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# Introductory Linear Algebra por Bernard Kolman 8/E

Module by: Julian Daza. E-mail the author

Summary: Algunos ejercicios resueltos sobre algebra lineal de Beernard Kolman

kol2005pgn245ejc21.tm por Julian Camilo Daza

21. Find de cosine of the angle betwen each pair of vectors u and v .

(a) u=(2,3,1)v=(3,-2,0)u=(2,3,1)v=(3,-2,0)

cos θ = | u v u v | cos θ = | u v u v |
(1)

sage: u=vector([2,3,1])

sage: v=vector([3,-2,0])

sage: u.dot_product(v)

0

sage: u.dot_product(u)

14

sage: v.dot_product(v)

13

cos θ = | 0 u v | cos θ = | 0 u v |
(2)

El angulo es 0.

(b) u=(1,2,-1,3)v=(0,0,-1,-2)u=(1,2,-1,3)v=(0,0,-1,-2)

sage: u=vector([1,2,-1,3])

sage: v=vector([0,0,-1,-2])

sage: u.dot_product(v)

-5

sage: u.dot_product(u)

15

sage: v.dot_product(v)

5

cosθ=|-553|cosθ=|-553|=|-13||-13|

θ = cos - 1 | - 1 3 | = 125 , 26 θ = cos - 1 | - 1 3 | = 125 , 26

(c) u=(2,0,1)v=(2,2,0)u=(2,0,1)v=(2,2,0)

sage: u=vector([2,0,1])

sage: v=vector([2,2,-1])

sage: u.dot_product(v)

3

sage: u.dot_product(u)

5

sage: v.dot_product(v)

9

cosθ=|335|cosθ=|335|=|15||15|

θ = cos - 1 | 1 5 | = 63 . 43 θ = cos - 1 | 1 5 | = 63 . 43

(d) u=(0,4,2,3)v=(0,-1,2,0)u=(0,4,2,3)v=(0,-1,2,0)

sage: u=vector([0,4,2,3])

sage: v=vector([0,-1,2,0])

sage: u.dot_product(v)

0

sage: u.dot_product(u)

29

sage: v.dot_product(v)

5

cos θ = | 0 u v | cos θ = | 0 u v |
(3)

El angulo es 0

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