cos d c d c d d c c α 1 1 2 2 1 2 2 2 1 2 2 2 3 2 3 1 3 3 2 1 3 3 10 10 10 71

Cos d c d c d d c c α 1 1 2 2 1 2 2 2 1 2 2 2 3 2 3

This preview shows page 207 - 212 out of 640 pages.

cos d c d c d d c c α = + + + = + 1 1 2 2 1 2 2 2 1 2 2 2 3 2 3 · · · · ( ) · · ( ) , 1 3 3 2 1 3 3 10 10 10 71 33 54 18 2 2 2 2 + + = = = α ° ' " r x y s x t y t : : = + = = − + 2 3 1 3 1 2 3 031 d A s Ax By C A B ( , ) · · ( ) = + + + = + + − 0 0 2 2 2 2 5 0 2 1 6 5 2 = = 4 29 4 29 29 u s x t y t x y x y : = = + = + = 2 3 5 2 3 5 5 2 6 0 s t t t t A s : 0 2 1 3 5 0 2 5 = = + = = r x y s x t y t : 2 1 5 2 3 5 = = = + : 030 d P r Ax By C A B ( , ) · · ( ) = + + + = + + − 0 0 2 2 2 2 0 1 0 8 2 1 2 8 5 8 5 5 = = u x t y t x y x y = = + − + = + = 3 2 2 3 2 2 2 8 0 x t y t = = + 3 2 2 029 d P r Ax By C A B ( , ) · · ( ) = + + + = + + 0 0 2 2 2 2 2 2 3 1 8 2 3 = = 7 13 7 13 13 u r x y x y : 2 = + = 1 3 2 2 3 8 0 r x y : = 1 3 2 2 028 5 SOLUCIONARIO
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208 A la vista de la siguiente figura, realiza las operaciones indicadas. a) AB + BI f) AB + 2 DC b) BC EF g) BF IE c) IH + 2 BC h) 2 HI + 2 CD d) AB + JF + DC i) AE AC e) HG + 2 CJ + 2 CB j) 2 IE + IB BC A G H I J B F C E D j) A G H I J B F C E D e) A G H I J B F C E D i) A G H I J B F C E D d) A G H I J B F C E D h) A G H I J B F C E D c) A G H I J B F C E D g) A G H I J B F C E D b) A G H I J B F C E D f) A G H I J B F C E D a) A G H I J B F C E D 032 Geometría analítica
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209 5 SOLUCIONARIO Haz las siguientes operaciones. 2 u + w e) v 2 u w d) 2 w u + 3 v c) 2 v + u w b) u + v a) a) u + v b) 2 v + u w c) 2 w u + 3 v d) v 2 u w e) 2 u + w u v w 033
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210 Geometría analítica Expresa los vectores AC , AF , EB , AE y FC de la figura como combinación lineal de los vectores AB y BC . AC = AB + BC AF = − AB + BC EB = 2 AB 2 BC AE = − AB + 2 BC FC = 2 AB Comprueba que los vectores u y v de la figura forman una base. Dibuja los vectores con coordenadas en esa base. a) (2, 1) b) (3, 1) c) ( 2, 3) Como los vectores tienen distinta dirección, forman una base. Razona qué pares de vectores forman una base. a) u = (2, 3) y v = (5, 4) b) u = (0, 2) y v = (4, 1) a) Los vectores u y v son linealmente independientes, puesto que no son proporcionales. b) De forma análoga, tenemos que: ( , ) ( , ) 0 2 4 1 0 2 = = = − t t t No son proporcionales. (2, No son p = = = 3 5 4 2 5 3 4 ) ( , ) t t t roporcionales. 036 u v c) u v b) u v a) u v 035 C D B A F E 034
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211 Si, respecto de una base, los vectores u y v son u = (2, 3) y v = (5, 4), halla las coordenadas de los siguientes vectores a) 2 u + v c) u + 2 v e) 3 u v b) 5 u + 2 v d) 2 u + v f) u + v a) 2 u + v = (4, 6) + (5, 4) = (9, 2) b) 5 u + 2 v = (10, 15) + (10, 8) = (20, 7) c) u + 2 v = ( 2, 3) + (10, 8) = (8, 11) d) 2 u + v = (4, 6) + (2,5; 2) = (6,5; 4) e) 3 u v = ( 6, 9) + ( 5, 4) = ( 11, 5) f) u + v = Calcula λ y μ para que los vectores u = (5, 1), v = ( 1, 4) y w = (13, 11) verifiquen que: λ u + μ v = w .
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