Le plar est rapporte à an repeire arthenemel \( (0, \vec{i} ; \vec{z}) \) O) riconsidére les paints \( M(4 ; 3), P(7 ; 4) \) ot \( R(1 ; 3) \) Sackant que: \[ \overrightarrow{M P} \cdot \overrightarrow{M R}=M P \times M R \times \cos (P \widehat{M R}) \] calculer la valeur à 0,1 digre prio de RिM

Step 1 ：\[ \overrightarrow{M P} = \begin{pmatrix} 7-4 \\ 4-3 \end{pmatrix}=\begin{pmatrix} 3 \\ 1 \end{pmatrix} , \overrightarrow{M R} = \begin{pmatrix} 1-4 \\ 3-3 \end{pmatrix}=\begin{pmatrix} -3 \\ 0 \end{pmatrix} \]

Step 2 ：\[ \overrightarrow{M P} \cdot \overrightarrow{M R} = 3 \times (-3) + 1 \times 0 = -9 \]

Step 3 ：\[ MP = \sqrt{(3)^2 + (1)^2} = \sqrt{10}, MR = \sqrt{(-3)^2 + (0)^2} = 3 \]

Step 4 ：\[ \cos(P \widehat{M R}) = \frac{-9}{(\sqrt{10}) \times (3)} \]

Step 5 ：\[ P \widehat{M R} = \arccos \left( \frac{-9}{3\sqrt{10}} \right) \]

Step 6 ：\[ P \widehat{M R} \approx 105.3^\circ \]