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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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