ÿþR +g a s T e h n i s k u n i v e r s i t t e . I n ~e n i e r m a t e m t i k a s k a t e d r a . U z d e v u m u
r i s i n j u m u p a r a u g i .
9 . n o d a r b +b a
1 . p i e m r s . S a s t d +t v i e n d o j u m u p l a k n e i , k a s i e t
1 ) c a u r p u n k t i e m M 1 ( 1 , 1 , - 1 ) , M 2 ( 1 , 0 , - 3 ) u n M 3 ( - 1 , - 1 , - 9 ) ;
2 ) c a u r p u n k t u M ( 3 , - 2 , - 4 ) p e r p e n d i k u l r i O z a s i j ;
3 ) c a u r p u n k t i e m M 1 ( 0 , 1 , 2 ) u n M 2 ( 1 , 2 , 0 ) p e r p e n d i k u l r i p l a k n e i x 7 = 0 ;
r
4 ) c a u r p u n k t u M ( 1 , - 2 , 1 ) p a r a l l i d i v i e m d o t i e m v e k t o r i e m a = ( 2 , - 1 , 3 ) u n
r
b = ( 1 , 2 , 5 ) ;
5 ) c a u r p u n k t u M 1 ( 2 , 2 , 1 ) p e r p e n d i k u l r i v e k t o r a m M 1 M , k u r M 2 ( 1 , 2 , 3 ) .
2
R i s i n j u m s .
1 ) I z m a n t o s i m f o r m u l u ( 3 ) , t . i . v i e n d o j u m u p l a k n e i c a u r t r i m d o t i e m p u n k t i e m , t a j
i e v i e t o s i m x 1 = 1 , y 1 = 1 , z 1 = - 1 - p u n k t a M 1 k o o r d i n t a s , x 2 = 1 , y 2 = 0 , z 2 = - 3 -
p u n k t a M 2 k o o r d i n t a s , x 3 = - 1 , y 3 = - 1 , z 3 = - 9 - p u n k t a M 3 k o o r d i n t a s , i e g ks i m :
x - 1 y - 1 z + 1 x - 1 y - 1 z + 1
1 - 1 0 - 1 - 3 - 1 = 0 , 0 - 1 - 2 = 0 ,
- 1 - 1 - 1 - 1 - 9 + 1 - 2 - 2 - 8
- 1 - 2 0 - 2 0 - 1
( x - 1 ) - ( y - 1 ) + ( z + 1 ) = 0 ,
- 2 - 8 - 2 - 8 - 2 - 2
4 ( x - 1 ) + 4 ( y - 1 ) - 2 ( z + 1 ) = 0 ,
4 x - 4 + 4 y - 4 - 2 z - 2 = 0 ,
4 x + 4 y - 2 z - 1 0 = 0 ,
2 x + 2 y - z - 5 = 0 .
r
2 ) I z m a n t o s i m f o r m u l u ( 1 ) u n p a r p l a k n e s n o r m l v e k t o r u Fe m s i m O z a s s o r t u k = ( 0 , 0 , 1 ) ,
r
t t a d f o r m u l ( 1 ) i e v i e t o s i m A = 0 , B = 0 , C = 1 - v e k t o r a k k o o r d i n t a s , x 0 = 3 ,
y 0 = - 2 , z 0 = - 4 - p u n k t a M k o o r d i n t a s :
0 ( x - 3 ) + 0 ( y + 2 ) + 1 ( z + 4 ) = 0
z + 4 = 0
r r
3 ) J a t r o d p l a k n e s n o r m l e s v e k t o r s n . V e k t o r s n v e i d o t a i s n u l e F7i a r
r
M 1 M = ( 1 ; 1 ; - 2 ) , k a s p i e d e r m e k l t a i p l a k n e i u n a r n 1 = ( 1 ; 0 ; 0 ) , k a s i r p e r p e n d i k u l r s
2
9 . n o d a r b +b a . 1 . l p p . A u g s t k m a t e m t i k a .
I . V o l o d k o
R +g a s T e h n i s k u n i v e r s i t t e . I n ~e n i e r m a t e m t i k a s k a t e d r a . U z d e v u m u
r i s i n j u m u p a r a u g i .
r
p l a k n e s n o r m l v e k t o r s . T a s n o z +m , k a v e k t o r u n v a r a m a t r a s t v e k t o r i l r e i z i n j u m a
r
M 1 M × i v e i d
2
r
r r
i j k
r 1 r
r - 2 1 - 2 1
r r
1
r r
n = M 1 M × n 1 = 1 1 - 2 = i - j + k = - 2 j - k
2
0 0 1 0 1 0
1 0 0
r
P c f o r m u l a s ( 1 ) , i e v i e t o j o t A = 0 , B = - 2 , C = - 1 - v e k t o r a n k o o r d i n t a s , x 0 = 0 ,
y 0 = 1 , z 0 = 2 - p u n k t a M 1 k o o r d i n t a s , i e g ks t
0 ( x - 0 ) - 2 ( y - 1 ) - ( z - 2 ) = 0
- 2 y + 2 - z + 2 = 0
2 y + z - 4 = 0
r
r
4 ) P l a k n e s n o r m l v e k t o r u m e k l s i m k v e k t o r u a u n b v e k t o r i l o r e i z i n j u m u , j o d i v u
v e k t o r u v e k t o r i l a i s r e i z i n j u m s i r p e r p e n d i k u l r s ai e m v e k t o r i e m , l +d z a r t o
p e r p e n d i k u l r s a r + m e k l t a j a i p l a k n e i .
r
r r
i j k
r r r
r - 1 3 2 3 2 - 1
r r r
r r
n = a × b = 2 - 1 3 = i - j + k = - 1 1 i - 7 j + 5 k
2 5 1 5 1 2
1 2 5
P l a k n e s v i e n d o j u m u i e g ks i m f o r m u l ( 1 ) i e v i e t o j o t A = - 1 1 , B = - 7 , C = 5 - v e k t o r a
r
n k o o r d i n t a s , x 0 = 1 , y 0 = - 2 , z 0 = 1 - p u n k t a M k o o r d i n t a s :
- 1 1 ( x - 1 ) - 7 ( y + 2 ) + 5 ( z - 1 ) = 0
- 1 1 x + 1 1 - 7 y - 1 4 + 5 z - 5 = 0
- 1 x - 7 y + 5 z - 8 = 0
1 1 x + 7 y - 5 z + 8 = 0
r
5 ) P l a k n e s n o r m l v e k t o r s n = M 1 M = ( - 1 ; 0 ; 2 ) u n p l a k n e s v i e n d o j u m ( 1 ) i e v i e t o j o t
r 2
A = - 1 , B = 0 , C = 2 - v e k t o r a n k o o r d i n t a s , x 0 = 2 , y 0 = 2 , z 0 = 1 - p u n k t a M 1
k o o r d i n t a s , i e g ks i m
- 1 ( x - 2 ) + 0 ( y - 2 ) + 2 ( z - 1 ) = 0
- x + 2 + 2 z - 2 = 0
x - 2 z = 0
9 . n o d a r b +b a . 2 . l p p . A u g s t k m a t e m t i k a .
I . V o l o d k o
R +g a s T e h n i s k u n i v e r s i t t e . I n ~e n i e r m a t e m t i k a s k a t e d r a . U z d e v u m u
r i s i n j u m u p a r a u g i .
2 . p i e m r s . A p r 7i n t l e F7i , k o v e i d o p l a k n e s 2 x + 3 y - 2 z - 8 = 0 u n
x + 2 y + 4 z + 1 = 0 .
R i s i n j u m s .
r r
P l a k n e s n o r m l v e k t o r i i r n 1 = ( 2 , 3 , - 2 ) u n n 2 = ( 1 , 2 , 4 ) . I z m a n t o j o t ao v e k t o r u s k a l r o
r e i z i n j u m u , a p r 7i n s i m l e F7i s t a r p ai e m v e k t o r i e m , k a s s a k r +t a r l e F7i s t a r p d o t a j m
p l a k n m :
r r
n Å" n 2 2 Å"1 + 3 Å" 2 + ( - 2 ) Å" 4 2 + 6 - 8
c o s Õ = = = = 0 ,
r 1 r
n 1 Å" n 2 2 2 + ( - 3 ) 2 + ( - 2 ) 2 1 2 + 2 2 + 4 2
4 + 9 + 4 Å" 1 + 4 + 1 6
Å"
t t a d d o t s p l a k n e s v e i d o t a i s n u l e F7i .
3 . p i e m r s . A p r 7i n t l e F7i , k o v e i d o p l a k n e x + 2 y - 2 z + 1 = 0 u n p l a k n e , k a s i e t c a u r
2 x
§# - y + z - 3 = 0
p u n k t u A ( 0 , 2 , - 1 ) p e r p e n d i k u l r i t a i s n e i
¨#
©#x + y - z + 1 = 0
R i s i n j u m s .
L e F7i , k o v e i d o p l a k n e s , a p r 7i n s i m k l e F7i s t a r p a t b i l s t o aa j i e m
r
n o r m l v e k t o r i e m . P i r m s p l a k n e s n o r m l v e k t o r s n 1 = ( 1 , 2 , - 2 ) . O t r s p l a k n e s
n o r m l v e k t o r s s a k r +t a r d o t s t a i s n e s v i r z i e n a v e k t o r u , s a v u k r t t a i s n e s v i r z i e n a v e k t o r s i r
v i e n d s a r p l a k Fu ( k u r u a7l u m a l +n i j a i r a+ t a i s n e ) n o r m l v e k t o r u v e k t o r i l o r e i z i n j u m u :
r
r r
i j k
r r
r - 1 1 2 1 2 - 1
r r
r r
n 2 = s = 2 - 1 1 = i - j + k = 3 j + 3 k
1 - 1 1 - 1 1 1
1 1 - 1
A t z +m s i m , k a m u m s i r v a j a d z +g s t i k a i o t r s p l a k n e s n o r m l v e k t o r s , n a v o b l i g t i
j z i n a o t r s p l a k n e s v i e n d o j u m s . T t a d
r r
n 1 Å" n 2 1 Å" 0 + 2 Å" 3 - 2 Å" 3 6 - 6
c o s Õ = = = = 0 ,
r r
n 1 Å" n 2 1 2 + 2 2 + ( - 2 ) 2 3 2 + 3 2
1 + 4 + 4 Å" 9 + 9
Å"
l +d z a r t o d o t s p l a k n e s i r p e r p e n d i k u l r a s .
r r 1 r
P i e z +m e . V e k t o r a n 2 v i e t v a r Fe m t m 2 = ( 0 , 1 , 1 ) = n 2 , t k v e k t o r a g a r u m s n e i e t e k m
3
l e F7a a p r 7i n aa n u .
9 . n o d a r b +b a . 3 . l p p . A u g s t k m a t e m t i k a .
I . V o l o d k o
R +g a s T e h n i s k u n i v e r s i t t e . I n ~e n i e r m a t e m t i k a s k a t e d r a . U z d e v u m u
r i s i n j u m u p a r a u g i .
2 x + y
x + 1 y z + 1 §# - z + 3 = 0
4 . p i e m r s . N o t e i k t l e F7i s t a r p t a i s n m = = u n
¨#x - 2 y + 2 z + 1 = 0
2 - 2 1
©#
R i s i n j u m s .
L e F7i s s t a r p d i v m t a i s n m i r v i e n d s a r l e F7i s t a r p t o v i r z i e n a v e k t o r i e m . P i r m s
r
t a i s n e s v i r z i e n a v e k t o r a k o o r d i n t a s v a r a m n o l a s +t n o v i e n d o j u m a : s 1 = ( 2 , - 2 , 1 ) . L a i
n o t e i k t u o t r s t a i s n e s v i r z i e n a v e k t o r u , v a j a d z s a t r a s t p l a k Fu , k u r a s a7e <o t i e s v e i d o o t r o
t a i s n i , n o r m l v e k t o r u v e k t o r i l o r e i z i n j u m u :
r
r r
i j k
r r
r - 1 2 - 1 2 1
r r
1
r r r
s 2 = n 1 × n 2 = 2 1 - 1 = i - j + k = - 5 j - 5 k
- 2 2 1 2 1 - 2
1 - 2 2
r r 1 r
M e k l j o t l e F7i , a i z v i e t o s i m s 2 a r s 3 = s 2 = ( 0 , - 1 , - 1 ) , t a d
5
r r
s 1 Å" s 3 2 Å" 0 + ( - 2 ) Å"( - 1 ) + 1 Å" ( - 1 ) 2 - 1 1 2
c o s ± = = = = = .
r r
s 1 Å" s 3 6
4 + 4 + 1 Å" 1 + 1 9 Å" 2 3 2
2
T d j d i l e F7i s s t a r p d o t a j m t a i s n m i r Õ = a r c c o s .
6
x = 6 + 3 t
§#
x - 1 y - 7 z - 3
ª#y
5 . p i e m r s . N o t e i k t l e F7i , k o v e i d o t a i s n e s = - 1 - 2 t u n = = .
¨#
2 1 4
ª#
z = - 2 + t
©#
R i s i n j u m s .
T k t a i s n e s u z d o t a s p a r a m e t r i s k a j u n k a n o n i s k a j f o r m , t s v i r z i e n a v e k t o r i i r
r r
n o l a s m i n o v i e n d o j u m i e m : s 1 = ( 3 , - 2 , 1 ) u n s 2 = ( 2 , 1 , 4 ) . L e F7i s t a r p t a i s n m , t p a t
k i e p r i e k aj u z d e v u m , v a r a p r 7i n t k l e F7i s t a r p a t b i l s t o aa j i e m v i r z i e n a v e k t o r i e m :
r r
s Å" s 2 3 Å" 2 + ( - 2 ) Å"1 + 1 Å" 4 6 - 2 + 4 8
c o s Õ = = = = ,
r 1 r
s 1 Å" s 2
9 + 4 + 1 Å" 4 + 1 + 1 6 1 4 Å" 2 1 7 6
8
t t a d m e k l t a i s l e F7i s i r Õ = a r c c o s .
7 6
9 . n o d a r b +b a . 4 . l p p . A u g s t k m a t e m t i k a .
I . V o l o d k o
R +g a s T e h n i s k u n i v e r s i t t e . I n ~e n i e r m a t e m t i k a s k a t e d r a . U z d e v u m u
r i s i n j u m u p a r a u g i .
6 . p i e m r s . N o t e i k t t a i s n e s u n p l a k n e s k r u s t p u n k t u :
x - 1 y - 1 z - 2
a ) = = u n 3 x - 2 y + z - 1 2 = 0
2 1 0 5
2 x + 3 y - 5 z - 5 = 0
§#
b ) u n x + 2 y - 3 z - 3 = 0 .
¨#
6 x - 2 y + z - 9 = 0
©#
R i s i n j u m s .
A t r a s t k r u s t p u n k t u n o z +m a t r a s t r i s i n j u m u v i e n d o j u m u s i s t m a i , k u r i e t i l p s t
p l a k n e s u n t a i s n e s v i e n d o j u m i .
a ) P r v e i d o s i m t a i s n e s v i e n d o j u m u s p a r a m e t r i s k a j f o r m , t a d , i e v i e t o j o t t o p l a k n e s
v i e n d o j u m , i e g ks t l i n e r u v i e n d o j u m u a r v i e n u n e z i n m u .
x - 1 y - 1 z - 2
= t Ò! x = 2 t + 1 , = t Ò! y = 1 0 t + 1 , = t Ò! z = 5 t + 2 .
2 1 0 5
x = 2 t + 1
§#
ª#y
T t a d = 1 0 t + 1 . `+s i z t e i k s m e s i e v i e t o s i m p l a k n e s v i e n d o j u m :
¨#
ª#
z = 5 t + 2
©#
3 ( 2 t + 1 ) - 2 ( 1 0 t + 1 ) + ( 5 t + 2 ) - 1 2 = 0
6 t + 3 - 2 0 t - 2 + 5 t + 2 - 1 2 = 0
- 9 t - 9 = 0
t = 1
K r u s t p u n k t a k o o r d i n t a s a t r o d n o t a i s n e s p a r a m e t r i s k a j i e m v i e n d o j u m i e m
x = 2 Å"1 + 1 = 3
§#
ª#y = 1 0 Å"1 + 1 = 1 1 Ò! A ( 3 , 1 1 , 7 )
¨#
ª#
z = 5 Å"1 + 2 = 7
©#
b ) D o t a j g a d +j u m v i s u t r i j u p l a k Fu v i e n d o j u m u s v a r a m a p v i e n o t v i e n s i s t m u n
a t r i s i n t t o :
2 x + 3 y - 5 z - 5 = 0
§#
ª#
6 x - 2 y + z - 9 = 0
¨#
ª#
x + 2 y - 3 z - 3 = 0
©#
I e g kt o s i s t m u a t r i s i n s i m p c G a u s a m e t o d e s :
9 . n o d a r b +b a . 5 . l p p . A u g s t k m a t e m t i k a .
I . V o l o d k o
R +g a s T e h n i s k u n i v e r s i t t e . I n ~e n i e r m a t e m t i k a s k a t e d r a . U z d e v u m u
r i s i n j u m u p a r a u g i .
›#2 3 - 5 5 ž# ›#1 2 - 3 3 ž#
œ# Ÿ# œ# Ÿ#
6
œ# - 2 1 9 ~ [ 1 . r i n d u m a i n +s i m v i e t m a r 3 . r i n d u ] ~ 6 - 2 1 9 ~
Ÿ# œ# Ÿ#
œ#1 2 - 3 3 Ÿ# œ#2 3 - 5 5 Ÿ#
# # # #
›#1 2 - 3 3 ž#
œ# Ÿ#
2 . r i n d a + 1 . r i n d a Å"( - 6 )
¡# ¤#
~
¢#3 . r i n d a + 1 . r i n d a Å"( - 2 ) ¥# ~ œ#0 - 1 4 1 9 - 9 Ÿ# ~ [ 2 . r i n d u m a i n +s i m v i e t m a r 3 . r i n d u ] ~
£# ¦# œ#0 - 1 1 - 1 Ÿ#
# #
›#1 2 - 3 3 ž# ›#1 2 - 3 3 ž#
œ# Ÿ# œ# Ÿ#
~ 0 - 1 1 - 1 ~ [ 3 . r i n d a + 2 . r i n d a Å"( - 1 4 ) ] ~ 0 - 1 1 - 1
œ# Ÿ# œ# Ÿ#
œ#0 - 1 4 1 9 - 9 Ÿ# œ#0 0 5 5 Ÿ#
# # # #
x + 2 y
§# - 3 z = 3
ª#
T t a d - y + z = - 1 , n o k u r i e n e s z = 1 , - y + 1 = - 1 Ò! y = 2 , x + 4 - 3 = 3 Ò! x = 2 .
¨#
ª#
5 z = 5
©#
E s a m i e g u v u ai , k a t a i s n e s u n p l a k n e s k r u s t p u n k t s i r B ( 2 , 2 , 1 ) .
7 . p i e m r s . S a s t d +t v i e n d o j u m u p l a k n e i , k u r a t r o d a s t a i s n e s
x = t + 5
§#
ª#y x - 1 y + 3 z
a ) = 3 t - 1 u n = =
¨#
5 3 2
ª#z = t + 2
©#
x = 2
§# - t
x + 1 y + 2 z + 1
ª#y
b ) = 3 + t u n = =
¨#
3 - 3 - 6
ª#z = 2 t
©#
R i s i n j u m s .
a ) N o t a i aFu v i e n d o j u m i e m i r z i n m i d i v i p l a k n e s p u n k t i : M 1 ( 5 , - 1 , 2 ) u n M ( 1 , - 3 , 0 ) .
2
r r
A t r a d +s i m p l a k n e s n o r m l v e k t o r u k t a i aFu v i r z i e n a v e k t o r u s 1 u n s 2 v e k t o r i l o
r e i z i n j u m u :
r
r r
i j k
r r
r r r r
3 1 1 1 1 3
r r r
n = s 1 × s 2 = 1 3 1 = i - j + k = 3 i + 3 j - 1 2 k
3 2 5 2 5 3
5 3 2
r
P l a k n e s v i e n d o j u m u i e g ks i m f o r m u l ( 1 ) i e v i e t o j o t A = 3 , B = 3 , C = - 1 2 - v e k t o r a n
k o o r d i n t a s , x 0 = 5 , y 0 = - 1 , z 0 = 2 - p u n k t a M 1 k o o r d i n t a s :
9 . n o d a r b +b a . 6 . l p p . A u g s t k m a t e m t i k a .
I . V o l o d k o
R +g a s T e h n i s k u n i v e r s i t t e . I n ~e n i e r m a t e m t i k a s k a t e d r a . U z d e v u m u
r i s i n j u m u p a r a u g i .
3 ( x - 5 ) + 3 ( y + 1 ) - 1 2 ( z - 2 ) = 0
3 x - 1 5 + 3 y + 3 - 1 2 z + 2 4 = 0
3 x + 3 y - 1 2 z + 1 2 = 0
x + y - 4 z + 4 = 0
r r
b ) T k d o t o t a i aFu v i r z i e n a v e k t o r i s 1 = ( - 1 ; 1 ; 2 ) u n s 2 = ( 3 ; - 3 ; - 6 ) i r k o l i n e r i ( j o
r r
s 2 = - 3 s 1 ) , t a d d o t s t a i s n e s i r p a r a l l a s , l +d z a r t o n e v a r a m a t r a s t p l a k n e s n o r m l v e k t o r u
r r
k t a i aFu v i r z i e n a v e k t o r u s 1 u n s 2 v e k t o r i l o r e i z i n j u m u . T p c u z p i r m s t a i s n e s
i z v l s i m i e s d i v u s p u n k t u s : Fe m o t t = 0 , i e g ks i m p u n k t u A ( 2 ; 3 ; 0 ) , Fe m o t t = 1 -
p u n k t u B ( 1 ; 4 ; 2 ) . U z o t r s t a i s n e s Fe m s i m t r e ao p u n k t u C ( - 1 ; - 2 ; - 1 ) u n s a s t d +s i m
p l a k n e s v i e n d o j u m u p c f o r m u l a s ( 3 ) v i e n d o j u m p l a k n e i c a u r t r i m d o t i e m p u n k t i e m
i e v i e t o s i m x 1 = 2 , y 1 = 3 , z 1 = 0 - p u n k t a A k o o r d i n t a s , x 2 = 1 , y 2 = 4 , z 2 = 2 -
p u n k t a B k o o r d i n t a s , x 3 = - 1 , y 3 = - 2 , z 3 = - 1 - p u n k t a M 3 k o o r d i n t a s :
x - 2 y - 3 z - 0 x - 2 y - 3 z
1 - 2 4 - 3 2 - 0 = 0 , - 1 1 2 = 0 ,
- 1 - 2 - 2 - 3 - 1 - 0 - 3 - 5 - 1
1 2 - 1 1
( x - 2 ) - ( y - 3 ) - 1 2 + z = 0
- 5 - 1 - 3 - 1 - 3 - 5
9 ( x - 2 ) - 7 ( y - 3 ) + 8 z = 0
9 x - 1 8 - 7 y + 2 1 + 8 z = 0
9 z - 7 y + 8 z + 3 = 0
x + 1 y - 2 z + 3
8 . p i e m r s . N o t e i k t , a r k d u k v r t +b u t a i s n e = = i r p a r a l l a p l a k n e i
3 k - 2
x - 3 y + 6 z + 7 = 0 .
R i s i n j u m s .
J a t a i s n e i e t p a r a l l i p l a k n e i , t a d t s v i r z i e n a v e k t o r s v e i d o t a i s n u l e F7i a r p l a k n e s
r r r
n o r m l v e k t o r u , t . i . s Å" n = 0 . T k t a i s n e s v i r z i e n a v e k t o r s s = ( 3 ; k ; - 2 ) , p l a k n e s
r
n o r m l v e k t o r s n = ( 1 ; - 3 ; 6 ) , t a d
3 Å"1 + ( - 3 ) Å" k + ( - 2 ) Å" 6 = 0 , 3 - 3 k - 1 2 = 0 , k = - 3 .
9 . n o d a r b +b a . 7 . l p p . A u g s t k m a t e m t i k a .
I . V o l o d k o
R +g a s T e h n i s k u n i v e r s i t t e . I n ~e n i e r m a t e m t i k a s k a t e d r a . U z d e v u m u
r i s i n j u m u p a r a u g i .
9 . p i e m r s . N o t e i k t p u n k t a M ( 4 ; - 1 ; 0 ) p r o j e k c i j u p l a k n x - 3 y - z + 4 = 0 .
R i s i n j u m s .
V i s p i r m s s a s t d +s i m v i e n d o j u m u s t a i s n e i , k a s i e t c a u r
p u n k t u M p e r p e n d i k u l r i d o t a j a i p l a k n e i . P l a k n e s
M r
n o r m l v e k t o r s n = ( 1 ; - 3 ; - 1 ) i r p e r p e n d i k u l r s d o t a j a i
p l a k n e i , l +d z a r t o i r p a r a l l s m e k l t a j a i t a i s n e i u n d e r p a r
t s v i r z i e n a v e k t o r u . T a i s n e s k a n o n i s k a j o s v i e n d o j u m o s
r
N ( 4 ) i e v i e t o s i m x 0 = 4 , y 0 = - 1 , z 0 = 0 - p u n k t a M
n
r
k o o r d i n t a s ; m = 1 , n = - 3 , p = - 1 - v e k t o r a n
k o o r d i n t a s :
x - 4 y + 1 z
= = .
1 - 3 - 1
P u n k t a M p r o j e k c i j a p l a k n i r a+s t a i s n e s u n d o t s
p l a k n e s k r u s t p u n k t s . L a i t o n o t e i k t u , t a i s n e s
v i e n d o j u m u s i z t e i k s i m p a r a m e t r i s k f o r m :
x - 4 y + 1 z
= t Ò! x = t + 4 , = t Ò! y = - 3 t - 1 , = t Ò! z = - t
1 - 3 - 1
u n i e g kt s i z t e i k s m e s i e v i e t o s i m p l a k n e s v i e n d o j u m :
( t + 4 ) - 3 ( - 3 t - 1 ) - ( - t ) + 4 = 0
t + 4 + 9 t + 3 + t + 4 = 0
1 1 t + 1 1 = 0
t = - 1
T t a d
x = t + 4 = - 1 + 4 = 3
§#
ª#y = - 3 t - 1 = - 3 Å"( - 1 ) - 1 = 2
¨#
ª#
z = - t = - ( - 1 ) = 1
©#
u n m e k l t p r o j e k c i j a i r p u n k t s N ( 3 ; 2 ; 1 ) .
9 . n o d a r b +b a . 8 . l p p . A u g s t k m a t e m t i k a .
I . V o l o d k o
R +g a s T e h n i s k u n i v e r s i t t e . I n ~e n i e r m a t e m t i k a s k a t e d r a . U z d e v u m u
r i s i n j u m u p a r a u g i .
x - 5 y z + 2 5
1 0 . p i e m r s . N o t e i k t p u n k t a M ( 2 ; 3 ; - 1 ) p r o j e k c i j u u z t a i s n e s = = .
3 2 - 2
R i s i n j u m s .
S a s t d +s i m v i e n d o j u m u p l a k n e i , k a s i e t c a u r p u n k t u M
p e r p e n d i k u l r i d o t a j a i t a i s n e i . T k t a i s n e s v i r z i e n a
r
v e k t o r s s = ( 3 ; 2 ; - 2 ) i r p a r a l l s d o t a j a i t a i s n e i , t a d t a s
M i r p e r p e n d i k u l r s m e k l t a j a i p l a k n e i u n d e r p a r t s
n o r m l v e k t o r u . T d j d i p l a k n e s v i e n d o j u m u i e g ks i m ,
v i e n d o j u m ( 1 ) i e v i e t o j o t A = 3 , B = 2 , C = - 2 -
r
v e k t o r a s k o o r d i n t a s , x 0 = 2 , y 0 = 3 , z 0 = - 1 -
N
r
s
p u n k t a M k o o r d i n t a s :
3 ( x - 2 ) + 2 ( y - 3 ) - 2 ( z + 1 ) = 0
3 x - 6 + 2 y - 6 - 2 z - 2 = 0
3 x + 2 y - 2 z - 1 4 = 0
P u n k t a M p r o j e k c i j a u z d o t s t a i s n e s s a k r +t a r t a i s n e s u n p l a k n e s k r u s t p u n k t u N . L a i t o
n o t e i k t u , t a i s n e s v i e n d o j u m u s i z t e i k s i m p a r a m e t r i s k f o r m
x - 5 y z + 2 5
= t Ò! x = 3 t + 5 , = t Ò! y = 2 t , = t Ò! z = - 2 t - 2 5
3 2 - 2
u n i e g kt s i z t e i k s m e s i e v i e t o s i m p l a k n e s v i e n d o j u m :
3 ( 3 t + 5 ) + 2 ( 2 t ) - 2 ( - 2 t - 2 5 ) - 1 4 = 0
9 t + 1 5 + 4 t + 4 t + 5 0 - 1 4 = 0
1 7 t + 5 1 = 0
t = - 3
I e g ks i m
x = 3 t + 5 = 3 Å"( - 3 ) + 5 = - 4
§#
ª#
y = 2 t = 2 Å"( - 3 ) = - 6
¨#
ª#z = - 2 t - 2 5 = - 2 Å"( - 3 ) - 2 5 = - 1 9
©#
t t a d p u n k t a M p r o j e k c i j a u z d o t s t a i s n e s i r p u n k t s N ( - 4 ; - 6 ; - 1 9 ) .
9 . n o d a r b +b a . 9 . l p p . A u g s t k m a t e m t i k a .
I . V o l o d k o
R +g a s T e h n i s k u n i v e r s i t t e . I n ~e n i e r m a t e m t i k a s k a t e d r a . U z d e v u m u
r i s i n j u m u p a r a u g i .
1 1 . p i e m r s . N o t e i k t p u n k t a m M ( 2 ; - 5 ; 7 ) s i m e t r i s k o p u n k t u a t t i e c +b p r e t t a i s n i
x - 5 y - 4 z - 6
= = .
1 3 2
R i s i n j u m s .
A p z +m s i m s i m e t r i s k o p u n k t u a r M 1 u n
a t z +m s i m , k a n o g r i e ~Fa M M 1 v i d u s p u n k t s P i r
p u n k t a M p r o j e k c i j a u z t a i s n e s . S a s t d +s i m
v i e n d o j u m u p l a k n e i , k a s i e t c a u r p u n k t u
M ( 2 ; - 5 ; 7 ) p e r p e n d i k u l r i d o t a j a i t a i s n e i . T a i s n e s
r
M 1
v i r z i e n a v e k t o r s s = ( 1 ; 3 ; 2 ) i r p a r a l l s d o t a j a i
P
M
t a i s n e i , l +d z a r t o i r p e r p e n d i k u l r s m e k l t a j a i p l a k n e i
u n d e r p a r t s n o r m l v e k t o r u . P c f o r m u l a s ( 1 )
i e g ks i m p l a k n e s v i e n d o j u m u :
1 ( x - 2 ) + 3 ( y + 5 ) + 2 ( z - 7 ) = 0
x - 2 + 3 y + 1 5 + 2 z - 1 4 = 0
x + 3 y + 2 z - 1 = 0
D o t s t a i s n e s v i e n d o j u m u s p r r a k s t +s i m p a r a m e t r i s k f o r m :
x - 5 y - 4 z - 6
= t Ò! x = t + 5 , = t Ò! y = 3 t + 4 , = t Ò! z = 2 t + 6
1 3 2
T t a d d o t s t a i s n e s p a r a m e t r i s k i e v i e n d o j u m i i r
x = 5 + t
§#
ª#y = 4 + 3 t
¨#
ª#z = 6 + 2 t
©#
T a g a d n o t e i k s i m t a i s n e s u n p l a k n e s k r u s t p u n k t u , i z t e i k s m e s n o t a i s n e s p a r a m e t r i s k a j i e m
v i e n d o j u m i e m i e v i e t o j o t p l a k n e s v i e n d o j u m :
( 5 + t ) + 3 ( 4 + 3 t ) + 2 ( 6 + 2 t ) - 1 = 0
5 + t + 1 2 + 9 t + 1 2 + 4 t - 1 = 0
1 4 t = - 2 8
t = - 2
A t t i e c +g i p u n k t a P k o o r d i n t a s :
9 . n o d a r b +b a . 1 0 . l p p . A u g s t k m a t e m t i k a .
I . V o l o d k o
R +g a s T e h n i s k u n i v e r s i t t e . I n ~e n i e r m a t e m t i k a s k a t e d r a . U z d e v u m u
r i s i n j u m u p a r a u g i .
§# - 2 = 3
x p = 5
ª#
¨#y = 4 - 6 = - 2
p
ª#z = 6 - 4 = 2
p
©#
T k P i r n o g r i e ~Fa M M 1 v i d u s p u n k t s , n o t e i k s i m M 1 k o o r d i n t a s n o v i e n d o j u m i e m :
x M + x M
§#
1
=
ª#x p
2
§#x M = 2 x p - x M = 4
ª#
1
ª#
y M + y M
ª#
ª#y
1
j e b = 2 y p - y M = 1
¨#y = ¨#
p M 1
2
ª# ª#
ª#z M = 2 z p - z M = - 3
z M + z M
ª#
©# 1
1
=
ª#z p
2
©#
T t a d p u n k t a M s i m e t r i s k a i s p u n k t s a t t i e c +b p r e t d o t o t a i s n i i r M 1 ( 4 , 1 , - 3 )
1 2 . p i e m r s . A p r 7i n t a t t l u m u n o k o o r d i n t u s i s t m a s s k u m p u n k t a l +d z p l a k n e i
2 x - y - 2 z - 6 = 0 .
R i s i n j u m s .
I z m a n t o s i m f o r m u l u a t t l u m a a p r 7i n aa n a i n o p u n k t a l +d z p l a k n e i :
2 Å" 0 - 0 - 2 Å" 0 - 6
6
d = = = 2
2 2 + ( - 1 ) 2 + ( - 2 ) 2 3
1 3 . p i e m r s . A p r 7i n t a t t l u m u s t a r p p a r a l l m p l a k n m 4 x + 5 y + 3 z - 1 2 = 0 u n
4 x + 5 y + 3 z + 8 = 0 .
R i s i n j u m s .
I z v l s i m i e s u z p l a k n e s 4 x + 5 y + 3 z - 1 2 = 0 v i e n u p u n k t u : p a t v a <+g i n o t e i k s i m
x A ; y A u n n o p l a k n e s v i e n d o j u m a i z s k a i t <o s i m t r e ao k o o r d i n t i . P i e m r a m , j a x A = y A =
1 , t a d 4 + 5 + 3 z A - 1 2 = 0 , 3 z A - 3 = 0 , z A = 1 u n t t a d p u n k t s A ( 1 , 1 , 1 ) i r v i e n s n o
p l a k n e s p u n k t i e m .
A p r 7i n s i m a t t l u m u n o p u n k t a A l +d z o t r a i p l a k n e i , t a s a r + b ks a t t l u m s s t a r p
d i v m p a r a l l m p l a k n m
4 Å"1 + 5 Å"1 + 3 Å"1 + 8
2 0 4
d = = = = 2 2
4 2 + 5 2 + 3 2 5 2 2
9 . n o d a r b +b a . 1 1 . l p p . A u g s t k m a t e m t i k a .
I . V o l o d k o
R +g a s T e h n i s k u n i v e r s i t t e . I n ~e n i e r m a t e m t i k a s k a t e d r a . U z d e v u m u
r i s i n j u m u p a r a u g i .
x = 3 t + 1
§#
ª#y
1 4 . p i e m r s . A p r 7i n t l e F7i s t a r p t a i s n i = t u n p l a k n i 2 x + 4 y - 5 z + 1 = 0 .
¨#
ª#z = 2 t - 5
©#
R i s i n j u m s .
L e F7i s t a r p t a i s n i u n p l a k n i v a r a p r 7i n t , i z m a n t o j o t t a i s n e s v i r z i e n a v e k t o r a
r r
s = ( 3 , 1 , 2 ) u n p l a k n e s n o r m l v e k t o r a n = ( 2 , 4 , - 5 ) s k a l r o r e i z i n j u m u :
r r
s Å" n 3 Å" 2 + 1 Å" 4 + 2 Å"( - 5 ) 6 + 4 - 1 0
s i n Õ = = = = 0 ,
r r
s Å" n
9 + 1 + 4 Å" 4 + 1 6 + 2 5 1 4 Å" 4 5
t t a d l e F7i s s t a r p t a i s n i u n p l a k n i i r Õ = a r c s i n 0 = 0 . T a i s n e s p u n k t s A ( 1 , 0 , - 5 ) n e p i e d e r
p l a k n e i , j o n e a p m i e r i n a p l a k n e s v i e n d o j u m u :
2 Å"1 + 4 Å" 0 - 5 Å"( - 5 ) + 1 `" 1 .
T a s n o z +m , k a t a i s n e i r p a r a l l a p l a k n e i ( p r e t j g a d +j u m t a i s n e p i e d e r t u p l a k n e i ) .
9 . n o d a r b +b a . 1 2 . l p p . A u g s t k m a t e m t i k a .
I . V o l o d k o
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