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Identitatea lui Jacobi

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Dacă \vec a, \vec b, \vec c \in V_3, \! atunci:

\vec a \times (\vec b \times \vec c) + \vec b \times (\vec c \times \vec a) + \vec c \times (\vec a \times \vec b) = \vec 0 \! (identitatea lui Jacobi)


Demonstraţie

\vec a\times (\vec b \times \vec c) + \vec b \times (\vec c \times \vec a ) + \vec c \times (\vec a \times \vec b) = \!
= (\vec a \cdot \vec c)  \vec b - (\vec a \cdot \vec b) \vec c + (\vec b \cdot \vec a) \vec c - (\vec b \cdot \vec c) \vec a + (\vec c \cdot \vec a) \vec b - (\vec c \cdot \vec b) \vec a= \vec 0. \!


Generalizare:

Dacă F este o formă biliniară: F: V \times V \rightarrow V \! pe spațiul vectorial V, \! atunci:

F(F(x,y),z) + F(F(y,z),x) + F(F(z,x),y) = 0 \; \forall \, x,y,z \in V. \!

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