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P
v
P1 P2
l P l
l1 l2
n P
v1 v2 l1 l2 P
P1 P2 P3
P l1 P l2
P l1 P l2
P P l
l1 l2
{x ,y ,z} {a,b,c,d} {l} {P, Q} {v, w} {n} {s, t} {P l} { , ,}
v1 v2 v1v2 =||v1|| ||v2|| cos()
v1v2 v1 v2
w v v( w) =
v wv v
v w
v
P1 P2 l:x ,y ,z= P1+t ( P2P1)
v= P2 P1
ax+by+cz = d n=a,b,c
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P v
l v P l:x ,y ,z= P+ t v v=1, 2, 1 P= (1, 2, 2)
x ,y ,z=1, 2, 2+t 1, 2, 1= t1, 2t+ 2, t+ 2
v=6, 6, 6 P= (4, 1, 100)
x ,y ,z=6t+ 4, 6t1, 6t+ 100
l1 (2, 2, 3) l2 :x ,y ,z =1, t1, 5t+ 2
l2 0, 1, 5 l1 l2 l1 0, 1, 5 l2 :x ,y ,z=2, t+ 2, 5t+ 3
P1 P2
P1 P2 v = P2
P1
l:x ,y ,z= P1+t v (0, 0, 0) (1, 2, 3)
v=10, 20, 30=1, 2, 3 l:x ,y ,z=t, 2t, 3t
(1, 1, 3) (2, 2, 2)
v=21, 2(1), 23=1, 1, 1 l:x ,y ,z=t+ 1, t1,
l P l
l
l:x ,y ,z=2t+ 1, t1, t x+ 2y+ 3z = 5 x=2t+ 1 y= t1 z =t 5 = (2t+ 1) + 2(t1) + 3(t) =3t1 t=2 t=2 (x ,y ,z) = (5, 3, 2)
5 + 2(3) + 3(2) = 5
l:
x ,y ,z
=
3t
1, 2t+ 2,
4t
2y+z = 1
x= 3t1 y= 2t+ 2 z=4t 1 = 2(2t+ 2) + (4t) = 4 1 = 4
l:x ,y ,z=3t1, 2t+ 2, 4t 2y+z = 4
x= 3t1 y= 2t+ 2 z=4t 4 = 2(2t+ 2) + (4t) = 4 4 = 4
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l1 l2
l1 :x ,y ,z = P1+ t v1 l2 :x ,y ,z = P2+ s v2 s t
s t
l1:
x ,y ,z
=
t, 2t, 3t
l2 :
x ,y ,z
=
s+ 2, s+ 1, 2s+
t, 2t, 3t=s+ 2, s+ 1, 2s+ 1 t=s+2 2t= s+1 3t= 2s+ 1
2s+ 4 =s+ 1 s= 1 t = 1 {s = 1, t = 1} t = 1
1, 2, 3 l1 (1, 2, 3)
l1 :x ,y ,z =2t+ 1, t1, t l2 :x ,y ,z =s+ 2, s+ 1, 2s+ 1
2t+ 1, t
1,
t
=
s+ 2, s+ 1, 2s+ 1
2t +1 =
s+ 2 t1 =s+ 1 t= 2s+ 1 s= 2t+ 1 t1 = 2t+ 2 t= 3 s= 7 {s= 7, t= 3}
n P
n=a,b,c ax + by+ cz= P
(1, 1, 1) n=1, 2, 1
x+ 2yz = d d (1, 1, 1) 1 + 21 =d d= 3 x+ 2yz = 3
(2, 3, 2) n=6, 6, 6
6x+ 6y+ 6z= d d (2, 3, 2) 1218 + 12 = d d= 6 6x+ 6y+ 6z= 6
x+y+z = 1
v1 v2 l1 l2 P
v1 v2 v1v2 v1 v2 n=v1v2
n= v1v2 v1 v2 l1 l2
l1 l2 l1 l2
l1 l2 n=v1v2 P l1
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v1 =1, 1, 1 v2 =1, 2, 3 (0, 0, 0)
n= v1v2 =
i j k
1 1 11 2 3
= 1 12 3
i 1 11 3
j+ 1 11 2
k=1, 2, 1 n v1 = (1)(1)+(2)(1) + (1)(1) = 0 n v2 = (1)(1)+(2)(2) +
(1)(3) = 0
x2y+z = d d d (x ,y ,z) = (0, 0, 0) d= 0 x2y+z = 0
(1, 1, 2) l1 :x ,y ,z=2t+ 1, t1, t l2 :x ,y ,z=s+ 2, s+ 1, 2s+ 1
v1 =2, 1, 1 v2 =1, 1, 2
n= v1v2 = i j k
2 1 11 1 2
= 1 11 2 i 2 11 2 j+ 2 11 1 k =3, 5, 1
n v1 = (3)(2) + (5)(1) + (1)(1) = 0 n v2 = (3)(1) + (5)(1) + (1)(2) = 0 3x+ 5yz = d d d (1, 1, 2) d= 3(1) + 5(1)(2) =4 3x+ 5yz =4
l1 :x ,y ,z=2t+ 1, t1, t l2 :x ,y ,z=s+ 2, s+ 1, 2s+ 1 l2 l1 l1 l2
l1 l2 3, 5, 1 3x+ 5yz = d d l1 l1 t= 0 (1, 1, 0) d= 3(1) + 5(1)0 =2 l1 l2 3x+ 5yz =2 l2 l2 s= 0 (2, 1, 1) d= 3(2) + 5(1)1 = 10 l2 l1 3x+ 5yz = 10
l1 l2
P1 P2 P3
n n v1 = P2 P1 v2 = P3 P1
P1 = (3, 0, 1) P2 = (1, 2, 2) P3 = (2, 1, 4)
v1 = P2P1 =2, 2, 3 v2= P3 P1 =5, 1, 5
n=v1v2 = 2 31 5
i 2 35 5
j+ 2 25 1
k =7, 5, 8
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n v1 = (7)(5) + (5)(1) + (8)(5) = 0 n v2 = (7)(2) + (5)(2) + (8)(3) = 0 7x5y+ 8z = d d P1 d= (7)(3)5(0) + 8(1) = 13 7x5y+ 8z = 13
7(3)5(0) +8(
1) = 13 7(1)
5(2) + 8(2) =
13 7(
2)
5(1) + 8(4) = 13
P1= (2, 2, 4) P2 = (1, 1, 1) P3 = (0, 1, 0)
v1 = P2P1 =3, 3, 5 v2 = P3P1 =2, 1, 4
n=v1v2 = 3 51 4
i 3 52 4
j+ 3 32 1
k=7, 2, 3 n v1 = 7(3) + 2(3) + 3(5) = 0 n v2= 7(2) + 2(1) + 3(4) = 0
7x+ 2y+ 3z = d d P1 d= 7(2) + 2(2) + 3(4) = 2 7x+ 2y+ 3z = 2
7(2)+2(2)+3(4) = 2 7(1) + 2(1) + 3(1) = 2 7(0) + 2(1) + 3(0) = 2
P l1 P l2
P l1 P l2 l n1 n2 l l v= n1n2 l (x ,y ,z)
xy+ 2z= 3 2x+yz = 0
n1 =1, 1, 2 n2 =2, 1, 1
v= n1
n2 =
1, 5, 3
n v1 =1(1) + 5(1) + 3(2) = 0 n v2=1(2) + 5(1) + 3(1) = 0 x = 0 y+ 2z = 3
yz = 0 y= z = 3 (0, 3, 3) l l:x ,y ,z=t, 5t+ 3, 3t+ 3
2x+y4z = 7 3x4y+ 5z= 2
n1 =2, 1, 4 n2 =3, 4, 5 v= n1 n2 =11, 2, 5
n v1 =11(2)2(1) + 5(4) = 0 n v2 =11(3)2(4) + 5(5) = 0 x = 0 y4z = 7 4y+z = 2 y=1 z =2 (0, 1, 2) l l:x ,y ,z=11t, 2t1, 5t2
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P l1 P l2
N1 N2
= n1 n2 n1n2 =||n1||||n2|| cos()
cos() = |n1
n2
|||n1||||n2|| 0
2
xy+ 2z = 3 2x+y+z = 0
n1 =1, 1, 2 n2 =2, 1, 1 n1n2 = 3 ||n1||=||n2||=
6
cos() = |n1n2|||n1||||n2|| = 3
6 6 =1
2
= 6
6
2x+y
4z = 7 3x
4y+ 5z = 2
n1 =2, 1, 4 n2 =3, 4, 5 n1n2 =30 ||n1|| = 21 ||n2|| =5
2
cos() = |n1n2|||n1||||n2|| = 30
5
42=
42
7
cos1
42
7
P P l
P
P
Q P l n P l v P w = Q P
n v= n( w) =
n wn n
n
v ||v||=|n
w
|||n||
n =a,b,c P =x0, y0, z0 n w =n P n Q= (ax0+by0+cz0)(d) ||n||=
a2 +b2 +c2
n Q= d Q P l
|ax0+by0+cz0d|a2 +b2 +c2
(x0, y0, z0) ax+by+cz = d
x+y+z = 1 (0, 0, 0)
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n=1, 1, 1 P =0, 0, 0 Q=1, 0, 0 w=1, 0, 0 n w= 1 ||n||= 3
||v||= 13
|1(0) + 1(0) + 1(0)1|12 + 12 + 12
= 1
3
2xy+ 2z= 12 (2, 1, 3) n =2, 1, 2 P =2, 1, 3 Q =6, 0, 0
w=4, 1, 3 n w= 3 ||n||= 3
||v||= 33
= 1
|2(2)1(1) + 2(3)12|22 + (1)2 + 22 =
3
3= 1
l1
l2
l1 l2 P l1 P l2
P P l1 P l2
n
P l1 : ax +by+cz = d1 P l2 : ax+by+cz = d2
|d2
d1
|a2 +b2 +c2
l1 :x ,y ,z =2t+ 1, t1, t l2 :x ,y ,z=s+ 2, s+ 1, 2s+ 1
v1 =2, 1, 1 v2 =1, 1, 2 n=3, 5, 1 n v1 = (3)(2) + (5)(1) + (1)(1) = 0 n v2 = (3)(1) + (5)(1) + (1)(2) = 0
P l1 P l2 3x+ 5yz = d d l1 l1 t = 0 (1, 1, 0)
d= 3(1) + 5(1)0 =2 P l1 : 3x+ 5yz =2 l2 l2 s = 0 (2, 1, 1)
d= 3(2) + 5(1)
1 = 10 P l2: 3x+ 5y
z = 10
|10(2)|
32 + 52 + (1)2 = 12
35
[(s+ 2)(2t+ 1)]2 +[(s+ 1)(t1)]2 + [(2s+ 1)(t)]2 6s2 2st+ 6t2 +6s+ 2t+ 6
144
35
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