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\[ \widehat{D}=\left(\begin{array}{cc} \cos \varphi & \sin \varphi \\ -\sin \varphi & \cos \varphi \end{array}\right) \] a) Calculate the matrix D for angles \( \phi \) of \( 45^{\circ}, 90^{\circ}, 180^{\circ} \), and \( 360^{\circ} \). b) Calculate the determinant of D for any angle \( \phi \). Notice: \[ \operatorname{det}\left(\begin{array}{ll} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \end{array}\right)=a_{1,1} \cdot a_{2,2}-a_{1,2} \cdot a_{2,1} . \] c) Show that the magnitude of a vect?r \( \vec{a}=\mathrm{D}-\overrightarrow{\mathrm{n}} \mathrm{s} \) the same as the magnitude of \( \overrightarrow{\mathrm{r}} \), so applying the rotation matrix to a vector leaves its magnitude unchanged. d) Show by calculating the scalar product that the new vector a indeed forms an angle \( \varphi \) with the old vector \( \vec{r} \).

Note :) According to Chegg's 2.0 answering guidelines, only (a) is eligible for answering. Solution :) i-) D=[cos?45sin?45?sin?45cos?45]=[1212?1212]