Répondre :
Bonjour Henri56
[tex]1)\ \overrightarrow{AB}.\overrightarrow{AC}=AB\times AC\times\cos(\overrightarrow{AB},\overrightarrow{AC})\\\\\overrightarrow{AB}.\overrightarrow{AC}=6\times6\times\cos(\dfrac{\pi}{3})\\\\\overrightarrow{AB}.\overrightarrow{AC}=36\times\dfrac{1}{2}\\\\\boxed{\overrightarrow{AB}.\overrightarrow{AC}=18}[/tex]
[tex]2)\ \overrightarrow{DE}.\overrightarrow{DC}=DE\times DE\times\cos(\overrightarrow{DE},\overrightarrow{DC})\\\\\overrightarrow{DE}.\overrightarrow{DC}=6\times6\times\cos(-\dfrac{\pi}{2})\\\\\overrightarrow{DE}.\overrightarrow{DC}=36\times0\\\\\boxed{\overrightarrow{DE}.\overrightarrow{DC}=0}[/tex]
[tex]3)\ \overrightarrow{BI}.\overrightarrow{BD}=BI\times BD\times\cos(\overrightarrow{BI},\overrightarrow{BD})\\\\\overrightarrow{BI}.\overrightarrow{BD}=3\sqrt{2}\times6\sqrt{2}\times\cos0\\\\\overrightarrow{BI}.\overrightarrow{BD}=18\times(\sqrt{2})^2\times1\\\\\overrightarrow{BI}.\overrightarrow{BD}=18\times2\times1\\\\\boxed{\overrightarrow{BI}.\overrightarrow{BD}=36}[/tex]
[tex]4)\ \overrightarrow{BA}.\overrightarrow{ED}=\overrightarrow{BA}.\overrightarrow{BC}=\\\\\overrightarrow{BA}.\overrightarrow{ED}=BA\times BC\times\cos(\overrightarrow{BA},\overrightarrow{BC})\\\\\overrightarrow{BA}.\overrightarrow{ED}=6\times6\times\cos(-\dfrac{\pi}{3})\\\\\overrightarrow{BA}.\overrightarrow{ED}=36\times\cos(\dfrac{\pi}{3})\\\\\overrightarrow{BA}.\overrightarrow{ED}=36\times\dfrac{1}{2}\\\\\boxed{\overrightarrow{BA}.\overrightarrow{ED}=18}[/tex]
[tex]5)\ \overrightarrow{BE}.\overrightarrow{BD}=BE\times BD\times\cos(\overrightarrow{BE},\overrightarrow{BD})\\\\\overrightarrow{BE}.\overrightarrow{BD}=6\times6\sqrt{2}\times\cos(\dfrac{\pi}{4})\\\\\overrightarrow{BE}.\overrightarrow{BD}=36\sqrt{2}\times\dfrac{\sqrt{2}}{2}\\\\\boxed{\overrightarrow{BE}.\overrightarrow{BD}=36}[/tex]
[tex]6)\ \overrightarrow{CA}.\overrightarrow{CD}=CA\times CD\times\cos(\overrightarrow{CA},\overrightarrow{CD})\\\\\overrightarrow{CA}.\overrightarrow{CD}=6\times6\times\cos(\dfrac{\pi}{2}+\dfrac{\pi}{3})\\\\\overrightarrow{CA}.\overrightarrow{CD}=36\times\cos(\dfrac{5\pi}{6})\\\\\overrightarrow{CA}.\overrightarrow{CD}=36\times[-\cos(\dfrac{\pi}{6})]\\\\\overrightarrow{CA}.\overrightarrow{CD}=36\times(-\dfrac{\sqrt{3}}{2})]\\\\\boxed{\overrightarrow{CA}.\overrightarrow{CD}=-18\sqrt{3}}[/tex]
[tex]7)\ \overrightarrow{IE}.\overrightarrow{IC}=IE\times IC\times\cos(\overrightarrow{IE},\overrightarrow{IC})\\\\\overrightarrow{IE}.\overrightarrow{IC}=3\sqrt{2}\times3\sqrt{2}\times\cos(-\pi)\\\\\overrightarrow{IE}.\overrightarrow{IC}=9\times(\sqrt{2})^2\times\cos\pi\\\\\overrightarrow{IE}.\overrightarrow{IC}=9\times2\times(-1)\\\\\boxed{\overrightarrow{IE}.\overrightarrow{IC}=-18}[/tex]
[tex]1)\ \overrightarrow{AB}.\overrightarrow{AC}=AB\times AC\times\cos(\overrightarrow{AB},\overrightarrow{AC})\\\\\overrightarrow{AB}.\overrightarrow{AC}=6\times6\times\cos(\dfrac{\pi}{3})\\\\\overrightarrow{AB}.\overrightarrow{AC}=36\times\dfrac{1}{2}\\\\\boxed{\overrightarrow{AB}.\overrightarrow{AC}=18}[/tex]
[tex]2)\ \overrightarrow{DE}.\overrightarrow{DC}=DE\times DE\times\cos(\overrightarrow{DE},\overrightarrow{DC})\\\\\overrightarrow{DE}.\overrightarrow{DC}=6\times6\times\cos(-\dfrac{\pi}{2})\\\\\overrightarrow{DE}.\overrightarrow{DC}=36\times0\\\\\boxed{\overrightarrow{DE}.\overrightarrow{DC}=0}[/tex]
[tex]3)\ \overrightarrow{BI}.\overrightarrow{BD}=BI\times BD\times\cos(\overrightarrow{BI},\overrightarrow{BD})\\\\\overrightarrow{BI}.\overrightarrow{BD}=3\sqrt{2}\times6\sqrt{2}\times\cos0\\\\\overrightarrow{BI}.\overrightarrow{BD}=18\times(\sqrt{2})^2\times1\\\\\overrightarrow{BI}.\overrightarrow{BD}=18\times2\times1\\\\\boxed{\overrightarrow{BI}.\overrightarrow{BD}=36}[/tex]
[tex]4)\ \overrightarrow{BA}.\overrightarrow{ED}=\overrightarrow{BA}.\overrightarrow{BC}=\\\\\overrightarrow{BA}.\overrightarrow{ED}=BA\times BC\times\cos(\overrightarrow{BA},\overrightarrow{BC})\\\\\overrightarrow{BA}.\overrightarrow{ED}=6\times6\times\cos(-\dfrac{\pi}{3})\\\\\overrightarrow{BA}.\overrightarrow{ED}=36\times\cos(\dfrac{\pi}{3})\\\\\overrightarrow{BA}.\overrightarrow{ED}=36\times\dfrac{1}{2}\\\\\boxed{\overrightarrow{BA}.\overrightarrow{ED}=18}[/tex]
[tex]5)\ \overrightarrow{BE}.\overrightarrow{BD}=BE\times BD\times\cos(\overrightarrow{BE},\overrightarrow{BD})\\\\\overrightarrow{BE}.\overrightarrow{BD}=6\times6\sqrt{2}\times\cos(\dfrac{\pi}{4})\\\\\overrightarrow{BE}.\overrightarrow{BD}=36\sqrt{2}\times\dfrac{\sqrt{2}}{2}\\\\\boxed{\overrightarrow{BE}.\overrightarrow{BD}=36}[/tex]
[tex]6)\ \overrightarrow{CA}.\overrightarrow{CD}=CA\times CD\times\cos(\overrightarrow{CA},\overrightarrow{CD})\\\\\overrightarrow{CA}.\overrightarrow{CD}=6\times6\times\cos(\dfrac{\pi}{2}+\dfrac{\pi}{3})\\\\\overrightarrow{CA}.\overrightarrow{CD}=36\times\cos(\dfrac{5\pi}{6})\\\\\overrightarrow{CA}.\overrightarrow{CD}=36\times[-\cos(\dfrac{\pi}{6})]\\\\\overrightarrow{CA}.\overrightarrow{CD}=36\times(-\dfrac{\sqrt{3}}{2})]\\\\\boxed{\overrightarrow{CA}.\overrightarrow{CD}=-18\sqrt{3}}[/tex]
[tex]7)\ \overrightarrow{IE}.\overrightarrow{IC}=IE\times IC\times\cos(\overrightarrow{IE},\overrightarrow{IC})\\\\\overrightarrow{IE}.\overrightarrow{IC}=3\sqrt{2}\times3\sqrt{2}\times\cos(-\pi)\\\\\overrightarrow{IE}.\overrightarrow{IC}=9\times(\sqrt{2})^2\times\cos\pi\\\\\overrightarrow{IE}.\overrightarrow{IC}=9\times2\times(-1)\\\\\boxed{\overrightarrow{IE}.\overrightarrow{IC}=-18}[/tex]
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