MATH 2510 Slides 15

# MATH 2510 Slides 15 - L e c t u r e N o t e 1 5 D r J e f f...

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Unformatted text preview: L e c t u r e N o t e 1 5 D r . J e f f C h a k- F u W O N G D e p a r t m e n t o f M a t h e m a t i c s C h i n e s e U n i v e r s i t y o f H o n g K o n g j w o n g @ m a t h . c u h k . e d u . h k M A T H 2 5 1 L i n e a r A l g e b r a a n d I t s A p p l i c a t i o n s W i n t e r , 2 1 1 P r o d u c e d b y J e f f C h a k- F u W O N G 1 D O T P R O D U C T O N R n W e s h a l l n o w d e fi n e t h e n o t i o n o f t h e l e n g t h o f a v e c t o r i n R n b y g e n e r a l i z i n g t h e c o r r e s p o n d i n g i d e a f o r R 2 . D O T P R O D U C T O N R n 2 D e fi n i t i o n : • T h e l e n g t h ( a l s o c a l l e d m a g n i t u d e o r n o r m ) o f t h e v e c t o r u = ( u 1 , u 2 , . . . , u n ) i n R n i s bardbl u bardbl = radicalBig u 2 1 + u 2 2 + . . . + u 2 n . T h e n o r m o f a v e c t o r c a n a l s o b e w r i t t e n i n t e r m s o f t h e d o t p r o d u c t ( s e e P a g e 2 6 ) bardbl u bardbl = √ u · u . • T h e d i s t a n c e b e t w e e n t h e p o i n t s ( u 1 , u 2 , . . . , u n ) a n d ( v 1 , v 2 , . . . , v n ) i s t h e n d e fi n e d a s t h e l e n g t h o f t h e v e c t o r u − v , w h e r e u = ( u 1 , u 2 , . . . , u n ) a n d v = ( v 1 , v 2 , . . . , v n ) . T h u s t h i s d i s t a n c e i s g i v e n b y bardbl u − v bardbl = radicalBig ( u 1 − v 1 ) 2 + ( u 2 − v 2 ) 2 + . . . + ( u n − v n ) 2 . W e c a n a l s o w r i t e t h i s d i s t a n c e a s f o l l o w s : d ( u , v ) = bardbl u − v bardbl . D O T P R O D U C T O N R n 3 S e e F i g u r e ? ? f o r d i s t a n c e i n R 2 , R 3 , a n d R n . ● ● ● ● ● ● u v u v u v R R R 2 3 n O O F i g u r e 1 : • R 2 : d ( u , v ) = bardbl u − v bardbl = radicalbig ( u 1 − v 1 ) 2 + ( u 2 − v 2 ) 2 . • R 3 : d ( u , v ) = bardbl u − v bardbl = radicalbig ( u 1 − v 1 ) 2 + ( u 2 − v 2 ) 2 + ( u 3 − v 3 ) 2 • R n : d ( u , v ) = bardbl u − v bardbl = radicalbig ( u 1 − v 1 ) 2 + ( u 2 − v 2 ) 2 + · · · + ( u n − v n ) 2 D O T P R O D U C T O N R n 4 E x a m p l e 1 L e t u = ( 2 , 3 , 2 , − 1 ) a n d v = ( 4 , 2 , 1 , 3 ) . T h e n bardbl u bardbl = radicalBig 2 2 + 3 2 + 2 2 + ( − 1 ) 2 = √ 1 8 , bardbl v bardbl = radicalbig 4 2 + 2 2 + 1 2 + 3 2 = √ 3 . T h e d i s t a n c e b e t w e e n t h e p o i n t s ( 2 , 3 , 2 , − 1 ) a n d ( 4 , 2 , 1 , 3 ) i s t h e l e n g t h o f t h e v e c t o r u − v . T h u s , f r o m e q u a t i o n ( 2 ) bardbl u − v bardbl = radicalBig ( 2 − 4 ) 2 + ( 3 − 2 ) 2 + ( 2 − 1 ) 2 + ( − 1 − 3 ) 2 = √ 2 2 . D O T P R O D U C T O N R n 5 I f u = ( u 1 , u 2 , . . . , u n ) a n d v = ( v 1 , v 2 , . . . , v n ) a r e v e c t o r s i n R n , t h e n t h e i r d o t v e c t o r i s d e fi n e d b y u · v = u 1 v 1 + u 2 v 2 + . . . + u n v n ....
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## This note was uploaded on 03/03/2012 for the course MATH 2510 taught by Professor Jeff during the Fall '10 term at CUHK.

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MATH 2510 Slides 15 - L e c t u r e N o t e 1 5 D r J e f f...

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