30. Provar a identidade de Lagrange:

Provar a identidade de Lagrange:

$\|\vec{u} \times \vec{v}\|^{2}=\|\vec{u}\|^{2}\|\vec{v}\|^{2}-(\vec{u} \bullet \vec{v})^{2} .$

Use a fórmula do produto vetorial duplo

$\vec{u} \times(\vec{v} \times \vec{w})=(\vec{u} \bullet \vec{w}) \vec{v}-(\vec{u} \bullet \vec{v}) \vec{w}$
para provar que
$[\vec{a} \times(\vec{b} \times \vec{c})]+[\vec{b} \times(\vec{c} \times \vec{a})]+[\vec{c} \times(\vec{a} \times \vec{b})]=0$

$30. Provar a identidade de Lagrange: \[ \|\vec{u} \times \vec{v}\|^{2}=\|\vec{u}\|^{2}\|\vec{v}\|^{2}-(\vec{u} \bullet \vec{v})^{2} . \] 31. Use a fórmula do produto vetorial duplo \[ \vec{u} \times(\vec{v} \times \vec{w})=(\vec{u} \bullet \vec{w}) \vec{v}-(\vec{u} \bullet \vec{v}) \vec{w} \] para provar que \[ [\vec{a} \times(\vec{b} \times \vec{c})]+[\vec{b} \times(\vec{c} \times \vec{a})]+[\vec{c} \times(\vec{a} \times \vec{b})]=0 \]$

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