ÿþA l g e b r a , A i R , A E i I R S B, X I I 2 0 1 3
G e o m e t r i a a n a l i t y c z n a
- ’!
Z a d . 1 . W y z n a c z y w s p ó Br z d n e w e k t o r a A B o r a z o b l i c z y j e g o d Bu g o [.
( a ) A = ( 1 , 2 , 3 ) , B = ( - 1 , - 3 , - 2 ) ( d ) A = ( 7 , - 8 , 9 ) , B = ( - 4 , 3 , - 2 )
( b ) A = ( - 1 , 0 , 1 ) , B = ( 2 , 4 , 2 ) ( e ) A = ( - 1 , 2 , - 3 ) , B = ( 2 , - 2 , - 3 )
( c ) A = ( 2 , - 1 , 7 ) , B = ( - 3 , - 2 , 3 ) ( f ) A = ( 7 , 6 , - 5 ) , B = ( 1 1 , 4 , - 9 )
- ’!
Z a d . 2 . D a n y j e s t p o c z t e k o r a z w p ó Br z d n e w e k t o r a A B . Z n a l e z w s p ó Br z d n e k o Dc a t e g o
w e k t o r a .
- ’!
( a ) A = ( 1 , 2 , 3 ) , A B = [ 3 , - 2 , 1 ]
- ’!
( b ) A = ( 7 , 6 , - 1 ) , A B = [ 4 , 5 , 6 ]
- ’!
( c ) A = ( 2 , 1 , 2 ) , A B = [ - 1 , - 2 , - 1 ]
-
’!
( d ) A = ( 2 À - 1 , À + 2 , 1 - 2 À) , A B = [ 1 - À, - À, 2 À]
-
’!
Z a d . 3 . D a n y j e s t k o n i e c o r a z w p ó Br z d n e w e k t o r a A B . Z n a l e z w s p ó Br z d n e p o c z t k u t e g o
w e k t o r a .
- - ’!
’!
( a ) B = ( 4 , 7 , 5 ) , A B = [ 2 , 5 , 2 ] ( c ) B = ( 2 , - 6 , - 8 ) , A B = [ - 4 , 6 , 1 0 ]
- - ’!
’!
1 2 1 2
( b ) B = ( - 1 , 1 0 , - 2 ) , A B = [ - 4 , 4 , - 4 ] ( d ) B = ( À - 1 , 2 , + 1 ) , A B = [ À , - 2 , - À ]
À
Z a d . 4 . D a n e s w e k t o r y a = [ 2 , 7 , - 1 ] i b = [ 3 , - 2 , 2 ] . O b l i c z y :
1
( a ) 2 a - 3 b , ( c ) a + 1 1 b , ( d ) | a | · b + | b | · a ,
2 2
( b ) a + 4 b , ( e ) | a | 2 · a - | b | 2 · b .
Z a d . 5 . P u n k t y A , B , C , D s w i e r z c h o Bk a m i r ó w n o l e g Bo b o k u . Z n a l e z w s p ó Br z d n e w i e r z -
c h o Bk a D .
( a ) A ( 4 , 5 , 6 ) , B ( 2 , 7 , 1 ) , C ( 3 , 6 , 2 ) ( b ) A ( - 1 , 4 , 2 ) , B ( 1 , 5 , 3 ) , C ( 4 , 1 , - 1 )
-
’!
Z a d . 6 . D a n e s p u n k t y A = ( 2 , 0 , - 3 ) i B = ( 1 6 , - 7 , 1 8 ) . Z n a l e z p u n k t C t a k i , |e 3 A C =
- ’!
-
4 C B .
- -
’! ’!
Z a d . 7 . D a n e s p u n k t y A = ( 1 2 , 4 , - 5 ) i B = ( 1 4 , 2 , 5 ) . Z n a l e z p u n k t C t a k i , |e 5 A B + 2 A C
j e s t w e k t o r e m z e r o w y m .
Z a d . 8 . Z n a l e z w e r s o r y p o n i |s z y c h w e k t o r ó w .
1
A l g e b r a , A i R , A E i I R S B, X I I 2 0 1 3
( a ) [ 3 , 0 , - 4 ] ( b ) [ - 1 , 2 , 2 ] ( c ) [ 4 , 1 , - 1 ] ( d ) [ 4 , 5 , - 2 ]
Z a d . 9 . W y z n a c z y c o s i n u s y k i e r u n k o w e p o n i |s z y c h w e k t o r ó w .
( a ) [ - 2 , 0 , 1 ] ( b ) [ 4 , 2 , - 4 ] ( c ) [ 5 , 8 , - 1 ] ( d ) [ 1 , 1 , - 5 ]
Z a d . 1 0 . D l a p o d a n y c h w e k t o r ó w a , b o b l i c z y i c h i l o c z y n s k a l a r n y i w e k t o r o w y .
" "
( a ) a = [ 2 , - 1 , 1 0 ] , b = [ - 4 , 2 , 1 ] ( c ) a = [ 2 , - 1 , 1 ] , b = [ - 2 2 , 2 , - 3 ]
( b ) a = [ 4 , - 4 , - 1 ] , b = [ 5 , - 6 , - 1 ] ( d ) a = [ À, 2 , - 1 ] , b = [ 1 , - 1 , À]
Z a d . 1 1 . D l a p o d a n y c h w e k t o r ó w a , b , c , o b l i c z y i c h i l o c z y n m i e s z a n y .
( a ) a = [ 1 , 1 , 2 ] , b = [ 1 , 2 , 1 ] , c = [ 2 , 1 , 1 ]
( b ) a = [ 3 , 4 , 5 ] , b = [ - 4 , 5 , 1 ] , c = [ 5 , - 1 , 3 ]
( c ) a = [ - 1 , - 1 , 7 ] , b = [ 3 , 3 , - 4 ] , c = [ 4 , 5 , 1 ]
( d ) a = [ 0 , - 2 , 3 ] , b = [ 5 , 3 , - 1 ] , c = [ - 7 , - 1 , - 4 ]
Z a d . 1 2 . O b l i c z y ( a b c ) , | b × c | , o r a z | 2 a - b + 3 c - 2 j + k | d l a p o d a n y c h w e k t o r ó w .
( a ) a = [ 2 , 4 , 1 ] , b = 2 i - j , c = j - k
( b ) a = [ 2 , 0 , - 1 ] , b = [ - 1 , 2 , 1 ] , c = 3 a + 4 b - 6 j
( c ) a = [ 1 , 0 , - 2 ] , b = - 2 i + j + k , c = ( b - a ) × [ 1 , 0 , 1 ]
Z a d . 1 3 . O b l i c z y ( a b c ) , a × c , | 3 a - b + c | d l a p o d a n y c h w e k t o r ó w .
( a ) a = [ 2 , 3 , - 3 ] , b = [ 2 , 5 , 1 ] , c = 4 i + 2 j - k
1
( b ) a = [ - 1 , 2 , 3 ] , b = [ 2 , 5 , 6 ] , c = a × b - a
3
1
( c ) a = [ 2 , 5 , - 1 ] , b = - i + j + 2 k , c = ( 2 a - b ) × ( a + b ) - [ 1 0 , - 4 , 5 ]
3
Z a d . 1 4 . N i e c h a = [ 2 , 5 , 1 ] , b = [ 7 , - 7 , 1 ] , c = [ 0 , 5 , - 1 ] , d = [ 4 , - 2 , - 1 ] . O b l i c z y :
( a ) a × b + c × d , ( c ) ( b × c ) · d - ( c × d ) · a ,
( b ) c × ( d × a ) + b , ( d ) ( a × c + c × d ) · b - d · c .
Z a d . 1 5 . Z n a l e z k t m i d z y w e k t o r a m i a , b .
( a ) a = [ - 1 , - 1 , 0 ] , b = [ 0 , - 1 , 0 ] ( c ) a = [ 5 , - 1 , - 2 ] , b = 2 a - [ 4 , - 2 , - 1 ]
( b ) a = [ 3 , - 2 , 1 3 ] , b = - 5 i - j + k ( d ) a = [ 4 , - 1 , 2 ] , b = a - [ 2 , 0 , 3 ]
Z a d . 1 6 . D l a j a k i c h w a r t o [c i p a r a m e t r u p p o d a n e w e k t o r y s p r o s t o p a d Be ?
( a ) a = [ 2 p - 1 , 1 , 3 ] , b = [ - 1 , 2 , 5 ] ( d ) a = [ p 2 , - p , 2 ] , b = [ p - 1 , p , p - 2 ]
( b ) a = [ 1 , 1 - 3 p , 2 ] , b = [ p , 4 , 2 p ]
( e ) a = [ p 2 , 2 , 1 - 2 p ] , b = [ p + 1 , 1 - p , 2 p ]
( c ) a = [ p - 1 , 2 , 6 ] , b = [ 2 p , 4 - p , 1 ]
2
A l g e b r a , A i R , A E i I R S B, X I I 2 0 1 3
( f ) a = [ 2 p - 1 , p 2 - 3 p , 1 8 - p ] , b = [ - 9 , p 2 + 3 p , p ]
Z a d . 1 7 . D l a j a k i e j w a r t o [c i p a r a m e t r u p k t m i d z y w e k t o r a m i a = [ 2 , p , - 1 ] i b = [ - 3 , 1 , - 2 ]
À
w y n o s i ?
3
Z a d . 1 8 . Z n a l e z w s p ó Br z d n e w e k t o r a ( w e k t o r ó w ) w , j e [l i w i a d o m o , |e j e s t o n p r o s t o p a d By d o
"
a = [ 3 , - 1 , 5 ] i b = [ 2 , 2 , 5 ] o r a z j e g o d Bu g o [ w y n o s i 3 1 4 .
Z a d . 1 9 . Z n a l e z w s p ó Br z d n e w e k t o r a ( w e k t o r ó w ) , k t ó r y j e s t p r o s t o p a d By d o w e k t o r ó w a =
"
[ - 3 , 2 , 2 ] i b = [ 1 , 0 , - 4 ] o r a z j e g o d Bu g o [ w y n o s i 4 2 .
Z a d . 2 0 . S p r a w d z i c z y p u k t y A , B , C , D l e | w j e d n e j p Ba s z c z y zn i e .
( a ) A = ( 1 , 7 , - 1 ) , B = ( 4 , 2 , - 3 ) , C = ( 2 , 2 , 0 ) , D = ( 1 , 0 , 1 )
( b ) A = ( 0 , 1 , 1 ) , B = ( 2 , 2 , 1 ) , C = ( - 2 , 3 , 3 ) , D = ( - 5 , 1 , 4 )
3
( c ) A = ( 1 1 , 0 , 1 ) , B = ( 1 , 1 , - 1 1 ) , C = ( - 1 , - 1 , 0 ) , D = ( 0 , 1 , 1 )
( d ) A = ( - 7 , 2 , 1 ) , B = ( - 7 , 6 , - 1 ) , C = ( - 5 , - 4 , 5 ) , D = ( 3 , 1 0 , 2 )
Z a d . 2 1 . D a n e s w i e r z c h o Bk i t r ó j k t a A B C . O b l i c z y j e g o p o l e o r a z d Bu g o [ w y s o k o [c i p o p r o -
w a d z o n e j z w i e r z c h o Bk a C .
( a ) A = ( 4 , 5 , 6 ) , B = ( - 1 , 2 , 3 ) , C = ( - 2 , 3 , 3 )
( b ) A = ( - 1 , - 4 , 2 ) , B = ( 5 , 1 , 6 ) , C = ( 8 , 2 , 7 )
( c ) A = ( - 3 , - 5 , 1 ) , B = ( - 1 , 2 , 5 ) , C = ( - 2 , - 1 , 4 )
Z a d . 2 2 . O b l i c z y o b j t o [ c z w o r o [c i a n u A B C D o r a z d Bu g o [ j e g o w y s o k o [c i p o p r o w a d z o n e j z
w i e r z c h o Bk a D .
( a ) A = ( 6 , 5 , 4 ) , B = ( 3 , 2 , - 1 ) , C = ( 3 , 3 , - 2 ) , D = ( 8 , 1 , 1 )
( b ) A = ( 4 , - 2 , 7 ) , B = ( 5 , 2 , 9 ) , C = ( 5 , 0 , 9 ) , D = ( 7 , - 5 , 4 )
( c ) A = ( 1 , 2 , 3 ) , B = ( 3 , 4 , 2 ) , C = ( - 2 , 3 , 2 ) , D = ( 4 , 3 , 7 )
Z a d . 2 3 . Z n a l e z d Bu g o [ d o w o l n e j w y s o k o [c i c z w o r o [c i a n u , z b u d o w a n e g o n a w e k t o r a c h a , b , c .
( a ) a = [ 2 , 4 , 1 ] , b = [ 5 , 2 , 2 ] , c = [ 3 , 3 , 1 ]
( b ) a = [ 2 , - 1 , 1 ] , b = [ 1 , 0 , 2 ] , c = [ 0 , 1 , 1 ]
( c ) a = [ 1 , 0 , 1 ] , b = [ - 1 , 1 , 3 ] , c = [ 2 , - 3 , 0 ]
Z a d . 2 4 . D a n e s w e k t o r y a = [ 3 , 6 , 1 ] , b = 2 i - k . O b l i c z y o b j t o [ r ó w n o l e g Bo [c i a n u z b u d o -
w a n e g o n a w e k t o r a c h p , q , r .
( a ) p = 2 a + k , q = b - 2 a , r = 3 a + b + 2 i - j
( b ) p = a - 4 j + k , q = a - b + i , r = ( 4 - | b | 2 ) · b
"1
( c ) p = | ( a - 5 j - i ) × ( b + j ) | b + 3 j , q = ( 4 j - a ) × b - k , r = b × ( b - a ) + [ 2 , - 3 , 8 ]
1 0
Z a d . 2 5 . D a n e s p u n k t y A = ( 1 , 1 , 6 ) , B = ( - 2 , 1 , 3 ) , C = ( 0 , - 2 , 1 ) . P u n k t D l e |y n a o s i
O X . Z n a l e z w s p ó Br z d n e p u n k t u D , j e [l i w i a d o m o , |e o b j t o [ c z w o r o [c i a n u A B C D
w y n o s i 6 .
3
A l g e b r a , A i R , A E i I R S B, X I I 2 0 1 3
Z a d . 2 6 . D a n e s p u n k t y A = ( 2 , 1 , 2 ) , B = ( 3 , - 2 , 1 ) , C = ( 5 , 0 , 1 ) , D = ( p , 2 p , 1 - 3 p ) .
W i a d o m o , |e o b j t o [ r ó w n o l e g Bo [c i a n u A B C D w y n o s i 1 0 . Z n a l e z w a r t o [ p a r a m e t r u
p .
Z a d . 2 7 . N a p i s a r ó w n a n i e p Ba s z c z y z n y p r z e c h o d z c e j p r z e z p u n k t A i p r o s t o p a d Be j d o w e k t o r a
n .
( a ) A = ( 3 , 2 , 1 ) , n = [ 4 , 5 , 6 ] ( c ) A = ( 5 , 5 , - 2 ) , n = [ - 1 , 2 , 4 ]
( b ) A = ( - 4 , 3 , 6 ) , n = [ 5 , 1 , - 1 ] ( d ) A = ( 3 , - 4 , 0 ) , n = [ 4 , 3 , 1 ]
Z a d . 2 8 . N a p i s a r ó w n a n i e p Ba s z c z y z n y p r z e c h o d z c e j p r z e z p u n k t A i r ó w n o l e g Be j d o p Ba s z -
c z y z n y À.
( a ) A ( 2 , - 3 , 1 ) , À : 2 x - 3 y + z = 0
( b ) A ( 1 , 2 , 4 ) , À : 4 x - y - 2 z + 1 = 0
Z a d . 2 9 . N a p i s a r ó w n a n i e p Ba s z c z y z n y p r z e c h o d z c e j p r z e z p u n k t y A , B i C .
( a ) A = ( 1 , 3 , 1 ) , B = ( 3 , 1 , 1 ) , C = ( - 3 , 3 , 1 )
( b ) A = ( 1 , 0 , - 1 ) , B = ( - 2 , - 6 , 1 ) , C = ( 3 , 1 7 , 2 )
( c ) A = ( - 6 , 1 , 2 ) , B = ( 3 , 2 , - 1 ) , C = ( 4 , 3 , 4 )
( d ) A = ( 0 , 0 , 1 ) , B = ( 1 , 2 , 0 ) , C = ( 0 , - 5 , 0 )
( e ) A = ( 1 , - 2 , 1 ) , B = ( 3 , 0 , 3 ) , C = ( - 1 , - 4 , - 1 )
( f ) A = ( - 2 , - 3 , - 4 ) , B = ( 1 , 3 , 2 ) , C = ( 2 , 5 , 4 )
Z a d . 3 0 . N a p i s a r ó w n a n i e p Ba s z c z y z n y p r z e c h o d z c e j p r z e z p u n k t y A ( 0 , 2 , 1 ) , B ( 2 , 6 , - 3 ) i
r ó w n o l e g Be j d o w e k t o r a v = [ 1 , 1 , - 3 ] .
Z a d . 3 1 . N a p i s a r ó w n a n i e p Ba s z c z y z n y p r z e c h o d z c e j p r z e z p u n k t y A ( - 3 , 1 , 1 ) , B ( - 8 , 2 , 0 ) i
y - 2
x + 1 z - 1
r ó w n o l e g Be j d o p r o s t e j = = .
3 0 1
Z a d . 3 2 . Z n a l e z r ó w n a n i e p Ba s z c z y z n y z a w i e r a j c e j p u n k t y A ( 1 , 0 , 0 ) , B ( 0 , 0 , 1 ) , k t ó r a z p Ba s z -
À
c z y z n À : x + y - z + 1 0 = 0 t w o r z y k t .
3
Z a d . 3 3 . D l a j a k i e j w a r t o [c i p a r a m e t r u p p Ba s z c z y z n y À1 : p x + 2 y + ( p - 1 ) z + 3 = 0 i
À2 : x + p y - z + 4 = 0 s
( a ) p r o s t o p a d Be , ( b ) r ó w n o l e g Be ?
Z a d . 3 4 . N a p i s a r ó w n a n i e p r o s t e j p r z e c h o d z c e j p r z e z p u n k t A i r ó w n o l e g Be j d o w e k t o r a v .
( a ) A = ( 2 , 0 , 5 ) , v = [ 4 , - 1 , 0 ] ( c ) A = ( 5 , 0 , 4 ) , v = [ - 1 , 2 , 3 ]
( b ) A = ( - 4 , 3 , 3 ) , v = [ 1 , 1 , - 2 ] ( d ) A = ( - 1 , 1 , 0 ) , v = [ 5 , 6 , - 4 ]
Z a d . 3 5 . N a p i s a r ó w n a n i e p r o s t e j p r z e c h o d z c e j p r z e z p u n k t y A i B .
4
A l g e b r a , A i R , A E i I R S B, X I I 2 0 1 3
( a ) A = ( 3 , 3 , 1 ) , B = ( 2 , 1 , 2 ) ( c ) A = ( - 5 , 4 , 4 ) , B = ( - 4 , 3 , 4 )
( b ) A = ( 4 , 4 , 5 ) , B = ( - 1 , 0 , 2 ) ( d ) A = ( 6 , 1 , - 5 ) , B = ( - 5 , 3 , 2 )
y - 3
x + 2 z - 6
Z a d . 3 6 . W j a k i c h p u n k t a c h p r o s t a l : = = p r z e b i j a p Ba s z c z y z n y u k Ba d u w s p ó B-
3 2 1
r z d n y c h ?
U w a g a : W e w s z y s t k i c h p o n i |s z y c h z a d a n i a c h , j e [l i p r o s t a m a p o s t a p a r a m e t r y c z n
ñø
ôø ±t + x 0
òøx =
y = ²t + y 0
ôø
óøz = ³t + z 0 ,
d l a u p r o s z c z e n i a n o t a c j i , n i e b d z i e m y z a k a |d y m r a z e m p i s a , |e t " R .
Z a d . 3 7 . N a p i s a r ó w n a n i e p Ba s z c z y z n y z a w i e r a j c e j p u n k t A o r a z p r o s t l :
ñø
1 - y
x - 2 z - 6
ôø - 1 ( c ) A = ( 4 , - 1 1 , 5 ) , l : = = ,
òøx = 5 t
3 4 7
x + 1 - y + 4
z + 3
( a ) A = ( 4 , - 3 , 3 ) , l : - t + 6
y =
( d ) A = ( - 4 , 0 , 5 ) , l : = = ,
ôø 3 2 4
óøz = 2 t + 3 ,
x + y - z = 0
( e ) A = ( 5 , 0 , 2 ) , l :
ñø
2 x + 2 y + 3 = 0 ,
ôø - 2 t + 1
òøx =
x + y - z + 3 = 0
( b ) A = ( 4 , 3 , - 3 ) , l : y = 5 t - 6
ôø
óøz = t - 3 , ( f ) A = ( 5 , 2 , 8 ) , l : 4 x - y - z + 9 = 0 .
Z a d . 3 8 . Z n a l e z p u n k t w s p ó l n y ( e w . p u n k t y w s p ó l n e l u b s p r a w d z i , |e i c h n i e m a ) p Ba s z c z y z n y
À i p r o s t e j l .
x - y = 0
( a ) À : 2 x + z - 1 = 0 , l :
x + y + z - 1 = 0
2 x + 5 y + 3 z - 2 9 = 0
( b ) À : x + 2 y + z - 5 = 0 , l :
x + y + 3 z + 9 = 0
ñø
ôø - t + 3
òøx =
( c ) À : 4 x - y - z - 2 = 0 , l : y = 2 t - 2
ôø
óøz = - 3 t + 1
ñø
ôø
òøx = 8
( d ) À : x - 6 y - z + 3 = 0 , l : y = 7 t - 6
ôø
óøz = 8 t + 7
y - 4
x - 2 z - 5
( e ) À : 2 x - 4 y + 3 z + 5 = 0 , l : = =
4 3 2
y - 6
x + 4 z + 3
( f ) À : 4 x - 3 y + 2 = 0 , l : = =
3 4 - 1
5
A l g e b r a , A i R , A E i I R S B, X I I 2 0 1 3
Z a d . 3 9 . D l a d o w o l n e g o p o d p u n k t u z z a d a n i a p o p r z e d n i e g o , n a p i s a r ó w n a n i e r z u t u p r o s t e j l
n a p Ba s z c z y z n À.
Z a d . 4 0 . Z n a l e z r z u t p u n k t u A n a p r o s t l .
ñø
ôø
x - y = 0
òøx = t + 3
( c ) A = ( 2 , 3 , - 1 0 ) , l :
( a ) A = ( 2 , 7 , 1 0 ) , l : - t + 7
y =
x + z + 1 = 0
ôø
óøz = t + 4
x + y + z = 0
( d ) A = ( 4 , 5 , - 2 ) , l :
ñø
x - 2 y + z = 0
ôø
òøx = t + 2
y + 1
x - 2 z - 5
( e ) A = ( 5 , - 1 , 1 ) , l : = =
2 1 - 1
( b ) A = ( 3 , 4 , 6 ) , l : y = t - 1
ôø
óøz = - t + 3 y - 1
x + 4 z - 3
( f ) A = ( 2 , 5 , - 1 ) , l : = =
1 2 - 1
Z a d . 4 1 . Z n a l e z r z u t p u n k t u A n a p Ba s z c z y z n À.
( a ) A = ( 7 , 7 , - 2 ) , À : x - y + z = 0
( b ) A = ( 4 , - 5 , 3 ) , À : 2 x + 4 y - z - 7 = 0
( c ) A = ( 2 , - 2 , - 1 ) , À : 3 x - y + 3 z - 5 = 0
( d ) A = ( 7 , - 1 , 3 ) , À : 3 x + 4 y + z - 2 = 0
Z a d . 4 2 . Z n a l e z p u n k t s y m e t r y c z n y d o p u n k t u A ( 2 , 1 , 7 ) w z g l d e m
( a ) p u n k t u B ( 3 , 5 , - 1 ) ,
y + 1
x - 3 z - 1
( b ) p r o s t e j l : = = ,
2 1 1
( c ) p Ba s z c z y z n y À : x + y - 3 z - 4 = 0 .
x + y + 2 z = 0
Z a d . 4 3 . * N a p i s a r ó w n a n i e p r o s t e j ( l u b p r o s t y c h ) r ó w n o l e g Be j d o p r o s t e j l 1 :
y - 3 z + 1 = 0
ñø
ôø
òøx = t + 4
i p r z e c i n a j c p r o s t l 2 : y = 3 t + 7 .
ôø
óøz = - 2 t - 7
Z a d . 4 4 . M a j c d a n j e d n z p o s t a c i r ó w n a n i a p r o s t e j , z n a l e z d w i e p o z o s t a Be .
ñø
ôø - 7 t + 5
x - y + 2 z = 0
òøx =
( b ) l :
( a ) l : y = 2 t - 7
5 x + 2 y - 3 z + 3 0 = 0
ôø
óøz = - 5 t + ,
y + 2
x - 1 z - 3
( c ) l : = =
2 3 4
Z a d . 4 5 . Z b a d a w z a j e m n e p o Bo |e n i e p a r p r o s t y c h .
ñø ñø
ôø - 8 ôø
òøx = 3 t òøx = s + 3
( a ) l 1 : y = - t + 3 , l 2 : y = - 2 s - 4
ôø ôø
óøz = 5 t - 1 6 óøz = - s - 3
6
A l g e b r a , A i R , A E i I R S B, X I I 2 0 1 3
ñø
ôø
òøx = 5 t
y
x z - 1
( b ) l 1 : y = - t + 2 , l 2 : = =
1 4 - 4
ôø
óøz = - 2 t + 3
2 x - y - 1 1 = 0 x + y + 3 z + 2 = 0
( c ) l 1 ; l 2 :
x - y - z - 8 = 0 x + z - 3 = 0
Z a d . 4 6 . O b l i c z y o d l e g Bo [ p u n k t u A o d p Ba s z c z y z n y À.
( a ) A = ( 1 4 , 0 , - 1 ) , À : 2 x - 7 y + z = 0 ( c ) A = ( 2 , 1 , - 3 ) , À : 5 x - 7 y + 2 z + 3 = 0
( b ) A = ( 3 , 5 , - 8 ) , À : 2 x - 4 y + z + 1 = 0
Z a d . 4 7 . O b l i c z y o d l e g Bo [ p u n k t u A o d p r o s t e j l .
ñø
ôø
x - y + z - 7 = 0
òøx = 2 t + 1
( b ) A = ( 3 , 1 , 2 ) , l :
( a ) A = ( 1 , 1 , 0 ) , l : - t + 2
y =
2 x - y - z - 5 = 0
ôø
óøz = 2 t - 1
y - 4
x z + 6
( c ) A = ( - 3 , 3 , 1 ) , l : = =
2 1 3
Z a d . 4 8 . N a p i s a r ó w n a n i e p r o s t e j p r z e c h o d z c e j p r z e z p u n k t A = ( 4 , 0 , - 2 ) i r ó w n o l e g Be j d o
x + y + z = 0
p r o s t e j .
x - 4 y + 2 z + 1 = 0 ,
Z a d . 4 9 . N a p i s a r ó w n a n i e p r o s t e j p r z e c h o d z c e j p r z e z p u n k t A = ( 4 , 2 , - 1 ) i :
( a ) p r o s t o p a d Be j d o p Ba s z c z y z n y 4 x + y - z = 0 ,
( b ) p r o s t o p a d Be j d o p Ba s z c z y z n y x - y - 2 z + 1 = 0 .
Z a d . 5 0 . N a p i s a r ó w n a n i e p Ba s z c z y z n y , z a w i e r a j c e j p u n k t A i r ó w n o l e g Be j d o w e k t o r ó w a i b .
( a ) A = ( 2 , 2 , 1 ) , a = [ 4 , 0 , 1 ] , b = [ 2 , 1 , 0 ]
( b ) A = ( 0 , 0 , 3 ) , a = [ 3 , 6 , - 1 ] , b = [ 2 , 0 , 0 ]
Z a d . 5 1 . N a p i s a r ó w n a n i e p Ba s z c z y z n y z a w i e r a j c e j p r o s t e l 1 i l 2 .
y - 2 y + 3
x + 1 z - 2 x - 2 z + 2
( a ) l 1 : = = , l 2 : = =
1 2 1 3 1 - 1
ñø
ôø
òøx = t + 1
y - 2
x + 3 z
( b ) l 1 : = = , l 2 : - t + 2
y =
3 - 2 2
ôø
óøz = 3 t
ñø
ôø - 6 t
òøx =
x + y - z = 0
( c ) l 1 : - 2 t + 4 l 2 :
y =
ôø
x - 2 y - 2 z = 0
óøz = 3 t ,
Z a d . 5 2 . Z n a l e z o d l e g Bo [ m i d z y p r o s t y m i .
y - 1 y - 1
x z - 2 x - 1 z - 2
( a ) l 1 : = = , l 2 : = = ;
4 1 2 4 1 2
7
A l g e b r a , A i R , A E i I R S B, X I I 2 0 1 3
x - y - 5 z - 4 = 0
y - 5
x - 4 z + 3
( b ) l 1 : l 2 : = = ;
3 - 2 1
x + y - z - 6 = 0 ,
Z a d . 5 3 . Z n a l e z o d l e g Bo [ m i d z y p a r a m i p Ba s z c z y z n .
( a ) À1 : x - 2 y + 2 z + 1 = 0 , À2 : - x + 2 y - 2 z + 4 = 0
( b ) À1 : 2 x - 3 y + z - 1 = 0 , À2 : 4 x - 6 y + 2 z + 7 = 0
( c ) À1 : - 3 x + 1 5 y - 6 z - 1 8 = 0 , À2 : 4 x - 2 0 y + 8 z + 2 4 = 0
Z a d . 5 4 . P u n k t A l e |y n a o s i O Y , n a t o m i a s t B i C m a j w s p ó Br z d n e B ( 2 , 4 , 3 ) i C ( 3 , 3 , 2 ) .
3 9
W i a d o m o , |e p o l e t r ó j k t a A B C w y n o s i . Z n a l e z w s p ó Br z d n e p u n k t u A .
2
Z a d . 5 5 . D l a j a k i e j w a r t o [c i p a r a m e t r u p p Ba s z c z y z n y :
À1 : x + p y - p = 0
À2 : x + y - p z + p 2 + 2 p - 1 = 0
À3 : 2 x - p z + p = 0
p r z e c i n a j s i w j e d n y m p u n k c i e ? J e [l i t a k a w a r t o [ p a r a m e t r u p i s t n i e j e , p o d a w s p ó B-
r z d n e t e g o p u n k t u .
x - p y + 1
z - 4
Z a d . 5 6 . D l a j a k i e j w a r t o [c i p a r a m e t r u p p r o s t a = = j e s t
p p + 2 3
( a ) r ó w n o l e g Ba d o p Ba s z c z y z n y À1 : 2 x - y + 3 z - 1 = 0 ,
( b ) p r o s t o p a d Ba d o p Ba s z c z y z n y À2 : 3 x + y - 3 z + 5 = 0 ,
( c ) z a w a r t a w p Ba s z c z y zn i e À3 : y - z + 5 = 0 ?
Z a d . 5 7 . N a p r o s t e j l z n a l e z p u n k t r ó w n o o d l e g By o d A i B .
y - 2
x - 1 z + 1
( a ) A = ( 3 , 2 , 0 ) , B = ( 1 , 1 , 1 ) , l : = =
2 1 2
x - y + 2 = 0
( b ) A = ( 2 , 3 , 0 ) , B = ( - 1 , 0 , 1 ) , l :
x + 2 y + z = 0
Z a d . 5 8 . J a k i e j e s t w z a j e m n e p o Bo |e n i e p r o s t e j l i p Ba s z c z y z n y À?
ñø
ôø
òøx = t + 1
( a ) À : x + y + 4 = 0 , l : - t
y =
ôø
óøz = 3 t + 1
y + 4
x - 5 z - 6
( b ) À : x + 2 z - 3 = 0 , l : = =
3 0 2
x - y + 2 = 0
( c ) À : 3 x + y - z + 1 = 0 , l :
2 y + z - 3 = 0
Z a d a n i a d o d a t k o w e
Z a d . 1 . D a n e s p u n k t y A ( 0 , 0 , 0 ) , B ( - 2 , - 1 , 2 ) , C ( 0 , 3 , 4 ) . W y b r a d o w o l n y z n i c h i w y z n a -
c z y r ó w n a n i e p r o s t e j z a w i e r a j c e j w y s o k o [ p o p r o w a d z o n z t e g o w i e r z c h o Bk a o r a z
r ó w n a n i e p r o s t e j z a w i e r a j c e j b o k p r z e c i w l e g By t e m u w i e r z c h o Bk o w i .
8
A l g e b r a , A i R , A E i I R S B, X I I 2 0 1 3
Z a d . 2 . D a n e s p u n k t y A ( 0 , 0 , 0 ) , B ( 1 , 2 , 1 ) , C ( 2 , 2 , 3 ) , D ( 4 , - 1 , 2 ) . W y b r a d o w o l n y z n i c h
i w y z n a c z y r ó w n a n i e p r o s t e j z a w i e r a j c e j w y s o k o [ o p u s z c z o n z t e g o p u n k t u n a
p r z e c i w l e g B [c i a n c z w o r o [c i a n u A B C D .
y - 2
x + 1 z + 3
Z a d . 3 . O p Ba s z c z y zn i e À w i a d o m o , |e z a w i e r a p r o s t l : = = i |e j e j o d l e g Bo [
2 1 1
o d p u n k t u A ( 4 , 3 , 0 ) w y n o s i 1 . Z n a l e z r ó w n a n i e p Ba s z c z y z n y À.
Z a d . 4 . O b l i c z y p o l e t r ó j k t a A B C , j e [l i w i a d o m o , |e A i B s p u n k t a m i p r z e c i c i a p r o s t e j
y - 2
x + 3 z
l : = = z p Ba s z c z y z n a m i O X Y i O Y Z , n a t o m i a s t C j e s t r z u t e m p u n k t u
3 1 - 1
( 6 , 0 , 2 ) n a p Ba s z c z y z n 2 x - 2 y + 3 z - 1 = 0 .
9
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