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ACTIVIDAD 13 - TALLER APLICADO: VECTORES Y PROYECCIONES EN R² Y R³.

MANUEL ALEJANDRO PABON

CORPORACIÓN UNIVERSITARIA IBEROAMERICANA COLOMBIA 2019

SOLUCIÓN a.

FALSO

b.

VERDADERO

c.

FALSO

d.

FALSO

e.

FALSO

2. -3(2v) Solución (2v) = 2 * v = (4, 8,-2,-1,3)

(8, 16,-4,-2,6)… -3(2v) = -3 * (8, 16,-4,-2,6) -3(2v) = (-24,-48, 12, 6,-18)

3. 2(u – 5v + w) Solución 5v= 5(4, 8,-2,-1,3) = (20, 40,-10,-5,15) (u – 5v) = (3, -4, 5, 0,0) - (20, 40,-10,-5,15) = (-17,-24, 15, 5,-15) 2(u -5v + w) = 2 (-17,-24, 15, 5,-15) = (-34, -48, 30, 10,-30)

4. 2 (p + 3 w) – u Solución 3 w = 3(-4, 7, -5) = (-12, 21,-15) (p + 3 w) = p = (0, 0, 1) + (-12, 21,-15) = (-12, 21, -14) (p + 3 w) – u = (-12, 21, -14) – (3, -4, 5, 0, 0) = -15, 25, -19) . 2 (p + 3 w) – u = 2 (-30, 50, -38)

5. 3p – 2 w 3p – 2 w = 3 (0, 0, 1) -2 (-4, 7, -5) = (8, -14, 13)

6. Solución (3, -1, 2, 5,6) * (4, 1, -1, -3, 2) = 12 -1 -2 -15 + 12= 6 Por lo tanto, los vectores no son ortogonales.

7. Solución Distancia: d= v – u

= (6, -6, 0, -1, 4) – (-3, 1, 1, -2, 3) =

9 -7 -1 +1+1

√92+ (-7)2+ (-1)2(1)2+ (1)2 = √81 + 49 + 1 + 1 + 1 = √133 Producto: (-3, 1, 1,-2, 3) * (6,-6, 0, -1, 4) = -18 – 6 + 0 + 2 + 12 = -10 Por lo tanto, los vectores no son ortogonales.

8. Solución U = (2, -1, 3, 4, 5)

V = (0, -1, -2, 3, 4)

CosƟ = U * V U * V

U * V = (2, -1, 3, 4, 5) * (0, -1, -2, 3, 4) = 0 + 1 – 6 + 12 + 20 = 27 Por lo tanto, los vectores no son ortogonales. 2 2 2 2 2 U = √2 + (-1) + 3 +4 +5 = √4 + 1 + 9 + 16 + 25 = √55 V

= √02 + (-1)2 + (-2)2 + 32 +42 = √0 + 1 + 4 + 9 + 16 = √30

CosƟ =

27 √55 * √30

CosƟ = 27 5 √66 Ɵ= 48. 34° v 9. proyvu= u * v 2 V U * V = (2, -1, 3, 4, 5) * (0, -1, -2, 3, 4) = 0 + 1 – 6 + 12 + 20 = 27 V

= √ 02 + (-1)2 + (-2)2 + 32 + 42 = √0 + 1 + 4 + 9 + 16 = √30

proyvu = 27 * (0, -1, -2, 3, 4) 30 0 27 54 81 108 proyvu= ( , - , - , , ) 30

30 30 30 30

proyvu= ( 0, -

9

9 27 18

,- ,

10

,

5 10 5

)

10. Solución

proyvu = u * v

v 2

V U * V = (1, 3, 5) * (5, -1, 1) = 5 - 3 + 5 = 7 2 2 2 V = √ 1 + 3 + 5 = √ 1 + 9 + 25 = √ 35 proyvu= 7/ 35 * ( 5, -1, 1) proyvu= (35/ 35 , - 7/35, 7/35) proyvu = (1, - 1/5, 1/5)