ 1
3
= 3
6
+ 2
6
= 5
6
[u ]
[v]
[u ] [v] O OAu
OBv OA+ OB =OC
[u ] [v]
OC
[u ] + [v] =OC
= [w ] .
[u ]
a
aOA
a [u ] =
aOA
= [a u ] .
pr[v][u ] =
prOBOA
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[u ] [v] = OA OB .
OA
OB
OA
OB
OA OA OB OB OA +OB OA +OB [u ] + [v]
[ ] u AB
(O,i ,j)
A, B, C, D
(a1, a2), (b1, b2), (c1, c2) (d1, d2) AB CD ABM CDN
b1 a1 = d1 c1 b2 a2=d2 c2
v1 = b1 a1 v2 = b2 a2
AB v = AB
v (O,
i ,
j)
v1=b1 a1 v2=b2 a2,
(a1, a2) (b1, b2) AB
v
v1= 0 v =
v2v1
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P1, P2, P P1P P P2 P=P2
P1P =
P P2.
(P1 P2 P)
(P1 P2 P) = (
P1P)
(P P2)
P1, P2 P
P1(x1, y1), P2(x2, y2) P(x, y)
x x1=(x2 x) y y1 = (y2 y),
x=x1+ x2
1 + y=
y1+ y21 +
.
>0 P P1 P2
P1P2 : 1
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P
P1P2
ABC (a1, a2)
(b1, b2) (c1, c2)
A(a1, a2) BC
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a1 =b1+ c1
2 a2=
b2+ c22
.
B(b1, b2)
b1 =a1+ c1
2 b2=
a2+ c22
.
AA BB
OA + AA
OB+
mBB m
OA+
AA=
OB+ m
BB .
a1+
b1+ c1
2 a1
= b1+ m
a1+ c1
2 b1
a2+
b2+ c2
2 a2
= b2+ m
a2+ c2
2 b2
, m
a1+b1
2
+c1
2 + b1
a1
2c1
2m= b1 a1
a2+ b22
+c2
2
+
b2 a2
2 c2
2
m= b2 a2
AA,
BB
= 23
, m= 23
.
AA BB G
(AAG) = (BB G) = 2 G AA CC
(AAG) = (CCG) = 2,
G= G
G
OG =
OA+
2
3
AA
= OA+23
(OA OA)
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= 1
3
OA+
2
3(
1
2(OB+
OC))
= 1
3(OA+
OB+
OC)
=
a1+ b1+ c1
3 ,
a2+ b2+ c23
.
(O,
i ,
j)
(P,u ,v)
(P,u ,v) (O,
i ,
j)
A (x, y) (O,
i ,
j)
(x, y) (P,u ,v) OA = x
i + y
j
P A= xu + yv
P A= OA OP
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P (O,
i ,
j) (p, q)
x
u + y
v = (x
p)
i + (y
q)
j
i ,
j u ,v
i ,
j ,u ,v
= (i ,u )
(j ,v) u = cos i + sin jv = sin i + cos j
x(cos i + sin
j ) +y( sin i + cos j) = (x p)i + (y q)j
i
j
x cos y sin = x px sin + y cos = y q .
x
y
x = (x p)cos + (y q)sin y = (x p)sin + (y q)cos .
(x, y) x, yp q
(x, y) x, y, p q
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x
y
=
cos sin
sin cos
x py q
.
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a + ib
a, b R i
C = {a + ib|a, b R} .
C
a + ib (a, b)
a + ib
ai + b
j .
z=a + ib
a= Re z
z
b= Im z
z Re z Im z
{z C | Im z = 0}
z= a +bi
|z| = z z=
a2 + b2 .
C
R {z C | Im z= 0}
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i i = 1
: (a + bi) + (c+ di) = (a + c) + (b+ d)i
: (a + bi) (c+ di) = (ac bd) + (ad + bc)i
a bi
z=a + bi
z
z+ z = 2a = 2Re z
z z = 2ib = 2iIm zz z = a2 + b2 =|z|2
z
z=
|z|2 z 1|z|2z= 1 z1 =
1
|z|2 z (a + bi)1 =
a
a2 + b2 b
a2 + b2i .
z
w=
zw
|w|2 a + bi
c+ di=
(a + bi)(c di)c2 + d2
z1+ z2 = z1+ z2
z1 z2 = z1 z2
(zn) = (z)n
|z| =|z| =| z||z1 z2| =|z1| |z2|
| |z1| |z2| | |z1+ z2| |z1| + |z2|
C
R
Ri
0
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z= 0 z
0
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z= 0 0
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A, B, C, D
z1, z2, z1 z2 1
ODB OAC z2
z1
z3 z1 z3
z2
z1 z2 z1 z2
z1 z2 : z1= z2 : 1 .
z2
z1 Arg(z2)
|z2
|
z |z| = 1 Arg(z)
i
/2
i2 =1 /2
i2z= z .
z
= 0 z
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z=r (cos +isin )
z1 =t(cos + i sin )
1 = z w
= r t (cos(+ ) + i( + ))
1
1
0
r t= 1 + = 0
z1 =1
r(cos() + i sin())
|z1| = 1|z| z1 arg z1 =
z1
Arg z1 =
2 Arg z Arg z= 00 Arg z= 0
z=r (cos + i sin )
z2 = r r (cos( + ) + i sin( + ))
= r2 (cos(2) + i sin(2))
z3
= r2
r (cos(2 + ) + i sin(2 + ))= r3 (cos(3) + i sin(3))
n
zn =rn (cos(n) + i sin(n)) .
De Moivre
z1
n N
zn =rn (cos(n) + i sin(n)) . De Moivre z= 0 n Z z= r(cos + i sin )
zn =rn (cos(n) + i sin(n))
z
S1
z1 = z .
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n 1
n
n
n
1
1
n
n n n
zn = 1 .
z
z=r(cos + i sin )
zn =rn (cos(n) + i sin(n)) = 1 ,
rn = 1
cos(n) = 1
sin(n) = 0
n
cos + i sin
cos(n) = 1 sin(n) = 0 n= 2k k Z
= k
n2, k Z .
k =
2kn
k+n= k+ 2
cos k+n+ i sin k+n = cos k+ i sin k k= 0,1, . . . , n 1
0= 0, 1=2
n, . . . , k=
2k
n , . . . , n1=
2(n 1)n
n n
cos
2k
n + i sin
2k
n k= 0, 1, . . . , n 1
wn= cos 2n
+ i sin 2n
k= 0, . . . , n 1
wkn = cos2k
n + i sin
2k
n .
k= 0
1
n S1 = {z C | |z| = 1} 0
2n
n
n 1
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a C
a= 0 n n a 0
zn = 0 z= 0 .
a = 0 a a= s(cos +isin ) s s1/n
n s z0 = s1/n
cos n
+ i sin n
zn0 =a z0 n a
zk n a zkz0
n= 1 zk
z0
n
a = 0 n n a= s(cos + i sin ) a n
zk =s1/n
cos
n+
2k
n
+ i sin
n+
2k
n
k= 0, 1, . . . , n 1
t+ i s+ i et(cos + i sin )
es(cos + i sin )
et(cos + i sin ) es(cos + i sin ) =et+s (cos( + ) + sin i( + ))
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et(cos + i sin )
n
=ent (cos(n) + i sin(n)) ,
t + i et(cos + i sin )
et+i =et(cos + i sin ) ,
ez =eRe(z) (cos(Im z) + i sin(Im z)) .
z= elog |z|+iarg z .
ez ew =ez+w
(ez)n =enz
(ez)1 =ez
ez =ez
|ez| =eRe z arg(ez) = Im z
cos , sin
z= cos + i sin 1z
= cos isin
2cos = z+1
z
2i sin = z 1z
.
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De Moivre zn = cos n+ i sin n zn = cos n isin n
2cos n = zn + 1zn
2i sin n = zn 1zn
cos6
26 cos6 =
z+
1
z
6
= z6 + 6z4 + 15z2 + 20 + 151
z2+ 6
1
z4+
1
z6
=
z6 +
1z6
+ 6
z4 +
1z4
+ 15
z2 +
1z2
+ 20
= 2 cos 6 + 12cos 4+ 30 cos2 + 20
cos6 = 1
32(cos6+ 6 cos 4 + 15 cos2 + 10)
(2i)5 sin5 =
z 1z
5
=
z5 1
z5
5
z3 1
z3
+ 10
z 1
z
25 sin5 = 2(sin5 5sin3 + 10 sin )
sin5
=
1
16 (sin5 5sin3 + 10sin ).
cos n, sin n
(cos6 + i sin 6) = (cos + i sin )6
= cos6 + 6i cos5 sin + 15i2 cos4 sin2
+20i3 cos3 sin3 + 15i4 cos2 sin4
+6i5 cos sin5 + i6 sin6 ,
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cos6= cos6
15cos4 sin2 + 15cos2 sin4
sin6
sin6= 6 cos5 sin 20cos3 sin3 + 6 cos sin5 .
nk
k= 0, 1, . . . , n
(1 +x)n =
nk=0
nk
xk ,
nk
=
n!
k!(n k)! =(n k+ 1) (n k+ 2) (n 1)n
1 2 k .
0! = 1 n0
=nk
= 1
C =n
k=0
nk
cos k
= 1 +n cos +n(n 1)
2 cos2+ . . .+ cos n
S =n
k=0
nk
sin k
= n sin + n(n 1)2
sin2 + . . .+ sin n.
C S
(1 +ei)n
(1 +ei)n =n
k=0
nk
eik
=n
k=0 n
k(cos k+ i sin k)
= C+ iS.
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1 + ein = (1 + cos + i sin )n=
2cos2
2+ 2i sin
2cos
2
n
=
2cos
2
ncos
2+ i sin
2
n
=
2cos
2
ncos
n
2 + i sin
n
2
.
C= 2cos2n
cosn
2
S=
2cos
2
nsin
n
2 .
n
Q= 1 + wn+ w2n+ . . .+ w
n1n .
wnQ = wn(1 + wn+ . . .+ wn1n )
= wn+ w2n+ . . .+ w
n1n + w
nn
= Q
(wn 1)Q= 0 wn= 1
1 +wn+ w2n+ . . .+ w
n1n = 0.
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E2
E3
AB A E3
AB
AB A
B
ABB A
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O, A, B, C
OA, OB, OC
A
A, B, C
AB, AC, AA
u = OA v = OB w = OC
z O z =OD a, b, c
R
z =au + bv + cw .
D z
u v u ,v ,w w OC C
c
OC = c
OC
A
B
OA
OB
a b OA = a OA OB = b
OB
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OA, OB , OC
D
OD =
OA+
OB +
OC
= a u + b v + c w .
u ,v ,w
u ,v ,w
u =OA v =OB
u v prvu u v
prvu =OA A A
OB
u v = |v |(prvu ) (prvu ) OB v
(u ,v) u =OA v = OB AOB O, A, B
u =OA v =OB OA OB OADB
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u v
u ,v
|OADB|
OADB
OC OG = OB+OC
OADB
OAEC
OAFG |OADB| + |OAEC| =|OAFG|
O
OA
OB
O
O
OA
OB
ON
ON
ON
ON ON
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ON
(OA,
OB)
OA OB 0< <
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(OB,
OA)
(OA,OB)
(OA,
OB)
(OA,
OB)
ON (
OA,
OB,
ON)
OX
(OA,
OB)
OA, OB
OA,OB
OA OB =
OO
OA,
OB
OX
OA,
OB
x ,y ,z O a
x y = y x
(a x ) y =a(x y)
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x (y + z) = (x y) + (x z)
E3 O
O u =OA v =OB w =OC
u ,v ,w (,u ), (,v), (,w ) O (,u ),(,v),(,w )
(O, u ,v ,w )
z O
z =au + bv + cw .
(a, b, c)
z (O,u ,v ,w )
au , bv , cw
z
(O,u ,v ,w ) D OD D
OD
D
(O,
i ,j ,
k)
i ,
j ,
k
(i ,
j ,
k)
z z
z = (z i )i + (z j)j + (z k)k ,
z =z1i + z2j + z3k y =y1i + y2j +y3
k
z y =z1y1+ z2y2+ z3y3
|z | = z21+ z22+ z23.
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(O,
i ,j ,
k)
i ,j
i j = k = j i
j k = i = k j
k i = j = i k .
z y = (z1i + z2j + z3k) (y1i + y2j + y3k)= z1y1
i i + z1y2i j + z1y3i k
+z2y1j i + z2y2j j + z2y3j k
+z3y1k i + z3y2k j + z3y3k k
= z1y1OO+ z1y2
k z1y3j
z2y1
k + z2y2
OO+ z2y3
i
+z3y1j z3y2i + z3y3OO ,
z y = (z2y3 z3y2)i + (z3y1 z1y3)j + (z1y2 z2y1)k .
33 a1, a2, a3 b1, b2, b3 c1, c2, c3
a1 a2 a3
b1 b2 b3
c1 c2 c3
=a1b2c3+ b1c2a3+ c1a2b3 a3b2c1 b3c2a1 c3a2b1 .
z y =
i
j
k
z1 z2 z3
y1 y2 y3
.
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x ,
y ,
z
x
(
y
z)
x (y z) = (x1i + x2j + x3k)
(y2z3 y3z2)i + (y3z1 y1z3)j + (y1z2 y2z1)k
= x1y2z3 x1y3z2+ x2y3z1 x2y1z3+ x3y1z2 x3y2z1
=
x1 x2 x3
y1 y2 y3
z1 z2 z3
.
x ,y ,z
x ,y ,z
(x ,y ,z) (x ,y ,z)
(x y) z = x (y z) .
(x y) z = z (x y)
=
z1 z2 z3
x1 x2 x3
y1 y2 y3
=
x1 x2 x3
y1 y2 y3
z1 z2 z3
= x (y z) .
[xyz] = x (y z) = (x y) z .
[xyz] = x (y z)=
x
(z
y)
= [zyx ] .
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b = x (y +z)x yx z b 2 = 0
b = 0
b b = b (x (y + z) x y x z)
=
b x (y + z) b x y b x z= (
b x ) (y + z) (b x) y (b x ) z
= (b x ) ((y + z) y z)
= 0 .
[xyz] = [yzx ] = [zx y] ,
[zyx ] = [yxz] = [xzy] .
x (y z) y z
y ,z x (y z)
y
z
x
(y
z) =
i
j
k
x1 x2 x3
y2z3 y3z2 y3z1 y1z3 y1z2 y2z1
=
i (x2(y1z2 y2z1) x3(y3z1 y1z3))+
j (x3(y2z3 y3z2) x1(y1z2 y2z1))
+k (x1(y3z1 y1z3) x2(y2z3 y3z2))
=
i (y1(x1z1+ x2z2+ x3z3) z1(x1y1+ x2y2+ x3y3))+
j (y2(x1z1+ x2z2+ x3z3) z2(x1y1+ x2y2+ x3y3))
+k (y3(x1z1+ x2z2+ x3z3)
z3(x1y1+ x2y2+ x3y3))
= (x z)y (x y)z .
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x (y z) = (x z)y (x y)z
(x y) z = (x z)y (y z)x .
u [u ]
u
[u ] = {v v u } .
OC=OA OB [
OA] [
OB]
[OA] [OB] = [OA OB]
= [OC] .
u OA,v OB w OC
[
u ]
[
v] = [
w ] .
(O,
i ,j ,
k)
(P,u ,v ,w )
(O,
i ,j ,
k) (O,u ,v ,w )
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X (x, y, z)
(O,
i ,j ,
k) X
OX = x
i + y
j + z
k .
(x, y, z) X
(O,u ,v ,w ) OX = xu + yv + zw .
i ,
j ,
k
u ,v ,w i = a1
u + b1v + c1wj = a2
u + b2v + c2w
k = a3u + b3v + c3w
OX = x(a1
u + b1v + c1w )+y(a2
u + b2v + c2w )+z(a3
u + b3v + c3w )= (xa1+ ya2+ za3)
u+(xb1+ yb2+ zb3)
v+(xc1+ yc2+ zc3)w
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x = a1x+ a2y+ a3z
y = b1x + b2y+ b3z
z = c1x + c2y+ c3z
x
y
z
=
a1 a2 a3b1 b2 b3
c1 c2 c3
xy
z
(O,
i ,
j ,
k)
(P,u ,v ,w ) OP =x0
i + y0
j + z0
k ,
P
(O,i ,j ,k)
(x0, y0, z0)
P X =
OX OP
= (x x0)i + (y y0)j + (z z0)k .
(x, y, z) X (P,u ,v ,w )
P X=xu + yv + zw ,
x = a1(x x0) +a2(y y0) +a3(z z0)y = b1(x x0) + b2(y y0) +b3(z z0) z = c1(x x0) + c2(y y0) +c3(z z0)
i ,
j ,
k
u ,v ,w
u ,v ,w i ,j ,k
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u u (O,
i ,
j ,
k) u
i ,
j ,
k
u = (
u
i )i + (
u
j)j + (
u
k)
k .
i u u i
1 u i
u j u k 2, 3 u j ,k
u i = cos 1 u j = cos 2 u k = cos 3
a1= i u = u i = cos 1
a2 = cos 2 a3 = cos 3
u =a1i + a2j + a3k .
(a1, a2, a3)
u (O,i ,j ,k)
v =b1i + b2j + b3k
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w =c1i + c2j + c3k (O,i ,j ,k) (P,u ,v ,w )
(O,
i ,
j ,
k)
u = a1i + a2j + a3kv = b1i + b2j + b3kw = c1i + c2j + c3k ,
i
j
k
u a1 a2 a3v b1 b2 b3w c1 c2 c3
(O,
i ,
j ,
k)
(P,u ,v ,w ) P (O,i ,j ,k) (x0, y0, z0)
X (x, y, z) (O,
i ,j ,
k)
(x, y, z)
X
(P,u ,v ,w )
x = a1(x x0) +a2(y y0) +a3(z z0)y = b1(x x0) + b2(y y0) +b3(z z0)z = c1(x x0) + c2(y y0) +c3(z z0)
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(x, y) f : R2
R
P(x, y)
f(x, y) = 0 {(x, y) R2 : f(x, y) = 0} f(x, y) = 0
{(x, y) R2 : f(x, y) = 0}
f(x, y) = 0
f(x, y) = ax+by + c
ax+by + c = 0
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{(x, y) : ax + by + c= 0}
=a/b y
c/b
f(x, y) =x2 +y2 r2 r >0
f(x, y) = 0
x2+y2 =r2 r (0, 0)
O r
C
(a, b) P (x, y)
P
C
CP
= 3
(x a)2 + (y b)2 = 3
f(x, y) =
(x a)2 + (y b)23 (x a)2 + (y b)2 x y (x a)2 + (y b)2 = 9 g(x, y) = (x a)2 +(y b)2 9
(x a)2 + (y b)2 = 0
P
(a, b)
f(x, y) g(x, y) R2
f(x, y)g(x, y) f
g f(x, y)g(x, y) = 0
f(x, y) = 0 g(x, y) = 0
y2 x2 = 0
(y+x)(y x) = 0
f(x, y) g(x, y) R2
f(x, y)2 +g(x, y)2
f g f(x, y)2 +
g(x, y)2 = 0 f(x, y) = 0
g(x, y) = 0
f(x, y) = 0
g(x, y) = 0
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P (x, y, z)
f : R3 R P(x, y, z)
f(x, y, z) = 0
C(a, b, c) r
(x a)2 + (y b)2 + (z c)2 =r2
g(x, y, z) = ax+by +c
f(x, y)
ax+ by +c = 0
P(x, y, z)
z R z
f(x, y, z)
z (x, y) ax+
by+ c= 0
x2
+ y2
+ z2
= rax+ by+ cz = d
y2 +z2 = 4
{(x, y, z) R3 : y2 + z2 = 4}
x (y, z)
O
2
{(x, y) R2 : 2x 3y+ 2 = 0}
{(x, y) R2 : y= 1}
{(x, y) R2 : (y 1)(2x y) = 0}
{(x, y) R2 : x2 + y2 2x 2y 2 = 0}
{(x, y) R2 : x2 + y2 = 2xy}
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{(x, y, z) R3 : x2 + y2 + z2 + 2x 2y 2 = 0}
{(x, y, z) R3 : x2 + y2 + z2 + 2x 2y+ 2 = 0}
{(x, y, z) R3 : 2x 3y= 0}
{(x, y, z) R3 : y= 1}
{(x, y, z) R3 : (y 1)(x2 + z2 1) = 0}
f : R R2 f(t) = (x(t), y(t)) t R P(x(t), y(t))
f : R R3 f(t) = (x(t), y(t), z(t)) t (x(t), y(t), z(t))
f : R2 R3 f(s, t) = (x(s, t), y(s, t), z(s, t))
(s, t) R2
P(x(s, t), y(s, t), z(s, t))
A(1,2)
B(1, 3) f(t) = (1 + 2t, 3 t)
(a, b)
r
0
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s t s2 + t2 1
x(t) = 1 t y(t) = 2 3t
t
t= 1x
t= y23
t
1 x= 13
(y 2) 3x+ y 5 = 0
|z w1| = |z w2|, > 0 .
w1 w2
z
|z w1| =|z w2|, >0 .
= 1
|z
w1
|=
|z
w2
|
z
w1 w2
= 1 z w1 w2
p w1 w2
p
w1 w2
(w1, w2, p) = p w1w2 p,
(w1, w2, p) = = (w1, w2,p) . p p
p=w1+ w2
1 + p =
w1 w21
z
|z
w1
|=
|z
w2
|
z w1, w2
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|z w1| = 2|z w2|
p p
c = p+p
2
= (w1+ w2)(1 ) + (w1 w2)(1 +)
2(1 2
)
= w1 2w2
1 2
r = |p p|
2
= |(w1 w2)(1 ) (w1 w2)(1 +)|
2|1 2
|=
|| |w1 w2||1 2|
c r
(z c)(z c) =r2
z(1 2) w1+ 2w2
z(1 2) w1+ 2w2
= 2(w1 w2)( w1 w2)
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zz
zw1
zw1+ w1w1 =
2(zz
zw2
zw2+ w2w2)
|z w1|2 =2|z w2|2 .
= 1
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P(x1, y1, z1) Q(x2, y2, z2) R(x3, y3, z3) .
X(x, y, z)
P X
P Q,
P R s t
P X=s
P Q + t
P R
(x x1, y y1, z z1) =s(x2 x1, y2 y1, z2 z1) +t(x3 x1, y3 y1, z3 z1)
(x, y, z) = (1
s
t)(x1, y1, z1) +s(x2, y2, z2) +t(x3, y3, z3) .
P Q
P R
P Q P R
P Q P R P X= 0 .
x2 x1 y2 y1 z2 z1x3
x1 y2
y1 z3
z1
x x1 y y1 z z1 = 0
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A = (y2
y1)(z3
z1)
(z2
z1)(y3
y1)
B = (z2 z1)(x3 x1) (x2 x1)(z3 z1)C = (x2 x1)(y3 y1) (y2 y1)(x3 x1)
Ax+ By + Cz (Ax1+ By1+ Cz1) = 0
P(x1, y1, z1)
n (k, , m) X(x, y, z) P X n
P X n = 0 ,
(x x1, y y1, z z1) (k, , m) = 0 ,
kx+ y+ mz (kx1+ y1+ mz1) = 0 .
Ax+ By + Cz+ D= 0 .
(x1, y1, z1)
Ax1+ By1+ Cz1+ D= 0 ,
(x, y, z)
A(x x1) +B(y y1) +C(z z1) = 0 ,
(x x1, y y1, z z1) (A, B, C) (x, y, z) (x1, y1, z1)
(A, B, C)
B=C= 0 A = 0
x= DA
O y z
A= 0, BC= 0
By + Cz+ D= 0
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Ox
O y z By + Cz+ D= 0
D= 0
Ax+ By + Cz= 0
O
ABCD= 0 (DA
, 0, 0)
(0,DB
, 0) (0, 0, DC
)
x, y, z , ,
= 0 x
+y
+z
= 1 .
P(x1, x2, x3) a = (u, v, w)
(x, y, z) = (x1, y1, z1) +s(u, v, w)
s uvw = 0x x1
u =
y y1v
=z z1
w
vx uy (vx1 uy1) = 0wy vz (wy1 vz1) = 0 .
w= 0 uv= 0 s
vx uy (vx1 uy1) = 0z = z1 .
A1x + B1y+ C1z+ D1= 0 ,
A2x + B2y+ C2z+ D2= 0 .
A1A2= B1B2
= C1C2= D1D2
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A1A2
=B1B2
=C1C2
= D1D2
(A1, B1, C1) =
k(A2, B2, C2)
A1A2
= B1B2
A1A2
= C1C2
,
(A1, B1, C1) (A2, B2, C2)
u = (A1, B1, C1) (A2, B2, C2)
(x, y, z) = (x0, y0, z0) +t(A1, B1, C1) (A2, B2, C2)
(x0, y0, z0)
A1x0+ B1y0+ C1z0 =
D1
A2x0+ B2y0+ C2z0 = D2 .
A1B2 B1A2= 0 z0= 0
A1x0+ B1y0 = D1A2x0+ B2y0 = D2
x0=(D1B2 B1D2)
A1B2 B1A2, y0=
(A1D2 D1A2)A1B2 B1A2
.
Ax+ By +C z+D = 0 n = (A, B, C) X0 X1
X1
X0X1
n
d = (prn X0X1) = X
0X
1 n
|n |
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= (x1 x0)A+ (y1 y0)B+ (z1 z0)C
A2 + B2 + C2
=
Ax1+ By1+ Cz1
(Ax0+ By0+ Cz0)
A2 + B2 + C2
Ax0+ By0+ Cz0 = D
d=Ax1+ By1+ Cz1+ D
A2 + B2 + C2
d X1
n = (A, B, C)
OX =
OX0 + ta (x, y, z) =
(x0, y0, z0) +t(a, b, c) X1 OX1= (x1, y1, z1)
X0X1 a
X1
e = (X0X1 a) a
X1 X0X1
e
d = |preX0X1| =
e X0X1
|e
|
= |(X0X1 a) a X0X1|
|(X0X1 a) a |.
a
X0X1 a
d = |(X0X1 a) (a X0X1)|
|X0X1 a | |a |
= |
X0X1
a
||a | .
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1, 2
OX =
OX1+ t
a1 t ROX =
OX2+ s
a2 s R
X1X2,
a1 a2
[X1X2,
a1 ,a2 ] = 0 .
u a1 a2
u = a1 a1.
X1X2
u
d = |pra1a2X1X2|
= |X1X2 a1 a2 ||a1 a2 |
= |[X1X2,a1 ,a2 ]|
|a1 a2 | .
T1 T2 1
2 T1 1
OT1 =
OX1+ t
a1
T2 2
OT2 =
OX2+ s
a2.
T1T2
a1 a2 T1T2 =
a1 a2.
T1T2=
OT2 OT1
OX2 OX1+ s a2 t a1 = a1 a2,
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(s, t, ) t, s
T1, T2
1 2 1
A(2,1,0)
u = (1, 3,1)
2
B(0,1,3) v = (2, 1, 1)
[AB,u ,v] =
2 2 31 3 12 1 1
= 29 .
u v = (1, 3,1) (2, 1,1) = (4,3,5) ,
|u v | = 52
d=|[AB, u ,v]|
|u
v|
=| 29|
5
2=
29
2
10 .
OB OA+ s v t u = (u v) ,
(2, 2, 3) +s (2, 1, 1) t (1, 3,1) = (4,3,5) ,
2s t 4 = 2s 3t + 3 = 2s + t + 5 = 3 ,
s t
t= 1
25 , s= 7
50.
T1 = (2,1, 0) + 125
(1, 3,1)
T2 = (0,1,3) 750
(2, 1, 1)
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|T1 T2| = 1
50 |116,87,145| =29
2
10 .
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r >0 C
X
|CX| =r
C
x2 + y2 =r2
C : (x0, y0)
(O,
i ,j )
(x x0)2 + (y y0)2 =r2
(x, y) = (x0, y0) +r(cos , sin ) 0
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(x, y) = (x0, y
0) +r1
t2
1 +t2,
2t
1 +t2
t (,) (x0 r, y0) t
1 t21 +t2
, 2t
1 +t2
t
t= sin
1 + cos .
r X1: (x1, y1)
X : (x, y)
(x, y) = (x1, y1) + t(a, b)
(x1+ ta)2 + (y1+ tb)
2 =r2 ,
x21+ y21+ 2t(ax1+ by1) + t
2(a2 + b2) r2 = 0 . t
= (ax1+ by1)2 (a2 + b2)(x21+ y21 r2) .
< 0
> 0
= 0
X1 x21+ y
21
r2 = 0 X1
ax1+ by1 = 0 ,
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X1
(a, b) = (y1, x1)
(x, y) = (x1 ty1, y1+ tx1) X1 : (x1, y1)
O r
x1x + y1y= r2
X1 X1
x21 +y21 r2 0 a, b X0 : (x0, y0)
X1
X0 x0x + y0y=r
2
X1 X0 x0x1+ y0y1 = r2
X1
xx1+ yy1 = r2 .
x21+ y21 r2 >0
X0 X0
X0, X0
X1
x1x+ y1y= r2 ,
X1
X1
X1 X1
X1
X1
X1
(x, y)
X1: (x1, y1)
X1
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Ax+ By + C= 0
(x1, y1) =
Ar
2
C ,Br
2
C
.
C : (x0, y0)
F1, F2
F1F2 i =
F1F2
|F1F2| F1, F2
(C, 0) (C, 0) (O,i ,j)
X
F1, F2
2a
a > c
|F1X| + |F2X| = 2a ,
(x+ c)2 + y2 +
(x c)2 + y2 = 2a
(x c)2 + y2 =
a ca
x
a2 c2a2
x2 + y2 =a2 c2
a > c a2 c2 >0 b2 =a2 c2 x2
a2+
y2
b2 = 1 .
c= 0 a2 =b2
a= c
F1F2
e= c
a
1 : x= a2
c 2 : x=
a2
c
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X r1, r2 X
F1, F2
r21 = (x + c)2 + y2 , r22 = (x c)2 + y2 .
r21 r22 = 4cx r1+r2 = 2a r1 r2 = 2cx
a r1 =a +
cx
a
r2=a cxa
X
d1=d(X, F1) =a2
c + x d2 = d(X, F2) =
a2
c x ,
r1d1
=a2 + cx
a
c
a2 + cx=
c
a=e
r2d2
X1 : (x1, y1) X2 : (x2, y2) X : (x, y)
X1X2 X
x=
x1+ tx21 +t , y=
y1+ ty21 +t ,
t= (X1X2X)
(X1X)
(XX2)
(x, y) t
(x1+ tx2)2
a2(1 +t)2 +
(y1+ ty2)2
b2(1 +t)2 = 1 ,
t = 1
x21+ 2tx1x2+ t2x21a2
+y21+ 2ty1y2+ t2y22
b2 = 1 + 2t+ t2 ,
x22a2
+y22b2
1
t2 + 2x1x2
a2 +
y1y2b2
1
t +
x21a2
+y21b2
1
= 0
L= x2
2
a2+
y22
b2 1 M= x1x2
a2 + y1y2
b2 1 N= x21
a2+
y21
b2 1
M2 LN= 0 .
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X1X2
X1
M= 0
X1
x1x
a2 +
y1y
b2 = 1 .
X1 x21a2
+y2
b >1
x1x
a2 +
y1y
b2 1
2
x21a2
+y21b2
1x2
a2+
y2
b2 1= 0
X1
X1
F1X1F2
n
Q1 cos(X1F1,
n )
cos(X1F2,
n )
F1, F2 x1x
a2 +
y1y
b2 = 1 (x1, y1) b
x2
a2 + y2
b2 = 1
(x, y) = (a cos , b sin ) .
(a,0)
(x, y) =
a(1 t2)
1 + t2 ,
2bt
1 +t2
.
t
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F1, F2
F1F2 i =
F1F2
|F1F2| F1, F2
(c, 0) (c, 0) X |F1X| |F2X| = 2a c > a
(x+ c)2 + y2
(x c)2 + y2 = 2a
(x c)2 + y2 = cxa a
a2 c2a2
x2 + y2 =a2 c2 .
c > a b2 =c2
a2
x2
a2 y
2
b2 = 1 .
x2
a2 1 = y
2
b2 >0
|x| > a
y2
b2 x
2
a2 = 1
(0, c) (0,c)
e= c
a
1 : x= a2
c 2 : x=
a2
c
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X : (x, y) =
x1+ tx2
1 +t ,
y1+ ty21 +t
X1 : (x1, y1) X2 : (x2, y2)
t
(x1+ tx2)2
a2(1 +t)2 (y1+ ty2)
2
b2(1 +t2) = 1 ,
x22
a2
y22
b2
1 t2 + 2
x1x2
a2
y1y2
b2
1 t +
x21
a2
y21
b2
1= 0 .
L= x2
2
a2 y22
b2 1 M= x1x2
a2 y1y2
b2 1 N= x21
a2 y21
b2 1
M2 LN= 0 X1X2
X1
M= 0
X1 x1x
a2 y1y
b2 = 1 .
X1 x21a2
y21
b2
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x2
a2y2
b2 = 1
cosh2 t sinh2 t = 1
(x, y) = (a cosh s, b sinh s) , s R x >0
(x, y) = (a cosh s, b sinh s) , s R
x
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(y22 2px2)t2 + 2(y1y2 p(x1+ x2))t + y21 2px1= 0 .
L= y22 2px2 , M=y1y2 p(x1+ x2) , N=y21 2px1 .
M2 LN= 0
X1X2
X1
M= 0
X1
y1y= p(x1+ x)2px1
(y1y p(x1+ x))2 (y21 2px1)(y2 2px) = 0
X1
X0
X0F X0
u X0 cos(
n ,
X0F) cos(
n ,
i )
x= 2pt2
(x, y) = 2p(t2, t) .
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f(x, y) =Ax2 + 2Bxy + Cy2 + 2Dx+ 2Ey + F= 0
A, B, C
(x, y) (0, 0)
(x,y)
Ax2 + 2Bxy + Cy2 + 2Dx+ 2Ey + F =
= A(x)2 + 2B(x)(y) +C(y)2 + 2D(x) + 2E(y) + F .
4Dx+ 4Ey = 0 .
(x1, y1), (x2, y2), (x3, y3)
D E
D= E= 0 .
y
(x, y) (x, y)
Ax2 + 2Bxy + Cy2 + 2Dx+ 2Ey + F =
= A(x)2 + 2B(x)y+ Cy2 + 2D(x) + 2Ey + F
4Bxy + 4Dx= 0 ,
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B=D = 0 .
P(x0, y0)
(x, y)
(x, y)
x = x0 + x cos y sin y = y0 + x
sin + y cos .
f(x, y)
f(x, y) =Ax2 + 2Bxy+ Cy2 + 2Dx+ 2Ey+ F = 0 ,
A, B, . . . , F A, B , . . . , F, x0, y0,
J1 = A+ C=A+ C
J2 = AC B2 =AC B2
J3=
A B D
B C F
D E F
=
A B D
B C E
D E F
.
J1, J2, J3
J2= 0 J2= 0 A C
A+ C =A+ C
J2= 0 J2= 0
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D = (Ax0+ By0+ D)cos + (Bx0+ Cy0+ E)sin
E = (Ax0+ By0+ D)sin + (Bx0+ Cy0+ E)cos
x0, y0 D
= E = 0
(x0, y0) sin cos
D =E = 0
Ax0+ By0+ D = 0
Bx0+ Cy0+ E = 0
J2 = AC
B2
= 0
x0= DC BEAC B2 y0 =
AE DBAC B2 .
(x0, y0) D=E = 0
Ax2 + 2Bxy+ Cy2 + F = 0
B = 0
B
B = A sin cos + B(cos2 sin2 ) +Csin cos
B = 0
B(cos2
sin2
) = (A C)sin cos
2B cos2= (A C)sin2
A= C
B= 0
cos2= 0
=
4
A =C
tan2=
2B
A C
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4
< 4
2
0 x2
a2+
y2
b2 = 1
AF < 0, CF > 0 x2
a2 y
2
b2 = 1 x
AF > 0, CF < 0 x2
a2 y
2
b2 = 1 y
F = 0
Ax2 + Cy2 = 0
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y = A
C
x
AC0 y= ix
F
A, F
C J1, J2, J3
FA FC = F2
AC = F2
AC B2 = F2
J2.
J3 =ACF =J2F
J3= 0 J2 >0 J1J3 < 0 J1J3 > 0
J3= 0 J2 0
J3 = 0 J2
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A C= 0
=
4
cos = sin =22
x= 1 +2
2 x
22
y =2
2 (x y
2)
y = 2 +
2
2 x+
2
2 y =
2
2 (x+ y+ 2
2)
1
2(x y
2)2 4
2
2 (x y
2)
2
2 (x+ y+ 2
2)+
+1
2(x+ y+ 2
2)2 + 10
2
2 (x y
2) 8
2
2 (x+ y+ 2
2) + 7 = 0 ,
x2 + 3y2 6 = 0 ,
y
x2
6 y2
2 = 1 .
J1=A+ C, J2 = AC
A, C
t2 J1t + J2
t2 2t 3 = 0
(t + 1)(t 3) = 0
A = 1, C = 3 A = 3, C= 1
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J2= 0
J2 = 0
xy B = 0
J2 =AC = 0 A, C
C= 0
Ax2 + 2Dx+ 2Ey+ F = 0 .
E
= 0
x+
D
A
2= 2E
A
y+
F
2E D
2
2AE
,
D
A, D
2AF2AE
x =DA
E
A
x =x+D
A, y = y D
2 AF2AE
.
p= EA
x, y x, y
x2 = 2py
B = C = 0 J3 = AE2 J1 = A
p2 =E2
A2 = J3
J13.
E = 0
x=D D2 AF
A
D2 AF > 0 D2 AF = 0 D2 AF
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D= 0
y+
E
C2
= 2D
C
x+
F
2D E2
2CD
E2CF2CD
,EC
y = EC
p= DC
y2 = 2px .
p2 = D2
C2 = J3
J13.
D = 0
y=E E2 CF
C
E2 CF > 0
E2 CF = 0
E2 CF0
J3= 0 J3= 0 J3= 0
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O OA
X r
O
= (OA, OX)
(r, ) X
(r, + 2k)
k
Z
x = r cos
y = r cos
r = x2 + y2
= arctany
x,
x y
(, ] (
3,1)
r=
3 + 1 = 2 , = arctan 1
3=
6.
(
3, 1),(
3,
1) (
3,
1)
2, 5
6
,
2,56
2,
6
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O a
r= a .
u
O
t
r= ut
= t
t
r= u
.
O
x2 + y2 + z2 =2
X : (x, y, z) X X
Oxy
X : (x, y, 0)
OX
x OX
= (Ox, OX)
= (OX, OX) , 2
2
X
X
(, , )
X
=
x2 + y2 + z2
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= arctany
x, <
= arcsin zx2
+ y2
+ z2
,
2
2
x = |OX| cos = cos cos y = |OX| sin = cos sin z = (XX) = sin
O a
= a
= 0
0 = 0 (x, y) 0= 0
z
2 0 0 >0 2 +0 0
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z
cos =
x2
+ y2
.
X
(x, y, z)
X X (x, y) (x, y, 0) (x, y)
X
(, )
r=
x2 + y2 , = (Ox, OX)
(r, , z) X
r =
x2 + y2
= arctany
x, <
z = z
x = |OX| cos = r cos y =
|OX
|sin = r sin
z = z
z a
r= a .
z=c
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(x, y) |c|
= 0
z
= a , = 0
z
= az , a = 0
z z
z
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z
(x, y, z)
x2 + y2 =a2 .
z
R
z
z=x y = 0
x2 + y2 =z2
2.
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z
(x, z)
f(x, z) = 0 , y= 0 , x 0 , (x, y, z) R3
f(
x2 + y2, z) = 0.
(x0, 0, z0)
z = z0
x0 z
(x, y, z0) x2 + y2 =x0
x2
a2+
y2
b2 = 1, z= 0
x
x2
a2+
y2 + z2
b2 = 1
z
(x a)2 + z2 =r2 , y = 0 , a > r >0 .
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x2 + y2 a2
+ z2 =r2
x2 + y2 2pz= 0
f(
x2 + y2, z) = 0
x2 = 2pz
y = 0
z
x
y
z
x2
a2+
y2
b2+
z2
c2 = 1 , a, b, c >0 .
f(x, y, z) =x2
a2+
y2
b2 +
z2
c 1
f(x, y, z) =f(x, y, z) =f(x,y, z) =f(x, y,z)
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(y, z) (x, z)
(x, y)
2a,2b,2c
A : (a,0, 0)
A : (a,0, 0)
B : (0, b, 0)
B : (0,b, 0) C : (0, 0, c) C(0, 0,c) (x, y)
x2
a + y
2
b = 1 , z= 0 .
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(x, y) z= k c k c
x2
a2+
y2
b2 = 1 k2
c2, z=k
|k| =c |k| < c
a
1 k
2
c2 , b
1 k
2
c2 .
(x, z) (y, z)
a, b, c
x2a2
+y2b2
z2c2
= 1 , a, b, c >0 .
(x, y) (x, z)
(y, z) x, y, z
(x, y)
x2
a2+
y2
b2 = 1 , z= 0 .
(x, z)
x2
a2+
z2
c2 = 1 , y= 0 ,
(y, z)
y2
b2+
z2
c2 = 1 , x= 0 .
z=k
x2
a2+y
2
b2 = 1 k
2
c2 , z=k .
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x= m |m| =a
y2
b2z2
c2 = 1 m2
a2 , x= m .
|m| =a 1 m2a2
= 0
y
b = z
c, x= m ,
(m,0,0)
a= b
x2
a2 y
2
b2 +
z2
c2 = 1 , a, b, c >0 .
(x, y) (x, z) (y, z)
x, y, z
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(x, y)
z=k
|k| > c |k| =c
(x, z) (y, z)
a= b
x2
a2
+y2
b2
= 2cz , a, b >0 , c
= 0 .
(x, z)
(y, z)
z
(x, y)
(x, y) z = k k > 0
x2
a2+
y2
b2 = 2ck , z =k .
k
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(x, z) (y, z)
(x, z)
x2 = 2a2cz , y = 0 ,
y=m
x2
a2 = 2cz m
2
b2 , y= m .
x2
a2 y
2
b2 = 2cz , a, b >0 , c = 0 .
(x, z) (y, z)
z
(x, y)
x2
a2 y
2
b2 = 0 , z= 0 ,
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(x, y) z=k
x2
a2 y
2
b2 = 2ck , z = k .
(x, z) (y, z)
x= m
y2
b2 = 2cz+m2
a2 , x= m .
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x2
a2+
y2
b2 z
2
c2 = 0
y2 = 2cz, c = 0
y2
b2 z
2
c2 = 1
y2
b2 +
z2
c2 = 1
x2
a2y2
b2 = 0
x2
a2+
y2
b2 z
2
c2 = 0
z2 = 0
x2
a2+
y2
b2 = 0
x2a2 +
y2
b2 + z2c2 = 0
x2
a2+
y2
b2+
z2
c2 = 1
x2 y2 z2 + 4x 8y+ 2z 17 = 0
(x + 2)2 4 (y+ 4)2 + 16 (z 1)2 + 1 17 = 0
(x+ 2)2 (y+ 4)2 (z 1)2 = 4 . (2,4, 1)
x2
4 y
2
4 z
2
4 = 1 .
(
2,
4, 1) x
(0,4,1) (4,4,1)
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(x, y), (x, z), (y, z)
x = 4 y = 4
z= 4
i 9x2 + 4z2 = 36y
ii 4y2 + 4z2
x2 = 0
iii 9x2 y2 = 4z
iv 2y2 + 4z2 =x2
v x2 + y
2
4 + z
2
9 = 1
vi x2
9 +
y2
16z
2
4 = 1
vii
x2
4 y2
9 + z2 = 1
viii
x2
4 y
2
9 = 0
x2 + y2 + z2 6x+ 4y 8 = 0