Hifumi's Study Notesnotesmathematicscegep3triple produit scalaire

triple produit scalaire

Tags

algebra/linear

cegep/3

Word count

168 words

Reading time

2 minutes

Opération vectorielle
Soit $\vec{u}, \vec{v}, \vec{w} \in R^{3}, k \in R$ ,

\vec{u} \cdot (\vec{v} \times \vec{w})

Propriétés

$\vec{u} \cdot (\vec{v} \times \vec{w}) = - \vec{v} \cdot (\vec{u} \times \vec{w}) = - \vec{w} \cdot (\vec{v} \times \vec{u}) = \dots$

$\vec{v} \cdot (\vec{v} \times \vec{w}) = \vec{v} \cdot (\vec{w} \times \vec{v}) = \vec{w} \cdot (\vec{v} \times \vec{v}) = 0$

Exemples

$\vec{u} \cdot (\vec{v} \times \vec{w}) = 8$

Calculer $(3 \vec{w}) \cdot ((2 \vec{v}) \times \vec{u})$ .

\begin{aligned} (3 \vec{w}) \cdot ((2 \vec{v}) \times \vec{u}) & = 6 (\vec{w} \cdot (\vec{v} \times \vec{u})) \\ = - 6 (\vec{u} \cdot (\vec{v} \times \vec{w})) \\ = - 6 \cdot 8 \\ = - 48 \end{aligned}

Calculer $(2 \vec{v} + \vec{w}) \cdot [(\vec{u} + \vec{v}) \times \vec{w}]$ .

\begin{aligned} (2 \vec{v} + \vec{w}) \cdot [(\vec{u} + \vec{v}) \times \vec{w}] & = (2 \vec{v} + \vec{w}) \cdot (\vec{u} \times \vec{w} + \vec{v} \times \vec{w}) \\ = 2 \vec{v} \cdot (\vec{u} \times \vec{w}) + 2 \vec{v} \cdot (\vec{v} \times \vec{w}) + \vec{w} \cdot (\vec{u} \times \vec{w}) + \vec{w} \cdot (\vec{v} \times \vec{w}) \\ = - 2 \vec{u} \cdot (\vec{v} \times \vec{w}) \\ = - 2 \cdot 8 \\ = - 16 \end{aligned}

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