# βιβλιο μαθηματικών β κατεύθυνσης

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διανύσματα, ευθεία, κωνικές τομές, μαθηματική επαγωγή, επαναλήπτικά θέματα

### Text of βιβλιο μαθηματικών β κατεύθυνσης

11.1 1.2 - 1.3 1.4 1.5 8 1 - 91.1 . . , . - .- .- .., u, v ., . 0. 0 = . .- - 1 = - 0 >, - 0 0 =- 0 0 = = , . () () 10 () . 0 = ( ) ( ) ( ) / / ' . . , . , / / () ., || () ., || , , . () 1 - 11 : `) = - = - == == - = : `) = = = , . = , = . ( ), ( ), .- 0 s s ( ) , = - . 12 -( ) , 0 || = ( ) , || = - , ( ),2= - , , 0 s s . , , . 1 - 13 = : = / = ( ) 1 = / = ( ) 2 ( ) 1 ( ) 2 := - = , . ,., .- , , = ) : = ( ) 1,= = . . 1 === , :) = ) = 3 . = . 2 14 : = ( ) 2) ( ) 1 ( ) 2 = = ) ) = ) = ( ) = .// = . = . . , . 4 1 - 151) ,. = = , :) = ) 2) ,,,. = = , . 3) 4) . () ().) || ) || ) = ) = 5) . :) ( ),. ) ( ),. ) ( ),. ) ( ),. 16 1- 171.2 ,. = = . + . + = ) .) .) . + + + + = + + = ) + 1. + = + ( )2.( ) ( ) + + = + + ( )3. 0 + = ( 0 )4.( ) 0 + = ( , ) + 18 - . , . ( ) = + x += x = x x += = - = , = = + = . . . . = + = + = = - : + = - : = - - 1- 19, + . s + s + - + = + || - + = || + + + 20 - ) :x OX = , = , = , = , = , = :x OX O = = + + + + = - - - = + + ) ) :x OX = , = , = , = :x OX O = = + + = - - = + . , = = : ++x=+ x=- + + = x : 1)xx) x x. = = , 2 x 1- 211 0 + + = + + + + = + 0 0 0+ = + + = = 0 0 = 1- + = - = 2 ( ) : = ( ) , , , , , . :++ ++ = 3 22 - 2 . : + + = + + + + = + + 0 0 = 1 0 0 0+ + + + + = + + + + + = + + + = + + 0 0 = = 2 0+ + + + + = + + + + 0 0 0 = = = 0 = , , , , , , = +++++ 0 41- 23 // = = . . . : + = + = = 22 3 3== = = == *+e) 0 3 ++= + = + = + = = + (1)( ) 1 +2 3 + = + = + = + || ) 0 ++= += + = + + = = (2)( ) 2 2-3 = = = || ,,, . +=+ , , . 5 ,, + + = 0 = = 2 3 ) || ) || 6 24 - 1) x :2) . :) .......... + = ) .......... = ) .......... + = ) .......... + = ) .......... + = ) .......... = 3) , , , , . : + = 4) ,,,, + = + . .5) + = . , .6) . : + = xxx 1- 257) . = + + . .8) . ( ) f 2 5 3 = + , . 26 - 1- 271.3 = 0 . : ,> 0 - , < 0 . = = 0 = 0 = 0 1-( )+ = + - ( ) + = + ,e- ( )( )= :- = = = 0 0 0-( ) ( ) ( ) = = -( ) = ,e- ( ) = - = = 0 = - = = 0 = 22 28 - , ,, u = + u e , . . = + u 5 3 -, = + v 2 3 , , ., 0 = / / = , e= = 0 / / . / / = 0 . =, = - || , = (:= )- || , = (:= )- = 0 , = 0 (:= 0 = : 2+ = = = = + 2 + = 2 1- 29 ) 2 3 2 5 3 = 2 3 2 5 3 0 + = ( ) ( )2 3 5 0 + = 2 3 5 0 + = 5 5 0 = 0 0 = ) 3 2 3 2 + + = + + ( ) ( )3 2 3 2 + + = + + 3 2 2 3 2 2 + + = + + 3 3 2 3 3 2 + = + 0 0 = 111 5 6 6 5 5 6 = + + = + 6 6 5 5 = ( ) ( )6 5 = 6 5 6 5 6 5 = = = ) 2 3 2 5 3 = ) 3 2 3 2 + + = + + 15 6 = 11 5 6 = + 2 -,- . 30 - 2 . :( ) ( )5 6 5 6 = = 5 5 6 6 = 5 6 6 5 + = + 11 5 6 = + 1 B = ( )B = B + = B 0 = 2B B = = B B = + = , , . u 2A+3B 5 = . 4B = . 3 - :) B 0 = ) = 1- 311 u 2A+3B 5 = = ( ) ( )2A+3 B 5 = = 2A+3B 3 5 5 = = 3B 5 = u2u 2A+3B 5 = = 2A+3B 3 2 = = ( ) ( )2 A +3 B = = 2A 3B = + u , , , , A+3 2 3 = + + ) , , ) , , . 5u u - - . :) -----) --- . - . - - -. 3B 5 2 3 = + 32 - ) / / :A 3 2 3 + = + + ( ) ( ) 3 2 3 = + + 3 2 2 3 3 = + + 3 = 3 3 + + 0 3 = + 3 = / / ) 3 = , 2 0 2 0 + + = + + = 2 2 0 + + = 2 2 = + 1 2 = + , , . : 2 0 + + = 6 ,,-- B, , . - - B/ / .B = - B/ / .B = - / / . = - . 12 --- .1- 33 = = = / / 1 6= 1 6= (1) ( ) ( )1 1 1 7 7 7= = + = + 1 1 7 7= + (2)( )( ) 121 1 1 1 1 7 7 6 42 7= = + = + ( ) 1=1 1 6 6= = = 1 1 42 642 742 6 716 6 6= + = + = = = + = + / / ( || ) , 1 6= 1 7= , , 7 -B = = - - ,. --- . 34 - () += += + + + = + + + + +`= +) (1) = (2) = (3) = (4) (1) (2), (3), (4) : + + = + + + + + + + = + + 0 + + = , , 2 2 = = 2 = + = + 2 = + 2+ = -,., . 8 . = = . . 9 - . 1- 35 : = + + (1) : = + + (2) (1) (2) :2 = + + + + + 2 = = = + + + + + 2 = + + + + 2 = + + + + 2 = + 2+ = x, y - , .( )() x 6 3y x 3y 6 2x 6y 12 2 26y 7x 11 66 7x 6y 66 11 7x 6y 66 11+ += + = = + = + = + + = + 9x 9 54 x 6 = = (2) ( ) ( ) 231 6 6 3y 2 12 3y 6 y2+ = + = = (3) (2) (3) 3x y2= x y || . , - ( )1 2= + 10 , x, , y x 6 3y + = + 6y 7x 11 66 + = x, y 11 (1) 36 - ) . = , = , r = || = = = = = ( ) ( ) = = r r = ( ) r r r r + + = + + = + =+ ) . = , = , r = || = = = BM= = ( ) ( ) = = r r = ( ) r r r r = = = , r, MA MB = , ) r +=+ ) r = 12 rr1- 37 0 = 0 + = = / /= 0 = 0 0 0 0=+ = == 0 = = ( ) 1/ / 2 2 2 2 2 = = + = + (1)( ) / / + + = = (2) (1) (2) : 1 2 2 2 2 = + = + 2 2 0 = ( ) ( ) 2 1 2 0 + = , : 2 2 0 211 2 0 2 1 2= = = = = = 2 = (1)( )1 2 2 2 2 2 / /22= + = + = , , ( )/ / 2 ( ) / / + ( ) / / 2 14 , - 0 + = , e. 0 = = 13 38 - , , / / = (1) : 4 1 23 3| | | |= = + + = ||\ . \ . 4 1 23 3| |+ = |\ . 4 3 3 3| | | |= + ||\ . \ . 2 2 23 3| | | |= = + + = ||\ . \ . 2 2 23 3| |+ = |\ . 2 4 3 3| | | |= + ||\ . \ . (1) 4 3 2 4 3 3 3 3 | | | | | | | | + = + |||| \ . \ . \ . \ . 4 3 2 4 3 3 3 3| | | | | | | | + = + ||||\ . \ . \ . \ . 4 3 2 4 03 3 3 3| | | | | | | | + = ||||\ . \ . \ . \ . 4 2 3 4 03 3 3 3| | | | + + + = ||\ . \ . () 4 2 012 6 03 312 6 9 12 0 9 12 03 4 03 3+ + = + = + = + = + =13 6 0 6 3 2+ = = = (1) 9 6 0 2 18 12 0 3 30 102 + = + = = = , 0 = . , , 23= + , 4 1 3| |= + |\ . , 2 2 3| |= + |\ . - , , -. 15(1) 13 37.1- 39 1) ) 2 3 2 3 + = + ) 3 5 8 16 3 5 8 + + = + + + ) 7 2 5 2 7 + = +) 2 3 5 3 2 + = 2) = + = + = 3) 3 2 = 2 = 4) 3 2 = 5 3 2 = + 5) = 3 = 2 3 = + 6) ( ) f K 2AB 3K = + () f 2AB 3 = + = ( ) f 2AB 3 = + = . 3 0 + = .7) ) = ) 2 = ) 3 + = ) + = 40 - 8) ,,, + = + . . 9) , , , , + + = + + . 10) 2 3 2 3 + + = + + . 11) . u 4 2 3 = + . 12) ,,, , , . ( ) ( ) u = + . 13) 2 3 2 + + = + + || 14) , , 2 3 2 3 = 15) , , 3 7 5 2 + = 16) , , , 3 4 0 + =) , , ) , , 1- 4117) , , , :3OA 4OB 7O + = ) ) , , ) 18) , , . 2A 0 + + = 19) . 2 3 2 0 + + = 20) . :)( )1 22= ) 4 = + + + 21) ,,. :22) ' '. ' ' 2 ' ' ' + = . 42 - 23) ,,- = = = . : 5 + + + = 24) - . :) 2 + + + = ) 2 + + + = 25) . 2 = . :) 2 = + ) = 26) ,,,, = , = , 2 = + 2 = . , , ,, - .27) 4v 2 3= + 3 3u 2 4= + . 28) u, v, w , , , u v 2w = + , B u 2v w = + u 5v 2w = , , .1- 4329) , , = , = , 3 2 = , -.) , .) , , .30) -2 5= . = = ( )2 7= 5 27+= .31) = , = 2 = .- , , , , , .32) = , = 2 = . , , :) , , , ) 2 = 33) ,1 4= = . = , = :) , ) 44 - 34) , , 3 = , 2 = 5 3 = , , . 35) - 66-. 2 = . - :) , .) .36) = = . ,, ,, ) , ) , . 37) . 3 = , 3 = , = = ) - , , ) , , -.38) . = + = + ,- / / . 1- 4539) , , , :( ) 4 2 4 3 + = , , - .40) : = + 1 + = , , . 41) ,, ( ) ( ) ( )2 2 2 2 2 0

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