[QUE/ME-02006] ME-PROBLEM

Let \({\bf A, B,..}\) be objects with components written as \(\vec{A}=(A_1,A_2,A_3), \vec{B}=(B_1,B_2,B_3)\). Introduce \(\vec{A}{'}=(A_1{'},A_2{'},A_3{'}), \vec{B}{'}=(B_1{'},B_2{'},B_3{'})\) etc. by means of equation \begin{equation}\label{EQ01} \vec{A}{'} = \vec{A} -\sin \alpha (\hat{n}\times\vec{A}) + (1-\cos\alpha) \hat{n}\times (\hat{n}\times\vec{A}). \end{equation} and with similar equations for other vectors.

• Using vector identities show that
  • \(\vec{A}{'}\cdot\vec{B}{'}=\vec{A}\cdot\vec{B}\);
  • If \(\vec{C}=\vec{A}\times\vec{B}\), then \(\vec{C}{'}\) is given by an equation similar to \eqRef{EQ01}.
• How is the expression related \( \vec{A}{'}\cdot(\vec{B}{'}\times\vec{C}{'})\) related \( \vec{A}\cdot(\vec{B}\times\vec{C})\)?