Tarea de repaso

Sea
m= (2,-1, 3)
v= (0,1,7)
w=(1,4,5)

1) Realizar las siguientes operaciones

a) (m x v) x (v x w)

i j k

(mxv) 2 -1 3 = (( -1) (7)-(3) (1) ) i – ((2) (7)- (3)(0))j +((2)(1)- (-1)(0))k

0 1 7

= (-7-3) i - (14)j + (2)k

= -10i - 14j + 2k

i j k

(vxw) 0 1 7 = ((1)(5) –(7)(4)) I – ((0)(5)-(7)(1))j + ((4)(0)-(1)(1))k

1 4 5

= (5-28) i – ( -7)j +(-1)k

= -23i +7j -1k

i j k

(m x v) x (v x w) -10 -14 2 =((-14)(-1)-(2)(7))i-((-10)(-1)-(2)(-23))j+((-10)(7)-

-23 7 -1 (-14)(-23))k

= (14-14)I –( 10+46)j +(-70-322)k

= I – 56j – 252k

b) m x (v-2w)

2w = 2(1,4,5)

=(2, 8,10,)

v-2w = (0-2)+(1-8)+(7-10)

= -2,-7,-3

i j k

m x (v-2w) 2 -1 3 = ((-1)(-3)-(3)(7))I –((2)(3)-(3)(-2))j +((2)(-7)-(-1)(-2))k

-2 -7 -3

= (3-21)i –(6+6)j +(-14-2)k

= -18i- 12j -16k

c) (mxv) -2w

i j k

(mxv) 2 -1 3 = (( -1) (7)-(3) (1) ) i – ((2) (7)- (3)(0))j +((2)(1)- (-1)(0))k

0 1 7

= (-7-3) i - (14)j + (2)k

= -10i - 14j + 2k

2w = 2(1,4,5)

=(2, 8,10,)

(mxv) -2w = (-10-2) + ( -12-8) +( 2-10)

= (-12, -20, -8)

2) Hallar el área del triangulo que tiene vértices P, Q, R.

Datos: Formula:

P( 2,0,-3) A= b x h

Q (1,4,5) 2

R (7,2,9) A= //P1P2 X P1P3//

2

P1P2= X2-X1 = (-1,4,8)

P1P3= X3-X1= (5,2,12)

i j k

P1P2 X P1P3 = -1 4 8 =((4)(12)-(8)(2))i-((-1)(12)-(8)(5))j+((-1)(2)-(4)(5))k

5 2 12

=(48-16)i –(-12-40)j+(-2-20)k

= 32i +52j -22k

//P1P2 X P1P3// = raiz de ( (32i)º2 +(52j)º2+(-22k)º2 = 64.9

A= 64.9 = 32.4 unidades cuadradas

2