Calcular los siguientes vectores: De acuerdo con las especificaciones...
Dados los vectores:
u = -i + 2j -4k
w = 2i-3j+k
v= -4i+3j+2k.
Calcular:
a) u . W, w . V
b) u x v , u x w
c) (u x w ). V
d) Cos ( u, w)
Respuesta de Valero Angel Serrano Mercadal
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¡Hola Carlos!
u = -i + 2j -4k, w = 2i-3j+k, v= -4i+3j+2k.
Calcular:
a)
u·w = (-1)·2 + 2(-3) + (-4)·1 = -2 - 6 - 4 = -12
w·v =2·(-4) + (-3)·3 + 1·2 = -8 - 9 + 2 = -15
b)
u x v=
| i j k |
|-1 2 -4 | = (4+12)i - (-2-16)j + (-3+8)k = 16i + 18j + 5k
|-4 3 2 |
·
u x w=
| i j k |
|-1 2 4 |= (2+12)i - (-1-8)j +(3-4)k = 14i + 9j - k
| 2 -3 1 |
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c) (u x w ). V
(14i + 9j - k)·(-4i+3j+2k) = 14·(-4) + 9·3 +(-1)·2 = -56+27-2=-31
d)
$$\begin{align}&\cos(u,w) =\frac{u·w}{||u||·||w||}=\\&\\& \frac{-12}{\sqrt{(-1)^2+2^2+(-4)^2}· \sqrt{2^2+(-3)^2+1^2}}=\\&\\&\frac{-12}{\sqrt{1+4+16} \sqrt{4+9+1}}=\frac{-12}{\sqrt{21}\sqrt{14}}=\\&\\&\frac{-12}{\sqrt{3·7·2·7}}= \frac{-12}{7 \sqrt 6}=\frac{-12 \sqrt 6}{42}=\frac{-2 \sqrt 6}{7}\approx\\&\\&-0.6998542122\\&\\&\\&\\&\end{align}$$:
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