1er. REPASO PARA EL 1-SEP-08

1).- Sea :
u = ( 2, -1, 3)

v = ( 0, 1, 7)
w = ( 1, 4, 5)
Realizar las siguientes operaciones:
a) ( u x v ) x ( v x w )

( u x v ) = = i[(-1)(7) -(3)(1)] - j[(2)(7) -(0)(3)] + k[(2)(1) -(0)(-1)]

= i[-7 -3] - j[14 -0] + k[2 -0]

= (-10i, -14j, 2k)

( v * w ) = = i[(1)(5) -(7)(4)] - j[(0)(5) -(1)(7)] + k[(0)(4) -(1)(1)]

= i[5 - 28] - j[0 -7] + k[0 -1)]

= (-23i, 7j, -1k)

( u x v ) x ( v x w ) =

= i[(-14)(1) -(7)(2)] - j[(-10)(1) -(23)(2)] + k[(-10)(7) - (23)(-14)]

= i[-14 - 14] - j[-10 -46] + k[-70 + 345]

= (0i, -56j, 275k)

b) u x ( v - 2w )
2w = (2, 8, 10)

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

= (-2, 7, 3)

u x ( v - 2w ) =

= i[(-1)(3) -(7)(3)] - j[(2)(3) -(-2)(3)] + k[(2)(7) - (-2)(-1)]

= i[-3 - 21] - j[6 + 6] + k[14 - 2]

= (-24i, -12j, 12k)




c) ( u x v ) - 2w
( u x v ) =

= i[(-1)(7) -(3)(1)] - j[(2)(7) -(0)(3)] + k[(2)(1) -(0)(-1)]

= i[-7 -3] - j[14 -0] + k[2 -0]

= (-10i, -14j, 2k)

2w = (2, 8, 10)

( u x v ) - 2w = (-10, -14, 2) - (2, 8, 10)

= (-8, -6, 8)

2) .- Hallar el area del triangulo que tiene vertives P, Q, R.
P (2, 0, -3)

Q (1, 4, 5)

R (7, 2, 9)

A = b * h / 2

P * Q =X2 - X1 = (-1, 4, 8)
P * R = X2 - X1 = (5, 2, 12)

= [(4)(12) - (8)(2)] i -[(-1)(12) - (5)(8)] j + [(-1)(2) - (5)(4)] k

= (48 - 16) i - (-12 - 40) j + (-2 -20) k

= 32 i + 52 j - 22 k





= 64.89

Aplicando la formula: A =
A = 32.445 u