Chapitre X Produit scalaire

D(3 ;5),E(-1 ;0), F(2 ;4)

\(\vec{DE}=\begin{pmatrix}-1-3\\0-5\end{pmatrix}=\begin{pmatrix}-4\\-5\end{pmatrix}\)

\(\vec{DF}=\begin{pmatrix}2-3\\4-5\end{pmatrix}=\begin{pmatrix}-1\\-1\end{pmatrix}\)

\(\vec{DE}\cdot\vec{DF}=(-4)\times(-1)+(-5)\times(-1)=4+5=9\)

\(\|\vec{DE}\|=\sqrt{(-4)^2+(-5)^2}=\sqrt{16+25}=\sqrt{41}\)

\(\|\vec{DF}\|=\sqrt{(-1)^2+(-1)^2}=\sqrt{1+1}=\sqrt{2}\)

\(\vec{DE}\cdot\vec{DF}=DE \times DF \times cos((\vec{DE},\vec{DF}))=9\)

donc \(\sqrt{41} \times \sqrt{2} \times cos((\vec{DE},\vec{DF}))=9\)

\(cos((\vec{DE},\vec{DF}))=\frac{9}{\sqrt{41} \times \sqrt{2}}\simeq 6,3°\)