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heure_3 - Produit Scalaire Travail dune force Soit F un...

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Produit Scalaire Travail d’une force I Soit F un vecteur force et D un vecteur d´ eplacement I Le travail de F est un scalaire W = F . D = F D cos ( θ ) F D O O v w Fig.: Direction et sens
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Produit Scalaire Produit Scalaire Attention I Le produit scalaire de deux vecteurs est un scalaire I eom´ etriquement : le produit scalaire de deux vecteurs v , w est : v w cos ( θ ) ( θ est l’angle entre les deux vecteurs) I Alg´ ebriquement : soit ( i , j , k ) une base de l’espace. v = v 1 i + v 2 j + v 3 k et w = w 1 i + w 2 j + w 3 k deux vecteurs de l’espace. On note le produit scalaire des deux vecteurs par : v . w = v 1 w 1 + v 2 w 2 + v 3 w 3 I v 2 = v 2 1 + v 2 2 + v 2 3 = v . v
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Produit Scalaire Propri´ et´ es du Produit Scalaire I v . w = w . v I v . ( λ w ) = ( λ v ) . w = λ ( v . w ) I u . ( v + w ) = u . v + u . w I Deux vecteurs v et w sont perpendiculaire ssi v . w = v w cos ( θ ) = 0
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Produit Scalaire Exemple I Soit u = i + 3 k , v = i + 3 j et w = 3 i + j - k I u . v = 1 . 1 + 0 . 3 + 3 . 0 = 1 les vecteurs ne sont pas perpendiculaires I v . w = 2 3 les vecteurs ne sont pas perpendiculaires I u . w = 1 . 3 + 0 . 1 +
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