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12/25/2023 · Primary School
122 Soient \( \vec{u} \), \( \vec{v} \) et \( \vec{w} \) trois vecteurs. Réduis les sommes suivantes : a) \( 2 \vec{u}+(\vec{v}-3 \vec{u})+2(7 \vec{u}+\vec{v}) \) b) \( -7(\vec{u}-\vec{w})+2(3 \vec{v}-3 \vec{u}+\vec{w})-2(\vec{w}-\vec{v}) \) c) \( 8(\vec{v}+3 \vec{u})+2(7 \vec{u}+\vec{v})-10 \vec{v} \)
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Step-by-step Solution
Pour réduire les sommes de vecteurs, nous allons distribuer les coefficients et regrouper les termes similaires.
### a) \( 2 \vec{u}+(\vec{v}-3 \vec{u})+2(7 \vec{u}+\vec{v}) \)
1. Distribuons les termes :
\[
2 \vec{u} + \vec{v} - 3 \vec{u} + 2(7 \vec{u}) + 2(\vec{v}) = 2 \vec{u} + \vec{v} - 3 \vec{u} + 14 \vec{u} + 2 \vec{v}
\]
2. Regroupons les termes similaires :
\[
(2 \vec{u} - 3 \vec{u} + 14 \vec{u}) + (\vec{v} + 2 \vec{v}) = (2 - 3 + 14) \vec{u} + (1 + 2) \vec{v} = 13 \vec{u} + 3 \vec{v}
\]
Donc, la somme réduite est :
\[
\boxed{13 \vec{u} + 3 \vec{v}}
\]
### b) \( -7(\vec{u}-\vec{w})+2(3 \vec{v}-3 \vec{u}+\vec{w})-2(\vec{w}-\vec{v}) \)
1. Distribuons les termes :
\[
-7 \vec{u} + 7 \vec{w} + 2(3 \vec{v}) - 2(3 \vec{u}) + 2(\vec{w}) - 2(\vec{w}) + 2(\vec{v}) = -7 \vec{u} + 7 \vec{w} + 6 \vec{v} - 6 \vec{u} + 2 \vec{w} - 2 \vec{w} + 2 \vec{v}
\]
2. Regroupons les termes similaires :
\[
(-7 \vec{u} - 6 \vec{u}) + (6 \vec{v} + 2 \vec{v}) + (7 \vec{w} + 2 \vec{w} - 2 \vec{w}) = (-7 - 6) \vec{u} + (6 + 2) \vec{v} + (7 + 2 - 2) \vec{w} = -13 \vec{u} + 8 \vec{v} + 7 \vec{w}
\]
Donc, la somme réduite est :
\[
\boxed{-13 \vec{u} + 8 \vec{v} + 7 \vec{w}}
\]
### c) \( 8(\vec{v}+3 \vec{u})+2(7 \vec{u}+\vec{v})-10 \vec{v} \)
1. Distribuons les termes :
\[
8 \vec{v} + 24 \vec{u} + 2(7 \vec{u}) + 2(\vec{v}) - 10 \vec{v} = 8 \vec{v} + 24 \vec{u} + 14 \vec{u} + 2 \vec{v} - 10 \vec{v}
\]
2. Regroupons les termes similaires :
\[
(24 \vec{u} + 14 \vec{u}) + (8 \vec{v} + 2 \vec{v} - 10 \vec{v}) = (24 + 14) \vec{u} + (8 + 2 - 10) \vec{v} = 38 \vec{u} + 0 \vec{v} = 38 \vec{u}
\]
Donc, la somme réduite est :
\[
\boxed{38 \vec{u}}
\]
Quick Answer
a) \( 13 \vec{u} + 3 \vec{v} \)
b) \( -13 \vec{u} + 8 \vec{v} + 7 \vec{w} \)
c) \( 38 \vec{u} \)
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