[QUE/ME-02006] Rotation of Coordinate System

\(\newcommand{\Prime}{^\prime}\)
Let \({\bf A, B,..}\) be vector objects with components written as \(\vec{A}=(A_1,A_2,A_3), \vec{B}=(B_1,B_2,B_3)\). Introduce \(\vec{A}\Prime=(A_1\Prime,A_2\Prime,A_3\Prime), \vec{B}\Prime=(B_1\Prime,B_2\Prime,B_3\Prime)\) etc. by means of equation \begin{equation}\label{EQ01} \vec{A}\Prime = \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

1. \(\vec{A}\Prime\cdot\vec{B}\Prime=\vec{A}\cdot\vec{B}\);
2. If \(\vec{C}=\vec{A}\times\vec{B}\), then \(\vec{C}\Prime\) is given by an equation similar to \eqref{EQ01}.

How is  is the expression \( \vec{A}\Prime\cdot(\vec{B}\Prime\times\vec{C}\Prime)\) related to \( \vec{A}\cdot (\vec{B}\times\vec{C}) \)?