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Produit scalaire
Propriétés de calculs du produit scalaire : symétrie, bilinéarité - Exercice 2
5 min
10
Soient
u
→
\overrightarrow{u}
u
et
v
→
\overrightarrow{v}
v
deux vecteurs tels que :
∥
u
→
∥
=
4
\left\| \overrightarrow{u} \right\| =4
∥
∥
u
∥
∥
=
4
;
∥
v
→
∥
=
5
\left\| \overrightarrow{v} \right\| =5
∥
∥
v
∥
∥
=
5
et
u
→
⋅
v
→
=
12
\overrightarrow{u} \cdot \overrightarrow{v} =12
u
⋅
v
=
12
Question 1
Calculer :
u
→
⋅
(
u
→
+
v
→
)
\overrightarrow{u} \cdot \left(\overrightarrow{u} +\overrightarrow{v} \right)
u
⋅
(
u
+
v
)
Correction
Soient trois vecteurs
u
→
\overrightarrow{u}
u
;
v
→
\overrightarrow{v}
v
et
w
→
\overrightarrow{w}
w
alors :
u
→
⋅
(
v
→
+
w
→
)
=
u
→
⋅
v
→
+
u
→
⋅
w
→
\overrightarrow{u} \cdot \left(\overrightarrow{v} +\overrightarrow{w} \right)=\overrightarrow{u} \cdot \overrightarrow{v} +\overrightarrow{u} \cdot \overrightarrow{w}
u
⋅
(
v
+
w
)
=
u
⋅
v
+
u
⋅
w
u
→
⋅
u
→
=
∥
u
→
∥
2
\overrightarrow{u} \cdot \overrightarrow{u} =\left\| \overrightarrow{u} \right\| ^{2}
u
⋅
u
=
∥
∥
u
∥
∥
2
u
→
⋅
(
u
→
+
v
→
)
=
u
→
⋅
u
→
+
u
→
⋅
v
→
\overrightarrow{u} \cdot \left(\overrightarrow{u} +\overrightarrow{v} \right)=\overrightarrow{u} \cdot \overrightarrow{u} +\overrightarrow{u} \cdot \overrightarrow{v}
u
⋅
(
u
+
v
)
=
u
⋅
u
+
u
⋅
v
équivaut successivement à :
u
→
⋅
(
u
→
+
v
→
)
=
∥
u
→
∥
2
+
u
→
⋅
v
→
\overrightarrow{u} \cdot \left(\overrightarrow{u} +\overrightarrow{v} \right)=\left\| \overrightarrow{u} \right\| ^{2} +\overrightarrow{u} \cdot \overrightarrow{v}
u
⋅
(
u
+
v
)
=
∥
∥
u
∥
∥
2
+
u
⋅
v
u
→
⋅
(
u
→
+
v
→
)
=
4
2
+
12
\overrightarrow{u} \cdot \left(\overrightarrow{u} +\overrightarrow{v} \right)=4 ^{2} +12
u
⋅
(
u
+
v
)
=
4
2
+
12
u
→
⋅
(
u
→
+
v
→
)
=
16
+
12
\overrightarrow{u} \cdot \left(\overrightarrow{u} +\overrightarrow{v} \right)=16 +12
u
⋅
(
u
+
v
)
=
16
+
12
u
→
⋅
(
u
→
+
v
→
)
=
28
\overrightarrow{u} \cdot \left(\overrightarrow{u} +\overrightarrow{v} \right)=28
u
⋅
(
u
+
v
)
=
28