Orthogonalité et distances dans l’espace - Spécialité

Base et repère orthonormés, coordonnées

Exercice 1 : Calculer une distance dans un repère orthonormé

Soit \( \left(H; \overrightarrow{ HG }, \overrightarrow{ HE }, \overrightarrow{ HJ }\right) \) un repère orthonormé.
Soit \( ABCDEFGH \) un parallélépipède rectangle tel que \( J \) est le milieu du segment \( [DH] \).

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On note \( I \) le milieu du segment \( [AF] \).

Donner les coordonées du point \( I \) dans ce repère.
On répondra sous la forme d'un triplet \( (x;y;z) \).

Donner les coordonées du point \( J \) dans ce repère.
On répondra sous la forme d'un triplet \( (x;y;z) \).

En déduire la distance \( IJ \)

Exercice 2 : Calculer une distance dans un repère orthonormé

Soit \( \left(B; \overrightarrow{ BC }, \overrightarrow{ BA }, \overrightarrow{ BJ }\right) \) un repère orthonormé.
Soit \( ABCDEFGH \) un parallélépipède rectangle tel que \( J \) est le milieu du segment \( [BF] \).

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On note \( I \) le milieu du segment \( [HA] \).

Donner les coordonées du point \( I \) dans ce repère.
On répondra sous la forme d'un triplet \( (x;y;z) \).

Donner les coordonées du point \( C \) dans ce repère.
On répondra sous la forme d'un triplet \( (x;y;z) \).

En déduire la distance \( IC \)

Exercice 3 : Calculer une distance dans un repère orthonormé

Soit \( \left(A; \overrightarrow{ AJ }, \overrightarrow{ AB }, \overrightarrow{ AD }\right) \) un repère orthonormé.
Soit \( ABCDEFGH \) un parallélépipède rectangle tel que \( J \) est le milieu du segment \( [AE] \).

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On note \( I \) le milieu du segment \( [CE] \).

Donner les coordonées du point \( I \) dans ce repère.
On répondra sous la forme d'un triplet \( (x;y;z) \).

Donner les coordonées du point \( E \) dans ce repère.
On répondra sous la forme d'un triplet \( (x;y;z) \).

En déduire la distance \( IE \)

Exercice 4 : Calculer une distance dans un repère orthonormé

Soit \( \left(C; \overrightarrow{ CJ }, \overrightarrow{ CD }, \overrightarrow{ CB }\right) \) un repère orthonormé.
Soit \( ABCDEFGH \) un parallélépipède rectangle tel que \( J \) est le milieu du segment \( [CG] \).

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On note \( I \) le milieu du segment \( [BE] \).

Donner les coordonées du point \( I \) dans ce repère.
On répondra sous la forme d'un triplet \( (x;y;z) \).

Donner les coordonées du point \( G \) dans ce repère.
On répondra sous la forme d'un triplet \( (x;y;z) \).

En déduire la distance \( IG \)

Exercice 5 : Calculer une distance dans un repère orthonormé

Soit \( \left(F; \overrightarrow{ FG }, \overrightarrow{ FJ }, \overrightarrow{ FE }\right) \) un repère orthonormé.
Soit \( ABCDEFGH \) un parallélépipède rectangle tel que \( J \) est le milieu du segment \( [BF] \).

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On note \( I \) le milieu du segment \( [AH] \).

Donner les coordonées du point \( I \) dans ce repère.
On répondra sous la forme d'un triplet \( (x;y;z) \).

Donner les coordonées du point \( C \) dans ce repère.
On répondra sous la forme d'un triplet \( (x;y;z) \).

En déduire la distance \( IC \)

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