ÿþW y d z i a B: W i L i Z, B u d o w n i c t w o i T r a n s p o r t , s e m . 2
d r J o l a n t a D y m k o w s k a
R a c h u n e k w e k t o r o w y
Z a d . 1 S p r a w d z i , c z y w e k t o r j e s t k o m b i n a c j l i n i o w w e k t o r ó w :
a x i
1 . 1 = [ 3 , 2 , - 5 ] , = [ 2 , 2 , 0 ] , = [ 1 , 0 , 0 ]
a x 1 x 2
1 . 2 = [ 4 , - 1 , 3 ] , = [ - 1 , 2 , 3 ] , = [ 2 , - 1 , - 2 ] , = [ 1 , 1 , 1 ]
a x 1 x 2 x 3
1 . 3 = [ - 1 , - 2 , 1 ] , = [ - 1 , - 1 , 0 ] , = [ 1 , 0 , - 1 ] , = [ 0 , - 1 , 1 ]
a x 1 x 2 x 3
Z a d . 2 Z b a d a j l i n i o w n i e z a l e |n o [ w e k t o r ó w :
2 . 1 = [ - 1 , - 1 , 0 ] , = [ 1 , 0 , - 1 ] , = [ 0 , - 1 , 1 ]
x 1 x 2 x 3
2 . 2 = [ 1 , 2 , 3 ] , = [ 2 , 3 , 1 ] , = [ 4 , 4 , 5 ]
x 1 x 2 x 3
2 . 3 = [ 3 , 2 , 3 ] , = [ 2 , 2 , 0 ] , = [ 1 , 0 , 0 ]
x 1 x 2 x 3
2 . 4 = [ 1 , 1 , 0 ] , = [ 1 , 1 , - 1 ] , = [ 0 , 0 , 1 ]
x 1 x 2 x 3
Z a d . 3 D o b r a s t a B a t a k , a b y w e k t o r y = [ 1 , 2 , 3 ] , = [ 0 , 3 , - 1 ] , = [ 2 , 5 , a ] b y By l i n i o w o z a l e |n e .
x 1 x 2 x 3
Z a d . 4 C z y w e k t o r y = [ 1 , 0 , 1 ] , = [ 1 , 1 , 0 ] , = [ 0 , 1 , 1 ] t w o r z b a z w R 3 , c z y j e s t t o b a z a o r t o g o n a l n a
e 1 e 2 e 3
( o r t o n o r m a l n a ) ? J e [l i t a k , t o z n a l e z w s p ó Br z d n e w e k t o r ó w = [ 1 , 1 , 1 ] , b = [ 3 , 5 , - 3 ] w t e j b a z i e .
a
Z a d . 5 C z y w e k t o r y = [ 1 , - 1 , 0 ] , = [ 1 , 1 , 1 ] , = [ 2 , - 1 , - 1 ] t w o r z b a z w R 3 , c z y j e s t t o b a z a o r t o g o n a l n a
e 1 e 2 e 3
( o r t o n o r m a l n a ) ? J e [l i t a k , t o z n a l e z w s p ó Br z d n e w e k t o r a = [ 3 , 4 , 3 ] w t e j b a z i e .
a
Z a d . 6 O b l i c z y i l o c z y n s k a l a r n y w e k t o r ó w = 2 - q i b = 5 + 2 , j e |e l i p , q s w e r s o r a m i w z a j e m n i e
a p p q
p r o s t o p a d By m i .
Z a d . 7 O b l i c z y i l o c z y n s k a l a r n y æ% b , j e |e l i = p + 2 + , b = 4 - 3 - i p , s w e r s o r a m i w z a j e m n i e
a a q r p q r q , r
p r o s t o p a d By m i .
Z a d . 8 Z n a l e z d Bu g o [ w e k t o r a = 2 - 3 , w i e d z c , |e p i s p r o s t o p a d Be o r a z | = 4 , | | = 2 .
a p q q p | q
À
Z a d . 9 O b l i c z y ( + , j e |e l i | = 1 , | = 5 i ( b ) = .
a b ) 2 a | b | a ,
3
Z a d . 1 0 O b l i c z y k t m i d z y w e k t o r a m i p i q , j e |e l i w i a d o m o , |e w e k t o r y = 2 + q i b = - 4 + 5 s w z a j e m n i e
a p p q
p r o s t o p a d Be o r a z | = | .
p | q |
Z a d . 1 1 O b l i c z y d Bu g o [c i p r z e k t n y c h r ó w n o l e g Bo b o k u z b u d o w a n e g o n a w e k t o r a c h = 5 + 2 i b = p - 3 , j e |e l i
a p q q
"
À
w i a d o m o , |e | = 2 2 , | | = 3 o r a z ( p , ) = .
p | q q
4
Z a d . 1 2 Z n a l e z 3 - 4 , æ% i | - , j e |e l i
a b a b a b |
1 2 . 1 = [ - 2 , 6 , 1 ] , b = [ 3 , - 3 , - 1 ]
a
1 2 . 2 = [ 3 , - 4 , 2 ] , b = [ 1 , 2 , - 5 ]
a
Z a d . 1 3 Z n a l e z c o s i n u s k t a m i d z y w e k t o r a m i
1
1 3 . 1 = [ - 4 , 8 , - 3 ] , b = [ 2 , 1 , 1 ]
a
1 3 . 2 = [ - 2 , - 3 , 0 ] , b = [ - 6 , 0 , 4 ]
a
Z a d . 1 4 Z n a l e z A B C , j e |e l i A ( 2 , 7 , 0 ) , B ( - 1 , - 1 , 4 ) , C ( 3 , 0 , 1 ) .
Z a d . 1 5 S p r a w d z i , c z y t r ó j k t A B C , g d z i e A ( 2 , 7 , 0 ) , B ( - 1 , - 1 , 4 ) , C ( 3 , 0 , 1 ) , j e s t p r o s t o k t n y . O b l i c z y j e g o
p o l e .
"
3
Z a d . 1 6 S p r a w d z i d l a j a k i c h w a r t o [c i p a r a m e t r ó w a i b w e k t o r p = [ , a , b ] j e s t w e r s o r e m p r o s t o p a d By m d o w e k t o r a
3
= [ 1 , 1 , 1 ] .
q
Z a d . 1 7 Z n a l e | w e k t o r w i e d z c , |e j e s t o n p r o s t o p a d By d o w e k t o r ó w b = [ 2 , 3 , - 1 ] , = [ 1 , - 2 , 3 ] o r a z
a c
æ% [ 2 , - 1 , 1 ] = - 6 .
a
Z a d . 1 8 D a n e s w e k t o r y = [ 3 , - 2 , 1 ] , b = [ 1 , 2 , 1 ] i = [ - 1 , 4 , 3 ] . O b l i c z y
a c
( æ% ( 2 × æ% ( - × ( +
b c ) c a ) a b ) a c )
Z a d . 1 9 W e k t o r = ( 2 - 4 + 5 × ( 3 + - z a p i s a j a k o k o m b i n a c j l i n i o w w e k t o r ó w p , q , , j e |e l i
a p q r ) p q r ) r
w i a d o m o , |e w e k t o r y t e t w o r z t r ó j k w e r s o r ó w w z a j e m n i e p r o s t o p a d By c h o o r i e n t a c j i z g o d n e j z o r i e n t a c j
u k Ba d u w s p ó Br z d n y c h .
Z a d . 2 0 O b l i c z y p o l e r ó w n o l e g Bo b o k u z b u d o w a n e g o n a w e k t o r a c h = p - 2 i b = 2 + 4 , j e |e l i | = 2 , | | = 3 i
a q p q p | q
À
( p , q ) = .
3
Z a d . 2 1 W i e d z c , |e p o l e r ó w n o l e g Bo b o k u z b u d o w a n e g o n a w e k t o r a c h p i q j e s t r ó w n e 2 o b l i c z y p o l e r ó w n o l e g Bo b o k u
z b u d o w a n e g o n a w e k t o r a c h = 2 - q i b = 2 + 3 .
a p p q
Z a d . 2 2 D a n e s w e k t o r y = i + 2 i b = 3 - 5 . O b l i c z y × .
a j k j a b
Z a d . 2 3 D a n e s w e k t o r y = [ 3 , - 1 , - 2 ] i b = [ 1 , 2 , - 1 ] . O b l i c z y ( 2 + × .
a a b ) b
Z a d . 2 4 Z n a l e z t a n g e n s k t a m i d z y w e k t o r a m i .
Z a d . 2 5 S p r a w d z i , c z y w e k t o r y = [ 3 , - 2 , 1 ] , b = [ 2 , 1 , 2 ] i = [ 3 , - 1 , - 2 ] s w s p ó Bp Ba s z c z y z n o w e ( l e | w j e d n e j
a c
p Ba s z c z y zn i e ) .
Z a d . 2 6 W y k a z a , |e p u n k t y A ( 1 , 2 , - 1 ) , B ( 0 , 1 , 5 ) , C ( - 1 , 2 , 1 ) i D ( 2 , 1 , 3 ) l e | w j e d n e j p Ba s z c z y zn i e .
Z a d . 2 7 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 = [ 2 , 3 , 4 ] , b = [ 0 , 4 , - 1 ] i = [ 5 , 1 , 3 ] .
a c
Z a d . 2 8 O b l i c z y o b j t o [ c z w o r o [c i a n u o w i e r z c h o Bk a c h A ( 4 , 0 , 0 ) , B ( 0 , 3 , 0 ) , C ( 0 , 0 , 2 ) i D ( 1 , 1 , 0 ) .
Z a d . 2 9 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 , i j e s t r ó w n a 3 . O b l i c z y o b j t o [ c z w o r o [c i a n u
q r
z b u d o w a n e g o n a w e k t o r a c h = p + - , b = 2 - q + i = p + 2 - 3 .
a q r p r c q r
Z a d . 3 0 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 i , j e |e l i w i a d o m o , |e o b j t o [
r
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 + , b = 2 - - i = p + q - 3 j e s t r ó w n a 4 8 .
a r p q r c r
2
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