Ivan Tec

**producto vectorial**

operaciones

1) Sea

u = (2,-1,3)

v = (0.,1,7)

w = (1,4,5)

realizar las siguientes:

a.) (uxv) x(vxw) =

$uxv= \begin{bmatrix}{2}&{-1}&{3}\\{0}&{1}&{7}\end{bmatrix}$
= $[(-1)(7)-(3)(1)]i -[(2)(7)-(0)(3)]j + [(2)(1)-(-1)(0)]$
= (-10,-14,2)

$vxw=\begin{bmatrix}{0}&{1}&{7}\\{1}&{4}&{5}\end{bmatrix}$
$= [(1)(5)-(7)(4)]i - [(0)(5)-(1)(7)]j + [ (0)(4)- (1)(1)]k$
$= (-23,7,-1)$

$(uxv) x(vxw)= \begin{bmatrix}{-10}&{-14}&{2}\\{-23}&{7}&{-1}\end{bmatrix}$

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

$=(0,-56,-392)$

b.) ux(v -2w)
$2w=2(1,4,5)= (2,8,10)$
$v-2w= (0,1,7)-(2,8,10)= (-2,-7-3)$
$ux(v-2w)=\begin{bmatrix}{2}&{-1}&{3}\\{-2}&{-7}&{-3}\end{bmatrix}$
$= (24,0,-16)$

c.) (uxv)-2w
$=(-10,-14,2)-(2,8,10)= (-12,-22,-8)$

2) Hallar el area de vertices P (2,0,-3), Q (1,4,5) y R (7,2,9).

$A=\frac{1}{2} \left\|{\vec{PQ}\ x \vec{PR}\}\right\|$

$\vec{PQ}\ = (-1,4,8)$

$\vec{PR}\ = (5,2,12)$

$\vec{PQ}\ x \vec{PR}\ = \begin{bmatrix}{-1}&{4}&{8}\\{5}&{2}&{12}\end{bmatrix}$

$= (32,52,-22)$

$\left\|{\vec{PQ}\ x \vec{PR}\}\right\|= \sqrt{1024 + 2704 + 484} = 64.89$

$A = \frac{64.9}{2}= 32.45 u^2$

Ivan Tec

jueves, 4 de septiembre de 2008

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