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Jose David (Kolman) Vectores

Module by: Jose David Rengifo Fuentes. E-mail the author

Summary: varios ejercicios que en donde se utilizan las formulas de los volumenes como también, ejemplos de planos

greyrgb0.75,0.75,0.75 orangergb1.0,0.5,0.5 brownrgb0.5,0.25,0.0 pinkrgb1.0,0.5,0.5

Exercise 1 Halle la distancia entre el punto P1=(2,3,5)P1=(2,3,5) y la recta

L 1 : x + 2 3 = y - 7 - 4 = z - 2 4 L 1 : x + 2 3 = y - 7 - 4 = z - 2 4
(1)

Tenemos que los siguientes datos:

P 1 = ( 2 , 3 , 5 ) ; P 0 = ( - 2 , 7 , 2 ) ; N = [ 3 , - 4 , 4 ] P 1 = ( 2 , 3 , 5 ) ; P 0 = ( - 2 , 7 , 2 ) ; N = [ 3 , - 4 , 4 ]
(2)

Reemplazamos en la formula de la distancia la cual es:

d = | ( P 1 - P 0 ) x N | N d = | ( 2 , 3 , 5 ) - ( - 2 , 7 , 2 ) x [ 3 , - 4 , 4 ] | 3 , - 4 , 4 d = | [ 4 , - 4 , 3 ] x [ 3 , - 4 , 4 ] | 41 d = | ( P 1 - P 0 ) x N | N d = | ( 2 , 3 , 5 ) - ( - 2 , 7 , 2 ) x [ 3 , - 4 , 4 ] | 3 , - 4 , 4 d = | [ 4 , - 4 , 3 ] x [ 3 , - 4 , 4 ] | 41
(3)

red sage]  blue x=var('x')

red sage]  blue y=var('y')

red sage]  blue z=var('z')

red sage]  blue A=matrix([[x,y,z],[4,-4,3],[3,-4,4]])

red sage]  blue A

x y z 4 - 4 3 3 - 4 4 x y z 4 - 4 3 3 - 4 4

red sage]  blue A.determinant()

- 4 z - 7 y - 4 x - 4 z - 7 y - 4 x

red sage]  blue

Por Ultimo se obtiene que

d = ( - 4 ) 2 + ( - 7 ) 2 + ( - 4 ) 2 41 = 81 41 = 9 41 = 1 . 405 d = ( - 4 ) 2 + ( - 7 ) 2 + ( - 4 ) 2 41 = 81 41 = 9 41 = 1 . 405
(4)

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