Orthogonalité et distances dans l’espace - Spécialité

Utiliser le produit scalaire : Calculer une longueur

Exercice 1 : Utiliser le produit scalaire dans un tétraèdre

Soit \( ABCD \) le tétraèdre suivant où \( AB \) vaut \( 4 \), \( AC \) vaut \( 9 \) et \( \widehat{CAD} = \dfrac{\pi }{3} \).

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Donner la mesure en radian de l'angle \( \widehat{BAC} \) sachant que \( \overrightarrow{AB} \cdot \overrightarrow{AC} = 6 \).
On donnera uniquement la valeur arrondie au centième, sans unité.

Donner la longueur de \( AD \) sachant que \( \overrightarrow{AC} \cdot \overrightarrow{AD} = 16 \).
On donnera uniquement la valeur arrondie au centième, sans unité.

Exercice 2 : Utiliser le produit scalaire dans un tétraèdre

Soit \( ABCD \) le tétraèdre suivant où \( AB \) vaut \( 7 \), \( AC \) vaut \( 9 \) et \( \widehat{CAD} = \dfrac{\pi }{6} \).

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Donner la mesure en radian de l'angle \( \widehat{BAC} \) sachant que \( \overrightarrow{AB} \cdot \overrightarrow{AC} = 32 \).
On donnera uniquement la valeur arrondie au centième, sans unité.

Donner la longueur de \( AD \) sachant que \( \overrightarrow{AC} \cdot \overrightarrow{AD} = 19 \).
On donnera uniquement la valeur arrondie au centième, sans unité.

Exercice 3 : Utiliser le produit scalaire dans un tétraèdre

Soit \( ABCD \) le tétraèdre suivant où \( AB \) vaut \( 6 \), \( AC \) vaut \( 9 \) et \( \widehat{CAD} = \dfrac{\pi }{4} \).

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Donner la mesure en radian de l'angle \( \widehat{BAC} \) sachant que \( \overrightarrow{AB} \cdot \overrightarrow{AC} = 20 \).
On donnera uniquement la valeur arrondie au centième, sans unité.

Donner la longueur de \( AD \) sachant que \( \overrightarrow{AC} \cdot \overrightarrow{AD} = 11 \).
On donnera uniquement la valeur arrondie au centième, sans unité.

Exercice 4 : Utiliser le produit scalaire dans un tétraèdre

Soit \( ABCD \) le tétraèdre suivant où \( AB \) vaut \( 3 \), \( AC \) vaut \( 8 \) et \( \widehat{CAD} = \dfrac{\pi }{3} \).

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Donner la mesure en radian de l'angle \( \widehat{BAC} \) sachant que \( \overrightarrow{AB} \cdot \overrightarrow{AC} = 20 \).
On donnera uniquement la valeur arrondie au centième, sans unité.

Donner la longueur de \( AD \) sachant que \( \overrightarrow{AC} \cdot \overrightarrow{AD} = 16 \).
On donnera uniquement la valeur arrondie au centième, sans unité.

Exercice 5 : Utiliser le produit scalaire dans un tétraèdre

Soit \( ABCD \) le tétraèdre suivant où \( AB \) vaut \( 7 \), \( AC \) vaut \( 5 \) et \( \widehat{CAD} = \dfrac{\pi }{6} \).

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Donner la mesure en radian de l'angle \( \widehat{BAC} \) sachant que \( \overrightarrow{AB} \cdot \overrightarrow{AC} = 24 \).
On donnera uniquement la valeur arrondie au centième, sans unité.

Donner la longueur de \( AD \) sachant que \( \overrightarrow{AC} \cdot \overrightarrow{AD} = 19 \).
On donnera uniquement la valeur arrondie au centième, sans unité.

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